Daily Papers

Hypergraphs offer a powerful framework for modeling higher-order interactions that traditional pairwise graphs cannot fully capture. However, practical constraints often lead to their simplification into projected graphs, resulting in substantial information loss and ambiguity in representing higher-order relationships. In this work, we propose MARIOH, a supervised approach for reconstructing the original hypergraph from its projected graph by leveraging edge multiplicity. To overcome the difficulties posed by the large search space, MARIOH integrates several key ideas: (a) identifying provable size-2 hyperedges, which reduces the candidate search space, (b) predicting the likelihood of candidates being hyperedges by utilizing both structural and multiplicity-related features, and (c) not only targeting promising hyperedge candidates but also examining less confident ones to explore alternative possibilities. Together, these ideas enable MARIOH to efficiently and effectively explore the search space. In our experiments using 10 real-world datasets, MARIOH achieves up to 74.51% higher reconstruction accuracy compared to state-of-the-art methods.

Prediction problems often admit competing models that perform almost equally well. This effect challenges key assumptions in machine learning when competing models assign conflicting predictions. In this paper, we define predictive multiplicity as the ability of a prediction problem to admit competing models with conflicting predictions. We introduce formal measures to evaluate the severity of predictive multiplicity and develop integer programming tools to compute them exactly for linear classification problems. We apply our tools to measure predictive multiplicity in recidivism prediction problems. Our results show that real-world datasets may admit competing models that assign wildly conflicting predictions, and motivate the need to measure and report predictive multiplicity in model development.

Predictive multiplicity refers to the phenomenon in which classification tasks may admit multiple competing models that achieve almost-equally-optimal performance, yet generate conflicting outputs for individual samples. This presents significant concerns, as it can potentially result in systemic exclusion, inexplicable discrimination, and unfairness in practical applications. Measuring and mitigating predictive multiplicity, however, is computationally challenging due to the need to explore all such almost-equally-optimal models, known as the Rashomon set, in potentially huge hypothesis spaces. To address this challenge, we propose a novel framework that utilizes dropout techniques for exploring models in the Rashomon set. We provide rigorous theoretical derivations to connect the dropout parameters to properties of the Rashomon set, and empirically evaluate our framework through extensive experimentation. Numerical results show that our technique consistently outperforms baselines in terms of the effectiveness of predictive multiplicity metric estimation, with runtime speedup up to 20times sim 5000times. With efficient Rashomon set exploration and metric estimation, mitigation of predictive multiplicity is then achieved through dropout ensemble and model selection.

Graph structure learning aims to learn connectivity in a graph from data. It is particularly important for many computer vision related tasks since no explicit graph structure is available for images for most cases. A natural way to construct a graph among images is to treat each image as a node and assign pairwise image similarities as weights to corresponding edges. It is well known that pairwise similarities between images are sensitive to the noise in feature representations, leading to unreliable graph structures. We address this problem from the viewpoint of statistical tests. By viewing the feature vector of each node as an independent sample, the decision of whether creating an edge between two nodes based on their similarity in feature representation can be thought as a {it single} statistical test. To improve the robustness in the decision of creating an edge, multiple samples are drawn and integrated by {it multiple} statistical tests to generate a more reliable similarity measure, consequentially more reliable graph structure. The corresponding elegant matrix form named B-Attention is designed for efficiency. The effectiveness of multiple tests for graph structure learning is verified both theoretically and empirically on multiple clustering and ReID benchmark datasets. Source codes are available at https://github.com/Thomas-wyh/B-Attention.

Graph generation is integral to various engineering and scientific disciplines. Nevertheless, existing methodologies tend to overlook the generation of edge attributes. However, we identify critical applications where edge attributes are essential, making prior methods potentially unsuitable in such contexts. Moreover, while trivial adaptations are available, empirical investigations reveal their limited efficacy as they do not properly model the interplay among graph components. To address this, we propose a joint score-based model of nodes and edges for graph generation that considers all graph components. Our approach offers two key novelties: (i) node and edge attributes are combined in an attention module that generates samples based on the two ingredients; and (ii) node, edge and adjacency information are mutually dependent during the graph diffusion process. We evaluate our method on challenging benchmarks involving real-world and synthetic datasets in which edge features are crucial. Additionally, we introduce a new synthetic dataset that incorporates edge values. Furthermore, we propose a novel application that greatly benefits from the method due to its nature: the generation of traffic scenes represented as graphs. Our method outperforms other graph generation methods, demonstrating a significant advantage in edge-related measures.

Machine learning models are often used to inform real world risk assessment tasks: predicting consumer default risk, predicting whether a person suffers from a serious illness, or predicting a person's risk to appear in court. Given multiple models that perform almost equally well for a prediction task, to what extent do predictions vary across these models? If predictions are relatively consistent for similar models, then the standard approach of choosing the model that optimizes a penalized loss suffices. But what if predictions vary significantly for similar models? In machine learning, this is referred to as predictive multiplicity i.e. the prevalence of conflicting predictions assigned by near-optimal competing models. In this paper, we present a framework for measuring predictive multiplicity in probabilistic classification (predicting the probability of a positive outcome). We introduce measures that capture the variation in risk estimates over the set of competing models, and develop optimization-based methods to compute these measures efficiently and reliably for convex empirical risk minimization problems. We demonstrate the incidence and prevalence of predictive multiplicity in real-world tasks. Further, we provide insight into how predictive multiplicity arises by analyzing the relationship between predictive multiplicity and data set characteristics (outliers, separability, and majority-minority structure). Our results emphasize the need to report predictive multiplicity more widely.

The Berge-Fulkerson conjecture states that every bridgeless cubic graph can be covered with six perfect matchings such that each edge is covered exactly twice. An equivalent reformulation is that it's possible to find a 6-cycle 4-cover. In this paper we discuss the oriented version (o6c4c) of the latter statement, pose it as a conjecture and prove it for the family of Isaacs flower snarks. Similarly to the case of oriented cycle double cover, we can always construct an orientable surface (possibly with boundary) from an o6c4c solution. If the o6c4c solution itself splits into two (not necessarily oriented) cycle double covers, then it's also possible to build another pair of orientable surfaces (also possibly with boundaries). Finally we show how to build a ribbon graph, and for some special o6c4c cases we show that this ribbon graph corresponds to an oriented 6-cycle double cover. Github: https://github.com/gexahedron/cycle-double-covers

Let G be a graph with adjacency matrix A(G). We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where n^+(G) and n^-(G) denote the number of positive and negative eigenvalues of A(G), respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torgasev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph G, its line graph L(G) satisfies n^+(L(G)) le n^-(L(G)) + 1 and obtain partial results.

The reciprocal distance Laplacian matrix of a connected graph G is defined as RD^L(G)=RT(G)-RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RD^L(G) is a real symmetric matrix, we denote its eigenvalues as λ_1(RD^L(G))geq λ_2(RD^L(G))geq dots geq λ_n(RD^L(G)). The largest eigenvalue λ_1(RD^L(G)) of RD^L(G) is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of n as a reciprocal distance Laplacian eigenvalue of RD^L(G) is exactly one less than the number of components in the complement graph G of G. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with n vertices. Also, we show that the star graph S_n is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with n vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph K_n, K_n-e, the star S_n, the complete balanced bipartite graph K_{n{2},n{2}} and the complete split graph CS(n,α) are all determined from the RD^L-spectrum.

Graph is powerful for representing various types of real-world data. The topology (edges' presence) and edges' features of a graph decides the message passing mechanism among vertices within the graph. While most existing approaches only manually define a single-value edge to describe the connectivity or strength of association between a pair of vertices, task-specific and crucial relationship cues may be disregarded by such manually defined topology and single-value edge features. In this paper, we propose the first general graph representation learning framework (called GRATIS) which can generate a strong graph representation with a task-specific topology and task-specific multi-dimensional edge features from any arbitrary input. To learn each edge's presence and multi-dimensional feature, our framework takes both of the corresponding vertices pair and their global contextual information into consideration, enabling the generated graph representation to have a globally optimal message passing mechanism for different down-stream tasks. The principled investigation results achieved for various graph analysis tasks on 11 graph and non-graph datasets show that our GRATIS can not only largely enhance pre-defined graphs but also learns a strong graph representation for non-graph data, with clear performance improvements on all tasks. In particular, the learned topology and multi-dimensional edge features provide complementary task-related cues for graph analysis tasks. Our framework is effective, robust and flexible, and is a plug-and-play module that can be combined with different backbones and Graph Neural Networks (GNNs) to generate a task-specific graph representation from various graph and non-graph data. Our code is made publicly available at https://github.com/SSYSteve/Learning-Graph-Representation-with-Task-specific-Topology-and-Multi-dimensional-Edge-Features.

This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function Delta(lambda) . This can be used to determine the multiplicities of eigenvalues for concrete operators.

For a graph G with n vertices and m edges, Lin et al. Lin define the atom--bond sum-connectivity (ABS) matrix of G such that the (i,j)^{th} entry is \[ 1 - \frac{2{d_i + d_j}} \] if vertex v_i is adjacent to the vertex v_j, and 0 otherwise. In this article, we determine the characteristic polynomial of the ABS matrix for certain specific classes of graphs. Furthermore, we compute the ABS eigenvalues and the ABS energy for these classes.

This is the second paper devoted to the numerical version of Signature-inverse Theorem in terms of the underlying joint invariants. Signature Theorem and its Inverse guarantee any application of differential invariant signature curves to the invariant recognition of visual objects. We first show the invalidity of Curvature-inverse and Signature inverse theorems, meaning non-congruent meshes may have the same joint invariant numerical curvature or signature. Then by classifying three and five point ordinary meshes respectively in the Euclidean and affine cases, we look for conditions in terms of the associated joint invariant signatures which make these theorems correct. Additionally, we bring forward The Host Theorem to provide a simpler version of Signature-inverse Theorem for closed ordinary meshes.

Gross, Mansour, and Tucker introduced the partial-duality polynomial of a ribbon graph [Distributions, European J. Combin. 86, 1--20, 2020], the generating function enumerating partial duals by the Euler genus. Chmutov and Vignes-Tourneret wondered if this polynomial and its conjectured properties would hold for general delta-matroids, which are combinatorial abstractions of ribbon graphs. Yan and Jin contributed to this inquiry by identifying a subset of delta-matroids-specifically, even normal binary ones-whose twist polynomials are characterized by a singular term. Building upon this foundation, the current paper expands the scope of the investigation to encompass even non-binary delta-matroids, revealing that none of them have width-changing twists.

The presence of linear paths in parameter space between two different network solutions in certain cases, i.e., linear mode connectivity (LMC), has garnered interest from both theoretical and practical fronts. There has been significant research that either practically designs algorithms catered for connecting networks by adjusting for the permutation symmetries as well as some others that more theoretically construct paths through which networks can be connected. Yet, the core reasons for the occurrence of LMC, when in fact it does occur, in the highly non-convex loss landscapes of neural networks are far from clear. In this work, we take a step towards understanding it by providing a model of how the loss landscape needs to behave topographically for LMC (or the lack thereof) to manifest. Concretely, we present a `mountainside and ridge' perspective that helps to neatly tie together different geometric features that can be spotted in the loss landscape along the training runs. We also complement this perspective by providing a theoretical analysis of the barrier height, for which we provide empirical support, and which additionally extends as a faithful predictor of layer-wise LMC. We close with a toy example that provides further intuition on how barriers arise in the first place, all in all, showcasing the larger aim of the work -- to provide a working model of the landscape and its topography for the occurrence of LMC.

Graph neural networks have recently achieved remarkable success in representing graph-structured data, with rapid progress in both the node embedding and graph pooling methods. Yet, they mostly focus on capturing information from the nodes considering their connectivity, and not much work has been done in representing the edges, which are essential components of a graph. However, for tasks such as graph reconstruction and generation, as well as graph classification tasks for which the edges are important for discrimination, accurately representing edges of a given graph is crucial to the success of the graph representation learning. To this end, we propose a novel edge representation learning framework based on Dual Hypergraph Transformation (DHT), which transforms the edges of a graph into the nodes of a hypergraph. This dual hypergraph construction allows us to apply message-passing techniques for node representations to edges. After obtaining edge representations from the hypergraphs, we then cluster or drop edges to obtain holistic graph-level edge representations. We validate our edge representation learning method with hypergraphs on diverse graph datasets for graph representation and generation performance, on which our method largely outperforms existing graph representation learning methods. Moreover, our edge representation learning and pooling method also largely outperforms state-of-the-art graph pooling methods on graph classification, not only because of its accurate edge representation learning, but also due to its lossless compression of the nodes and removal of irrelevant edges for effective message-passing.

An n-vertex tree T is said to be graceful if there exists a bijective labelling φ:V(T)to {1,ldots,n} such that the edge-differences {|φ(x)-φ(y)| : xyin E(T)} are pairwise distinct. The longstanding graceful tree conjecture, posed by Rósa in the 1960s, asserts that every tree is graceful. The gracesize of an n-vertex tree T, denoted gs(T), is the maximum possible number of distinct edge-differences over all bijective labellings φ:V(T)to {1,ldots,n}. The graceful tree conjecture is therefore equivalent to the statement that gs(T)=n-1 for all n-vertex trees. We prove an asymptotic version of this conjecture by showing that for every varepsilon>0, there exists n_0 such that every tree on n>n_0 vertices satisfies gs(T)geqslant (1-varepsilon)n. In other words, every sufficiently large tree admits an almost graceful labelling.

We study orbits in a family of Markoff-like surfaces with extra off-diagonal terms over prime fields F_p. It is shown that, for a typical surface of this form, every non-trivial orbit has size divisible by p. This extends a theorem of W.Y. Chen from the Markoff surface itself to others in this family. The proof closely follows and elaborates on a recent argument of D.E. Martin. We expect that there is just one orbit generically. For some special parameters, we prove that there are at least two or four orbits. Cayley's cubic surface plays a role in parametrising the exceptional cases and dictating the number of solutions mod p.

In this paper, we study 6-dimensional GKM manifolds with 4 fixed points. We classify all possible GKM graphs, and for each type of graph we construct a manifold, proving the existence. We show that six types occur. (P1) complex projective space C P^3 with standard complex structure (P2) blow up of S^6 at a fixed point, diffeomorphic to C P^3 (P3) C P^3 as the homogeneous space Sp(2)/(U(1) times Sp(1)) with non-standard almost complex structure (Q1) complex quadric Q_3 with standard complex structure (Q2) blow up of S^6 along isotropy 2-sphere, diffeomorphic to Q_3 (S) S^2 times S^4, obtained as equivariant gluing along orbits of two S^6's

In this paper, we study the ideas of composition and decomposition in the context of vector spaces, graphs and matroids. For vector spaces V_{AB}, treated as collection of row vectors, with specified column set Auplus B, we define V_{SP}lrarv V_{PQ}, Scap Q= emptyset, to be the collection of all vectors (f_S,f_Q) such that (f_S,f_P)in V_{SP}, (f_P,f_Q)in V_{PQ}. An analogous operation G_{SP}lrarg G_{PQ}equivd G_{PQ} can be defined in relation to graphs G_{SP}, G_{PQ}, on edge sets Suplus P, Puplus Q, respectively in terms of an overlapping subgraph G_P which gets deleted in the right side graph (see for instance the notion of k-sum oxley). For matroids we define the `linking' M_{SP}lrarm M_{PQ} equivd (M_{SP}vee M_{PQ})times (Suplus Q), denoting the contraction operation by 'times'. In each case, we examine how to minimize the size of the `overlap' set P, without affecting the right side entity. In the case of vector spaces, there is a polynomial time algorithm for achieving the minimum, which we present. Similar ideas work for graphs and for matroids under appropriate conditions. Next we consider the problem of decomposition. Here, in the case of vector spaces, the problem is to decompose V_{SQ} as V_{SP}lrarv V_{PQ}, with minimum size P. We give a polynomial time algorithm for this purpose. In the case of graphs and matroids we give a solution to this problem under certain restrictions.

Learning visual representations with self-supervised learning has become popular in computer vision. The idea is to design auxiliary tasks where labels are free to obtain. Most of these tasks end up providing data to learn specific kinds of invariance useful for recognition. In this paper, we propose to exploit different self-supervised approaches to learn representations invariant to (i) inter-instance variations (two objects in the same class should have similar features) and (ii) intra-instance variations (viewpoint, pose, deformations, illumination, etc). Instead of combining two approaches with multi-task learning, we argue to organize and reason the data with multiple variations. Specifically, we propose to generate a graph with millions of objects mined from hundreds of thousands of videos. The objects are connected by two types of edges which correspond to two types of invariance: "different instances but a similar viewpoint and category" and "different viewpoints of the same instance". By applying simple transitivity on the graph with these edges, we can obtain pairs of images exhibiting richer visual invariance. We use this data to train a Triplet-Siamese network with VGG16 as the base architecture and apply the learned representations to different recognition tasks. For object detection, we achieve 63.2% mAP on PASCAL VOC 2007 using Fast R-CNN (compare to 67.3% with ImageNet pre-training). For the challenging COCO dataset, our method is surprisingly close (23.5%) to the ImageNet-supervised counterpart (24.4%) using the Faster R-CNN framework. We also show that our network can perform significantly better than the ImageNet network in the surface normal estimation task.

The paper is a continuation of the study started in Yorzh1. Schrodinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of delta type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of L_1 and C^M edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.

The Markov numbers are positive integers appearing as solutions to the Diophantine equation x^2 + y^2 + z^2 = 3xyz. These numbers are very well-studied and have many combinatorial properties, as well as being the source of the long-standing unicity conjecture. In 2018, Canakc{\i} and Schiffler showed that the Markov number m_{a{b}} is the number of perfect matchings of a certain snake graph corresponding to the Christoffel path from (0,0) to (a,b). Based on this correspondence, Schiffler in 2023 introduced two orderings on lattice paths. For any path omega, associate a snake graph G(omega) and a continued fraction g(omega). The ordering <_M is given by the number of perfect matchings on G(omega), and the ordering <_L is given by the Lagrange number of g(omega). In this work, we settle two conjectures of Schiffler. First, we show that the path omega(a,b) = RRcdots R UU cdots U is the unique maximum over all lattice paths from (0,0) to (a,b) with respect to both orderings <_M and <_L. We then use this result to prove that sup L(omega) over all lattice paths is exactly 1+sqrt5.

We propose a graph clustering formulation based on multicut (a.k.a. weighted correlation clustering) on the complete graph. Our formulation does not need specification of the graph topology as in the original sparse formulation of multicut, making our approach simpler and potentially better performing. In contrast to unweighted correlation clustering we allow for a more expressive weighted cost structure. In dense multicut, the clustering objective is given in a factorized form as inner products of node feature vectors. This allows for an efficient formulation and inference in contrast to multicut/weighted correlation clustering, which has at least quadratic representation and computation complexity when working on the complete graph. We show how to rewrite classical greedy algorithms for multicut in our dense setting and how to modify them for greater efficiency and solution quality. In particular, our algorithms scale to graphs with tens of thousands of nodes. Empirical evidence on instance segmentation on Cityscapes and clustering of ImageNet datasets shows the merits of our approach.

We study the joint image of lines incident to points, meaning the set of image tuples obtained from fixed cameras observing a varying 3D point-line incidence. We prove a formula for the number of complex critical points of the triangulation problem that aims to compute a 3D point-line incidence from noisy images. Our formula works for an arbitrary number of images and measures the intrinsic difficulty of this triangulation. Additionally, we conduct numerical experiments using homotopy continuation methods, comparing different approaches of triangulation of such incidences. In our setup, exploiting the incidence relations gives both a faster point reconstruction and in three views more accurate.

For a positive integer k, a graph property H, and a graph parameter P, let ex_{P}(n, H; δgeq k) denote the maximum value of P over all n-vertex graphs with minimum degree at least k that do not possess the property H. The corresponding extremal families are denoted by EX_{P}(n, H; δgeq k). For two disjoint graphs H_1 and H_2, let H_1 cup H_2 denote their (disjoint) union, i.e., the graph with vertex set V(H_1) cup V(H_2) and edge set E(H_1) cup E(H_2); and let H_1 vee H_2 denote their join. In 1962, Erdős established a classical theorem on the maximum number of edges in a non-Hamiltonian graph of given order and minimum degree. Motivated by recent work on feasible graph parameters in Ai2023, we prove several extensions of Erdős's 1962 theorem on non-Hamiltonian graphs. The first result gives a common generalization of the extremal theorem due to Erdős and its spectral analogs. As direct applications, we obtain complete solutions to open problems raised in the literature since 2016, thereby improving nearly all related prior results in this direction. Our proof technique differs somewhat from those in MR3539577,MR3556876. We also prove an analog theorem for the Hamiltonian-connected property and obtain a result which extends the theorem of Füredi, Kostochka, and Luo MR3843180 on Hamilton cycles.

Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.

Networks provide a powerful formalism for modeling complex systems by using a model of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once; for example, communication within a group rather than person-to person, collaboration among a team rather than a pair of coauthors, or biological interaction between a set of molecules rather than just two. Such higher-order interactions are ubiquitous, but their empirical study has received limited attention, and little is known about possible organizational principles of such structures. Here we study the temporal evolution of 19 datasets with explicit accounting for higher-order interactions. We show that there is a rich variety of structure in our datasets but datasets from the same system types have consistent patterns of higher-order structure. Furthermore, we find that tie strength and edge density are competing positive indicators of higher-order organization, and these trends are consistent across interactions involving differing numbers of nodes. To systematically further the study of theories for such higher-order structures, we propose higher-order link prediction as a benchmark problem to assess models and algorithms that predict higher-order structure. We find a fundamental differences from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.

We study the approximability of an existing framework for clustering edge-colored hypergraphs, which is closely related to chromatic correlation clustering and is motivated by machine learning and data mining applications where the goal is to cluster a set of objects based on multiway interactions of different categories or types. We present improved approximation guarantees based on linear programming, and show they are tight by proving a matching integrality gap. Our results also include new approximation hardness results, a combinatorial 2-approximation whose runtime is linear in the hypergraph size, and several new connections to well-studied objectives such as vertex cover and hypergraph multiway cut.

Assume n, m are positive integers and G is a graph. Let P_{n,m} be the graph obtained from the path with vertices {-m, -(m-1), ldots, 0, ldots, n} by adding a loop at vertex 0. The double cone Delta_{n,m}(G) over a graph G is obtained from the direct product G times P_{n,m} by identifying V(G) times {n} into a single vertex (star, n), identifying V(G) times {-m} into a single vertex (star, -m), and adding an edge connecting (star, -m) and (star, n). This paper determines the fractional chromatic number of Delta_{n,m}(G). In particular, if n < m or n=m is even, then chi_f(Delta_{n,m}(G)) = chi_f(Delta_n(G)), where Delta_n(G) is the nth cone over G. If n=m is odd, then chi_f(Delta_{n,m}(G)) > chi_f(Delta_n(G)). The chromatic number of Delta_{n,m}(G) is also discussed.

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

Visual recognition relies on understanding both the semantics of image tokens and the complex interactions among them. Mainstream self-attention methods, while effective at modeling global pair-wise relations, fail to capture high-order associations inherent in real-world scenes and often suffer from redundant computation. Hypergraphs extend conventional graphs by modeling high-order interactions and offer a promising framework for addressing these limitations. However, existing hypergraph neural networks typically rely on static and hard hyperedge assignments, leading to excessive and redundant hyperedges with hard binary vertex memberships that overlook the continuity of visual semantics. To overcome these issues, we present Soft Hypergraph Neural Networks (SoftHGNNs), which extend the methodology of hypergraph computation, to make it truly efficient and versatile in visual recognition tasks. Our framework introduces the concept of soft hyperedges, where each vertex is associated with hyperedges via continuous participation weights rather than hard binary assignments. This dynamic and differentiable association is achieved by using the learnable hyperedge prototype. Through similarity measurements between token features and the prototype, the model generates semantically rich soft hyperedges. SoftHGNN then aggregates messages over soft hyperedges to capture high-order semantics. To further enhance efficiency when scaling up the number of soft hyperedges, we incorporate a sparse hyperedge selection mechanism that activates only the top-k important hyperedges, along with a load-balancing regularizer to ensure balanced hyperedge utilization. Experimental results across three tasks on five datasets demonstrate that SoftHGNN efficiently captures high-order associations in visual scenes, achieving significant performance improvements.

We study the set of networks, which consist of sources, sinks and neutral points, bijective to the permutations. The set of directed edges, which characterizes a network, is constructed from a polyomino or a Rothe diagram of a permutation through a Dyck tiling on a ribbon. We introduce a new combinatorial object similar to a tree-like tableau, which we call a forest. A forest is shown to give a permutation, and be bijective to a network corresponding to the inverse of the permutation. We show that the poset of networks is a finite graded lattice and admits an EL-labeling. By use of this EL-labeling, we show the lattice is supersolvable and compute the M\"obius function of an interval of the poset.

<<<This is a pre-acceptance version, please, go through Pattern Recognition Journal on Sciencedirect to read the final version>>>. Edge detection is the basis of many computer vision applications. State of the art predominantly relies on deep learning with two decisive factors: dataset content and network's architecture. Most of the publicly available datasets are not curated for edge detection tasks. Here, we offer a solution to this constraint. First, we argue that edges, contours and boundaries, despite their overlaps, are three distinct visual features requiring separate benchmark datasets. To this end, we present a new dataset of edges. Second, we propose a novel architecture, termed Dense Extreme Inception Network for Edge Detection (DexiNed), that can be trained from scratch without any pre-trained weights. DexiNed outperforms other algorithms in the presented dataset. It also generalizes well to other datasets without any fine-tuning. The higher quality of DexiNed is also perceptually evident thanks to the sharper and finer edges it outputs.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

Convolutional neural networks (CNNs) have made great breakthroughs in 2D computer vision. However, their irregular structure makes it hard to harness the potential of CNNs directly on meshes. A subdivision surface provides a hierarchical multi-resolution structure, in which each face in a closed 2-manifold triangle mesh is exactly adjacent to three faces. Motivated by these two observations, this paper presents SubdivNet, an innovative and versatile CNN framework for 3D triangle meshes with Loop subdivision sequence connectivity. Making an analogy between mesh faces and pixels in a 2D image allows us to present a mesh convolution operator to aggregate local features from nearby faces. By exploiting face neighborhoods, this convolution can support standard 2D convolutional network concepts, e.g. variable kernel size, stride, and dilation. Based on the multi-resolution hierarchy, we make use of pooling layers which uniformly merge four faces into one and an upsampling method which splits one face into four. Thereby, many popular 2D CNN architectures can be easily adapted to process 3D meshes. Meshes with arbitrary connectivity can be remeshed to have Loop subdivision sequence connectivity via self-parameterization, making SubdivNet a general approach. Extensive evaluation and various applications demonstrate SubdivNet's effectiveness and efficiency.

We algorithmically determine the regions and facets of all dimensions of the canonical polyhedral complex, the universal object into which a ReLU network decomposes its input space. We show that the locations of the vertices of the canonical polyhedral complex along with their signs with respect to layer maps determine the full facet structure across all dimensions. We present an algorithm which calculates this full combinatorial structure, making use of our theorems that the dual complex to the canonical polyhedral complex is cubical and it possesses a multiplication compatible with its facet structure. The resulting algorithm is numerically stable, polynomial time in the number of intermediate neurons, and obtains accurate information across all dimensions. This permits us to obtain, for example, the true topology of the decision boundaries of networks with low-dimensional inputs. We run empirics on such networks at initialization, finding that width alone does not increase observed topology, but width in the presence of depth does. Source code for our algorithms is accessible online at https://github.com/mmasden/canonicalpoly.

This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erdos, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.

We present an algorithm for efficiently exploring inequivalent Calabi-Yau threefold hypersurfaces in toric varieties. A direct enumeration of fine, regular, star triangulations (FRSTs) of polytopes in the Kreuzer-Skarke database is foreseeably impossible due to the large count of distinct FRSTs. Moreover, such an enumeration is needlessly redundant because many such triangulations have the same restrictions to 2-faces and hence, by Wall's theorem, lead to equivalent Calabi-Yau threefolds. We show that this redundancy can be circumvented by finding a height vector in the strict interior of the intersection of the secondary cones associated with each 2-face triangulation. We demonstrate that such triangulations are generated with orders of magnitude fewer operations than the naive approach of generating all FRSTs and selecting only those differing on 2-faces. Similar methods are also presented to directly generate (the support of) the secondary subfan of all fine triangulations, relevant for random sampling of FRSTs.

Complex networks provide a means to describe cities through their street mesh, expressing characteristics that refer to the structure and organization of an urban zone. Although other studies have used complex networks to model street meshes, we observed a lack of methods to characterize the relationship between cities by using their topological features. Accordingly, this paper aims to describe interactions between cities by using vectors of topological features extracted from their street meshes represented as complex networks. The methodology of this study is based on the use of digital maps. Over the computational representation of such maps, we extract global complex-network features that embody the characteristics of the cities. These vectors allow for the use of multidimensional projection and clustering techniques, enabling a similarity-based comparison of the street meshes. We experiment with 645 cities from the Brazilian state of Sao Paulo. Our results show how the joint of global features describes urban indicators that are deep-rooted in the network's topology and how they reveal characteristics and similarities among sets of cities that are separated from each other.

The complexity of a graph is the number of its labeled spanning trees. In this work complexity is studied in settings that admit regular graphs. An exact formula is established linking complexity of the complement of a regular graph to numbers of closed walks in the graph by way of an infinite alternating series. Some consequences of this result yield infinite classes of lower and upper bounds on the complexity of such graphs. Applications of these mathematical results to biological problems on neuronal activity are described.

In this paper, we study the flow of signals through linear paths with the nonlinear condition that a node emits a signal when it receives external stimuli or when two incoming signals from other nodes arrive coincidentally with a combined amplitude above a fixed threshold. Sets of such nodes form a polychrony group and can sometimes lead to cascades. In the context of this work, cascades are polychrony groups in which the number of nodes activated as a consequence of other nodes is greater than the number of externally activated nodes. The difference between these two numbers is the so-called profit. Given the initial conditions, we predict the conditions for a vertex to activate at a prescribed time and provide an algorithm to efficiently reconstruct a cascade. We develop a dictionary between polychrony groups and graph theory. We call the graph corresponding to a cascade a chinampa. This link leads to a topological classification of chinampas. We enumerate the chinampas of profits zero and one and the description of a family of chinampas isomorphic to a family of partially ordered sets, which implies that the enumeration problem of this family is equivalent to computing the Stanley-order polynomials of those partially ordered sets.

Understanding the origins of militarized conflict is a complex, yet important undertaking. Existing research seeks to build this understanding by considering bi-lateral relationships between entity pairs (dyadic causes) and multi-lateral relationships among multiple entities (systemic causes). The aim of this work is to compare these two causes in terms of how they correlate with conflict between two entities. We do this by devising a set of textual and graph-based features which represent each of the causes. The features are extracted from Wikipedia and modeled as a large graph. Nodes in this graph represent entities connected by labeled edges representing ally or enemy-relationships. This allows casting the problem as an edge classification task, which we term dyad classification. We propose and evaluate classifiers to determine if a particular pair of entities are allies or enemies. Our results suggest that our systemic features might be slightly better correlates of conflict. Further, we find that Wikipedia articles of allies are semantically more similar than enemies.

We study the implications of the modeling choice to use a graph, instead of a hypergraph, to represent real-world interconnected systems whose constituent relationships are of higher order by nature. Such a modeling choice typically involves an underlying projection process that maps the original hypergraph onto a graph, and is common in graph-based analysis. While hypergraph projection can potentially lead to loss of higher-order relations, there exists very limited studies on the consequences of doing so, as well as its remediation. This work fills this gap by doing two things: (1) we develop analysis based on graph and set theory, showing two ubiquitous patterns of hyperedges that are root to structural information loss in all hypergraph projections; we also quantify the combinatorial impossibility of recovering the lost higher-order structures if no extra help is provided; (2) we still seek to recover the lost higher-order structures in hypergraph projection, and in light of (1)'s findings we propose to relax the problem into a learning-based setting. Under this setting, we develop a learning-based hypergraph reconstruction method based on an important statistic of hyperedge distributions that we find. Our reconstruction method is evaluated on 8 real-world datasets under different settings, and exhibits consistently good performance. We also demonstrate benefits of the reconstructed hypergraphs via use cases of protein rankings and link predictions.

Modeling and estimating mixed memberships for overlapping unipartite un-weighted networks has been well studied in recent years. However, to our knowledge, there is no model for a more general case, the overlapping bipartite weighted networks. To close this gap, we introduce a novel model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. Our model allows an adjacency matrix to follow any distribution as long as its expectation has a block structure related to node membership. In particular, BiMMDF can model overlapping bipartite signed networks and it is an extension of many previous models, including the popular mixed membership stochastic blcokmodels. An efficient algorithm with a theoretical guarantee of consistent estimation is applied to fit BiMMDF. We then obtain the separation conditions of BiMMDF for different distributions. Furthermore, we also consider missing edges for sparse networks. The advantage of BiMMDF is demonstrated in extensive synthetic networks and eight real-world networks.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Computer vision researchers are embracing two promising paradigms: Vision Transformers (ViTs) and Multi-task Learning (MTL), which both show great performance but are computation-intensive, given the quadratic complexity of self-attention in ViT and the need to activate an entire large MTL model for one task. M^3ViT is the latest multi-task ViT model that introduces mixture-of-experts (MoE), where only a small portion of subnetworks ("experts") are sparsely and dynamically activated based on the current task. M^3ViT achieves better accuracy and over 80% computation reduction but leaves challenges for efficient deployment on FPGA. Our work, dubbed Edge-MoE, solves the challenges to introduce the first end-to-end FPGA accelerator for multi-task ViT with a collection of architectural innovations, including (1) a novel reordering mechanism for self-attention, which requires only constant bandwidth regardless of the target parallelism; (2) a fast single-pass softmax approximation; (3) an accurate and low-cost GELU approximation; (4) a unified and flexible computing unit that is shared by almost all computational layers to maximally reduce resource usage; and (5) uniquely for M^3ViT, a novel patch reordering method to eliminate memory access overhead. Edge-MoE achieves 2.24x and 4.90x better energy efficiency comparing with GPU and CPU, respectively. A real-time video demonstration is available online, along with our open-source code written using High-Level Synthesis.

This paper generalizes Kocay's lemma, with particular applications to graph reconstruction, as well as discussing and proving aspects around the power of these generalizations and Kocay's original lemma, with a result on the reconstruction of the multiplicity of a tree T as a subgraph of G.

With their meaningful geometry and their omnipresence in the 3D world, edges are extremely useful primitives in computer vision. 3D edges comprise of lines and curves, and methods to reconstruct them use either multi-view images or point clouds as input. State-of-the-art image-based methods first learn a 3D edge point cloud then fit 3D edges to it. The edge point cloud is obtained by learning a 3D neural implicit edge field from which the 3D edge points are sampled on a specific level set (0 or 1). However, such methods present two important drawbacks: i) it is not realistic to sample points on exact level sets due to float imprecision and training inaccuracies. Instead, they are sampled within a range of levels so the points do not lie accurately on the 3D edges and require further processing. ii) Such implicit representations are computationally expensive and require long training times. In this paper, we address these two limitations and propose a 3D edge mapping that is simpler, more efficient, and preserves accuracy. Our method learns explicitly the 3D edge points and their edge direction hence bypassing the need for point sampling. It casts a 3D edge point as the center of a 3D Gaussian and the edge direction as the principal axis of the Gaussian. Such a representation has the advantage of being not only geometrically meaningful but also compatible with the efficient training optimization defined in Gaussian Splatting. Results show that the proposed method produces edges as accurate and complete as the state-of-the-art while being an order of magnitude faster. Code is released at https://github.com/kunalchelani/EdgeGaussians.

Let G be a graph with adjacency matrix A(G) and degree matrix D(G), and let L_μ(G):=A(G)-μD(G). Two graphs G_1 and G_2 are called degree-similar if there exists an invertible matrix M such that M^{-1} A(G_1) M =A(G_2) and M^{-1} D(G_1) M =D(G_2). In this paper, we address three problems concerning degree-similar graphs proposed by Godsil and Sun. First, we present a new characterization of degree-similar graphs using degree partition, from which we derive methods and examples for constructing cospectral graphs and degree-similar graphs. Second, we construct infinite pairs of non-degree-similar trees G_1 and G_2 such that tI- L_μ(G_1) and tI-L_μ(G_2) have the same Smith normal form over Q(μ)[t], which provides a negative answer to a problem posed by Godsil and Sun. Third, we establish several invariants of degree-similar graphs and obtain results on unicyclic graphs that are degree-similar determined. Lastly we prove that for a strongly regular graph G and any two edges e and f of G, G backslash e and G backslash f have identical μ-polynomial, i.e., det(tI-L_μ(G backslash e))=det(tI-L_μ(G backslash f)), which enables the construction of pairs of non-isomorphic graphs with same μ-polynomial, where G backslash e denotes the graph obtained from G by deleting the edge e.

With the growing volume of CT examinations, there is an increasing demand for automated tools such as organ segmentation, abnormality detection, and report generation to support radiologists in managing their clinical workload. Multi-label classification of 3D Chest CT scans remains a critical yet challenging problem due to the complex spatial relationships inherent in volumetric data and the wide variability of abnormalities. Existing methods based on 3D convolutional neural networks struggle to capture long-range dependencies, while Vision Transformers often require extensive pre-training on large-scale, domain-specific datasets to perform competitively. In this work of academic research, we propose a 2.5D alternative by introducing a new graph-based framework that represents 3D CT volumes as structured graphs, where axial slice triplets serve as nodes processed through spectral graph convolution, enabling the model to reason over inter-slice dependencies while maintaining complexity compatible with clinical deployment. Our method, trained and evaluated on 3 datasets from independent institutions, achieves strong cross-dataset generalization, and shows competitive performance compared to state-of-the-art visual encoders. We further conduct comprehensive ablation studies to evaluate the impact of various aggregation strategies, edge-weighting schemes, and graph connectivity patterns. Additionally, we demonstrate the broader applicability of our approach through transfer experiments on automated radiology report generation and abdominal CT data.

Feature-based image matching has extensive applications in computer vision. Keypoints detected in images can be naturally represented as graph structures, and Graph Neural Networks (GNNs) have been shown to outperform traditional deep learning techniques. Consequently, the paradigm of image matching via GNNs has gained significant prominence in recent academic research. In this paper, we first introduce an innovative adaptive graph construction method that utilizes a filtering mechanism based on distance and dynamic threshold similarity. This method dynamically adjusts the criteria for incorporating new vertices based on the characteristics of existing vertices, allowing for the construction of more precise and robust graph structures while avoiding redundancy. We further combine the vertex processing capabilities of GNNs with the global awareness capabilities of Transformers to enhance the model's representation of spatial and feature information within graph structures. This hybrid model provides a deeper understanding of the interrelationships between vertices and their contributions to the matching process. Additionally, we employ the Sinkhorn algorithm to iteratively solve for optimal matching results. Finally, we validate our system using extensive image datasets and conduct comprehensive comparative experiments. Experimental results demonstrate that our system achieves an average improvement of 3.8x-40.3x in overall matching performance. Additionally, the number of vertices and edges significantly impacts training efficiency and memory usage; therefore, we employ multi-GPU technology to accelerate the training process. Our code is available at https://github.com/songxf1024/GIMS.

The eternal vertex cover game is played between an attacker and a defender on an undirected graph G. The defender identifies k vertices to position guards on to begin with. The attacker, on their turn, attacks an edge e, and the defender must move a guard along e to defend the attack. The defender may move other guards as well, under the constraint that every guard moves at most once and to a neighboring vertex. The smallest number of guards required to defend attacks forever is called the eternal vertex cover number of G, denoted evc(G). For any graph G, evc(G) is at least the vertex cover number of G, denoted mvc(G). A graph is Spartan if evc(G) = mvc(G). It is known that a bipartite graph is Spartan if and only if every edge belongs to a perfect matching. We show that the only König graphs that are Spartan are the bipartite Spartan graphs. We also give new lower bounds for evc(G), generalizing a known lower bound based on cut vertices. We finally show a new matching-based characterization of all Spartan graphs.

Diffusion-based generative graph models have been proven effective in generating high-quality small graphs. However, they need to be more scalable for generating large graphs containing thousands of nodes desiring graph statistics. In this work, we propose EDGE, a new diffusion-based generative graph model that addresses generative tasks with large graphs. To improve computation efficiency, we encourage graph sparsity by using a discrete diffusion process that randomly removes edges at each time step and finally obtains an empty graph. EDGE only focuses on a portion of nodes in the graph at each denoising step. It makes much fewer edge predictions than previous diffusion-based models. Moreover, EDGE admits explicitly modeling the node degrees of the graphs, further improving the model performance. The empirical study shows that EDGE is much more efficient than competing methods and can generate large graphs with thousands of nodes. It also outperforms baseline models in generation quality: graphs generated by our approach have more similar graph statistics to those of the training graphs.

We consider a number of graph kernels and proximity measures including commute time kernel, regularized Laplacian kernel, heat kernel, exponential diffusion kernel (also called "communicability"), etc., and the corresponding distances as applied to clustering nodes in random graphs and several well-known datasets. The model of generating random graphs involves edge probabilities for the pairs of nodes that belong to the same class or different predefined classes of nodes. It turns out that in most cases, logarithmic measures (i.e., measures resulting after taking logarithm of the proximities) perform better while distinguishing underlying classes than the "plain" measures. A comparison in terms of reject curves of inter-class and intra-class distances confirms this conclusion. A similar conclusion can be made for several well-known datasets. A possible origin of this effect is that most kernels have a multiplicative nature, while the nature of distances used in cluster algorithms is an additive one (cf. the triangle inequality). The logarithmic transformation is a tool to transform the first nature to the second one. Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property. In our experiments, the leader is usually the logarithmic Communicability measure. However, we indicate some more complicated cases in which other measures, typically, Communicability and plain Walk, can be the winners.

Graphs or networks are a very convenient way to represent data with lots of interaction. Recently, Machine Learning on Graph data has gained a lot of traction. In particular, vertex classification and missing edge detection have very interesting applications, ranging from drug discovery to recommender systems. To achieve such tasks, tremendous work has been accomplished to learn embedding of nodes and edges into finite-dimension vector spaces. This task is called Graph Representation Learning. However, Graph Representation Learning techniques often display prohibitive time and memory complexities, preventing their use in real-time with business size graphs. In this paper, we address this issue by leveraging a degeneracy property of Graphs - the K-Core Decomposition. We present two techniques taking advantage of this decomposition to reduce the time and memory consumption of walk-based Graph Representation Learning algorithms. We evaluate the performances, expressed in terms of quality of embedding and computational resources, of the proposed techniques on several academic datasets. Our code is available at https://github.com/SBrandeis/kcore-embedding

Seminal research in the field of graph neural networks (GNNs) has revealed a direct correspondence between the expressive capabilities of GNNs and the k-dimensional Weisfeiler-Leman (kWL) test, a widely-recognized method for verifying graph isomorphism. This connection has reignited interest in comprehending the specific graph properties effectively distinguishable by the kWL test. A central focus of research in this field revolves around determining the least dimensionality k, for which kWL can discern graphs with different number of occurrences of a pattern graph P. We refer to such a least k as the WL-dimension of this pattern counting problem. This inquiry traditionally delves into two distinct counting problems related to patterns: subgraph counting and induced subgraph counting. Intriguingly, despite their initial appearance as separate challenges with seemingly divergent approaches, both of these problems are interconnected components of a more comprehensive problem: "graph motif parameters". In this paper, we provide a precise characterization of the WL-dimension of labeled graph motif parameters. As specific instances of this result, we obtain characterizations of the WL-dimension of the subgraph counting and induced subgraph counting problem for every labeled pattern P. We additionally demonstrate that in cases where the kWL test distinguishes between graphs with varying occurrences of a pattern P, the exact number of occurrences of P can be computed uniformly using only local information of the last layer of a corresponding GNN. We finally delve into the challenge of recognizing the WL-dimension of various graph parameters. We give a polynomial time algorithm for determining the WL-dimension of the subgraph counting problem for given pattern P, answering an open question from previous work.

With the increasing prevalence of graph-structured data, multi-view graph clustering has been widely used in various downstream applications. Existing approaches primarily rely on a unified message passing mechanism, which significantly enhances clustering performance. Nevertheless, this mechanism limits its applicability to heterophilous situations, as it is fundamentally predicated on the assumption of homophily, i.e., the connected nodes often belong to the same class. In reality, this assumption does not always hold; a moderately or even mildly homophilous graph is more common than a fully homophilous one due to inevitable heterophilous information in the graph. To address this issue, in this paper, we propose a novel SiMilarity-enhanced Homophily for Multi-view Heterophilous Graph Clustering (SMHGC) approach. By analyzing the relationship between similarity and graph homophily, we propose to enhance the homophily by introducing three similarity terms, i.e., neighbor pattern similarity, node feature similarity, and multi-view global similarity, in a label-free manner. Then, a consensus-based inter- and intra-view fusion paradigm is proposed to fuse the improved homophilous graph from different views and utilize them for clustering. The state-of-the-art experimental results on both multi-view heterophilous and homophilous datasets collectively demonstrate the strong capacity of similarity for unsupervised multi-view heterophilous graph learning. Additionally, the consistent performance across semi-synthetic datasets with varying levels of homophily serves as further evidence of SMHGC's resilience to heterophily.

State-of-the-art Graph Neural Networks (GNNs) have limited scalability with respect to the graph and model sizes. On large graphs, increasing the model depth often means exponential expansion of the scope (i.e., receptive field). Beyond just a few layers, two fundamental challenges emerge: 1. degraded expressivity due to oversmoothing, and 2. expensive computation due to neighborhood explosion. We propose a design principle to decouple the depth and scope of GNNs -- to generate representation of a target entity (i.e., a node or an edge), we first extract a localized subgraph as the bounded-size scope, and then apply a GNN of arbitrary depth on top of the subgraph. A properly extracted subgraph consists of a small number of critical neighbors, while excluding irrelevant ones. The GNN, no matter how deep it is, smooths the local neighborhood into informative representation rather than oversmoothing the global graph into "white noise". Theoretically, decoupling improves the GNN expressive power from the perspectives of graph signal processing (GCN), function approximation (GraphSAGE) and topological learning (GIN). Empirically, on seven graphs (with up to 110M nodes) and six backbone GNN architectures, our design achieves significant accuracy improvement with orders of magnitude reduction in computation and hardware cost.

Grid-based approaches to approximate nearest neighbor (ANN) search have been absent from modern scaling analyses. We present a systematic characterization of a multiprobe grid algorithm with respect to dataset size N and dimensionality d. Our experiments reveal a previously unreported d-scaling crossover on the GloVe embedding family, in which multiprobe grid search maintains an approximately constant dimensional scaling exponent while other graph-, tree-, and partitioning-based methods exhibit degrading throughput. The advantage comes with near-linear query scaling in N, but also with lower indexing cost than competing ANN methods. Our results suggest that grid-based methods such as multiprobe grid may be competitive in rebuild-heavy or high-dimensional settings where indexing cost and dimensional robustness dictate performance. More broadly, recent work has formalized self-attention as an ANN operation. Thus, the N- and d-scaling properties of ANN algorithms may guide cost analysis of efficient transformer architectures. Code is available at: https://github.com/weiz345/MultiProbeANN.

Quantifying the similarity between two mathematical structures or datasets constitutes a particularly interesting and useful operation in several theoretical and applied problems. Aimed at this specific objective, the Jaccard index has been extensively used in the most diverse types of problems, also motivating some respective generalizations. The present work addresses further generalizations of this index, including its modification into a coincidence index capable of accounting also for the level of relative interiority between the two compared entities, as well as respective extensions for sets in continuous vector spaces, the generalization to multiset addition, densities and generic scalar fields, as well as a means to quantify the joint interdependence between two random variables. The also interesting possibility to take into account more than two sets has also been addressed, including the description of an index capable of quantifying the level of chaining between three structures. Several of the described and suggested eneralizations have been illustrated with respect to numeric case examples. It is also posited that these indices can play an important role while analyzing and integrating datasets in modeling approaches and pattern recognition activities, including as a measurement of clusters similarity or separation and as a resource for representing and analyzing complex networks.

The problems of detecting and recovering planted structures/subgraphs in Erdős-Rényi random graphs, have received significant attention over the past three decades, leading to many exciting results and mathematical techniques. However, prior work has largely focused on specific ad hoc planted structures and inferential settings, while a general theory has remained elusive. In this paper, we bridge this gap by investigating the detection of an arbitrary planted subgraph Γ= Γ_n in an Erdős-Rényi random graph G(n, q_n), where the edge probability within Γ is p_n. We examine both the statistical and computational aspects of this problem and establish the following results. In the dense regime, where the edge probabilities p_n and q_n are fixed, we tightly characterize the information-theoretic and computational thresholds for detecting Γ, and provide conditions under which a computational-statistical gap arises. Most notably, these thresholds depend on Γ only through its number of edges, maximum degree, and maximum subgraph density. Our lower and upper bounds are general and apply to any value of p_n and q_n as functions of n. Accordingly, we also analyze the sparse regime where q_n = Θ(n^{-α}) and p_n-q_n =Θ(q_n), with αin[0,2], as well as the critical regime where p_n=1-o(1) and q_n = Θ(n^{-α}), both of which have been widely studied, for specific choices of Γ. For these regimes, we show that our bounds are tight for all planted subgraphs investigated in the literature thus farand many more. Finally, we identify conditions under which detection undergoes sharp phase transition, where the boundaries at which algorithms succeed or fail shift abruptly as a function of q_n.

Triggered by limitations of graph-based deep learning methods in terms of computational expressivity and model flexibility, recent years have seen a surge of interest in computational models that operate on higher-order topological domains such as hypergraphs and simplicial complexes. While the increased expressivity of these models can indeed lead to a better classification performance and a more faithful representation of the underlying system, the computational cost of these higher-order models can increase dramatically. To this end, we here explore a simplicial complex neural network learning architecture based on random walks and fast 1D convolutions (SCRaWl), in which we can adjust the increase in computational cost by varying the length and number of random walks considered while accounting for higher-order relationships. Importantly, due to the random walk-based design, the expressivity of the proposed architecture is provably incomparable to that of existing message-passing simplicial neural networks. We empirically evaluate SCRaWl on real-world datasets and show that it outperforms other simplicial neural networks.

Attention-based graph neural networks (GNNs), such as graph attention networks (GATs), have become popular neural architectures for processing graph-structured data and learning node embeddings. Despite their empirical success, these models rely on labeled data and the theoretical properties of these models have yet to be fully understood. In this work, we propose a novel attention-based node embedding framework for graphs. Our framework builds upon a hierarchical kernel for multisets of subgraphs around nodes (e.g. neighborhoods) and each kernel leverages the geometry of a smooth statistical manifold to compare pairs of multisets, by "projecting" the multisets onto the manifold. By explicitly computing node embeddings with a manifold of Gaussian mixtures, our method leads to a new attention mechanism for neighborhood aggregation. We provide theoretical insights into generalizability and expressivity of our embeddings, contributing to a deeper understanding of attention-based GNNs. We propose both efficient unsupervised and supervised methods for learning the embeddings. Through experiments on several node classification benchmarks, we demonstrate that our proposed method outperforms existing attention-based graph models like GATs. Our code is available at https://github.com/BorgwardtLab/fisher_information_embedding.

The {K_{1,1}, K_{1,2},C_m: mgeq3}-factor of a graph is a spanning subgraph whose each component is an element of {K_{1,1}, K_{1,2},C_m: mgeq3}. In this paper, through the graph spectral methods, we establish the lower bound of the signless Laplacian spectral radius and the upper bound of the distance spectral radius to determine whether a graph admits a {K_2}-factor. We get a lower bound on the size (resp. the spectral radius) of G to guarantee that G contains a {K_{1,1}, K_{1,2},C_m: mgeq3}-factor. Then we determine an upper bound on the distance spectral radius of G to ensure that G has a {K_{1,1}, K_{1,2},C_m: mgeq3}-factor. Furthermore, by constructing extremal graphs, we show that the above all bounds are best possible.

Federated learning is an active research topic since it enables several participants to jointly train a model without sharing local data. Currently, cross-silo federated learning is a popular training setting that utilizes a few hundred reliable data silos with high-speed access links to training a model. While this approach has been widely applied in real-world scenarios, designing a robust topology to reduce the training time remains an open problem. In this paper, we present a new multigraph topology for cross-silo federated learning. We first construct the multigraph using the overlay graph. We then parse this multigraph into different simple graphs with isolated nodes. The existence of isolated nodes allows us to perform model aggregation without waiting for other nodes, hence effectively reducing the training time. Intensive experiments on three public datasets show that our proposed method significantly reduces the training time compared with recent state-of-the-art topologies while maintaining the accuracy of the learned model. Our code can be found at https://github.com/aioz-ai/MultigraphFL

We present GRAPE (Group RepresentAtional Position Encoding), a unified framework for positional encoding based on group actions. GRAPE brings together two families of mechanisms: (i) multiplicative rotations (Multiplicative GRAPE) in SO(d) and (ii) additive logit biases (Additive GRAPE) arising from unipotent actions in the general linear group GL. In Multiplicative GRAPE, a position n in Z (or t in R) acts as G(n)=exp(n,ω,L) with a rank-2 skew generator L in R^{d times d}, yielding a relative, compositional, norm-preserving map with a closed-form matrix exponential. RoPE is recovered exactly when the d/2 planes are the canonical coordinate pairs with log-uniform spectrum. Learned commuting subspaces and compact non-commuting mixtures strictly extend this geometry to capture cross-subspace feature coupling at O(d) and O(r d) cost per head, respectively. In Additive GRAPE, additive logits arise as rank-1 (or low-rank) unipotent actions, recovering ALiBi and the Forgetting Transformer (FoX) as exact special cases while preserving an exact relative law and streaming cacheability. Altogether, GRAPE supplies a principled design space for positional geometry in long-context models, subsuming RoPE and ALiBi as special cases. Project Page: https://github.com/model-architectures/GRAPE.

In this study, we tackle the challenging fine-grained edge detection task, which refers to predicting specific edges caused by reflectance, illumination, normal, and depth changes, respectively. Prior methods exploit multi-scale convolutional networks, which are limited in three aspects: (1) Convolutions are local operators while identifying the cause of edge formation requires looking at far away pixels. (2) Priors specific to edge cause are fixed in prediction heads. (3) Using separate networks for generic and fine-grained edge detection, and the constraint between them may be violated. To address these three issues, we propose a two-stage transformer-based network sequentially predicting generic edges and fine-grained edges, which has a global receptive field thanks to the attention mechanism. The prior knowledge of edge causes is formulated as four learnable cause tokens in a cause-aware decoder design. Furthermore, to encourage the consistency between generic edges and fine-grained edges, an edge aggregation and alignment loss is exploited. We evaluate our method on the public benchmark BSDS-RIND and several newly derived benchmarks, and achieve new state-of-the-art results. Our code, data, and models are publicly available at https://github.com/Daniellli/ECT.git.

We develop a Hellmann--Feynman type perturbation theory for the discrete signed p-Laplacian and apply it to a parametrized perturbation by edge cuts. We show that the eigenvalues of the signed p-Laplacian can be characterized as citical values of the parameter-dependent eigenvalues of a simpler graph.

The drug development pipeline for a new compound can last 10-20 years and cost over 10 billion. Drug repurposing offers a more time- and cost-effective alternative. Computational approaches based on biomedical knowledge graph representations have recently yielded new drug repurposing hypotheses. In this study, we present a novel, disease-specific hypergraph representation learning technique to derive contextual embeddings of biological pathways of various lengths but that all start at any given drug and all end at the disease of interest. Further, we extend this method to multi-disease hypergraphs. To determine the repurposing potential of each of the 1,522 drugs, we derive drug-specific distributions of cosine similarity values and ultimately consider the median for ranking. Cosine similarity values are computed between (1) all biological pathways starting at the considered drug and ending at the disease of interest and (2) all biological pathways starting at drugs currently prescribed against that disease and ending at the disease of interest. We illustrate our approach with Alzheimer's disease (AD) and two of its risk factors: hypertension (HTN) and type 2 diabetes (T2D). We compare each drug's rank across four hypergraph settings (single- or multi-disease): AD only, AD + HTN, AD + T2D, and AD + HTN + T2D. Notably, our framework led to the identification of two promising drugs whose repurposing potential was significantly higher in hypergraphs combining two diseases: dapagliflozin (antidiabetic; moved up, from top 32% to top 7%, across all considered drugs) and debrisoquine (antihypertensive; moved up, from top 76% to top 23%). Our approach serves as a hypothesis generation tool, to be paired with a validation pipeline relying on laboratory experiments and semi-automated parsing of the biomedical literature.

We study the Stochastic Boolean Function Certification (SBFC) problem, where we are given n Bernoulli random variables {X_e: e in U} on a ground set U of n elements with joint distribution p, a Boolean function f: 2^U to {0, 1}, and an (unknown) scenario S = {e in U: X_e = 1} of active elements sampled from p. We seek to probe the elements one-at-a-time to reveal if they are active until we can certify f(S) = 1, while minimizing the expected number of probes. Unlike most previous results that assume independence, we study correlated distributions p and give approximation algorithms for several classes of functions f. When f(S) is the indicator function for whether S is the spanning set of a given matroid, our problem reduces to finding a basis of active elements of a matroid by probing elements. We give a non-adaptive O(log n)-approximation algorithm for arbitrary distributions p, and show that this is tight up to constants unless P = NP, even for partition matroids. For uniform matroids, we give constant factor 4.642-approximation ([BBFT20]) that can be further improved to a 2-approximation if additionally the random variables are negatively correlated for the case of 1-uniform matroid. We also give an adaptive O(log k)-approximation algorithm for SBFC for k-uniform matroids for the Graph Probing problem, where we seek to probe the edges of a graph one-at-a-time until we find k active edges. The underlying distribution on edges arises from (hidden) independent vertex random variables, with an edge being active if at least one of its endpoints is active. This significantly improves over the information-theoretic lower bound on Ω(poly(n)) ([JGM19]) for adaptive algorithms for k-uniform matroids with arbitrary distributions.

This paper proposes a Deep Learning based edge detector, which is inspired on both HED (Holistically-Nested Edge Detection) and Xception networks. The proposed approach generates thin edge-maps that are plausible for human eyes; it can be used in any edge detection task without previous training or fine tuning process. As a second contribution, a large dataset with carefully annotated edges has been generated. This dataset has been used for training the proposed approach as well the state-of-the-art algorithms for comparisons. Quantitative and qualitative evaluations have been performed on different benchmarks showing improvements with the proposed method when F-measure of ODS and OIS are considered.

A subgraph is constructed by using a subset of vertices and edges of a given graph. There exist many graph properties that are hereditary for subgraphs. Hence, researchers from different communities have paid a great deal of attention in studying numerous subgraph problems, on top of the ordinary graph problems. Many algorithms are proposed in studying subgraph problems, where one common approach is by extracting the patterns and structures of a given graph. Due to the complex structures of certain types of graphs and to improve overall performances of the existing frameworks, machine learning techniques have recently been employed in dealing with various subgraph problems. In this article, we present a comprehensive review on five well known subgraph problems that have been tackled by using machine learning methods. They are subgraph isomorphism (both counting and matching), maximum common subgraph, community detection and community search problems. We provide an outline of each proposed method, and examine its designs and performances. We also explore non-learning-based algorithms for each problem and a brief discussion is given. We then suggest some promising research directions in this area, hoping that relevant subgraph problems can be tackled by using a similar strategy. Since there is a huge growth in employing machine learning techniques in recent years, we believe that this survey will serve as a good reference point to relevant research communities.

Generating multiview images from a single view facilitates the rapid generation of a 3D mesh conditioned on a single image. Recent methods that introduce 3D global representation into diffusion models have shown the potential to generate consistent multiviews, but they have reduced generation speed and face challenges in maintaining generalizability and quality. To address this issue, we propose EpiDiff, a localized interactive multiview diffusion model. At the core of the proposed approach is to insert a lightweight epipolar attention block into the frozen diffusion model, leveraging epipolar constraints to enable cross-view interaction among feature maps of neighboring views. The newly initialized 3D modeling module preserves the original feature distribution of the diffusion model, exhibiting compatibility with a variety of base diffusion models. Experiments show that EpiDiff generates 16 multiview images in just 12 seconds, and it surpasses previous methods in quality evaluation metrics, including PSNR, SSIM and LPIPS. Additionally, EpiDiff can generate a more diverse distribution of views, improving the reconstruction quality from generated multiviews. Please see our project page at https://huanngzh.github.io/EpiDiff/.

The integration of multi-omics single-cell data remains challenging due to high-dimensionality and complex inter-modality relationships. To address this, we introduce MoRE-GNN (Multi-omics Relational Edge Graph Neural Network), a heterogeneous graph autoencoder that combines graph convolution and attention mechanisms to dynamically construct relational graphs directly from data. Evaluations on six publicly available datasets demonstrate that MoRE-GNN captures biologically meaningful relationships and outperforms existing methods, particularly in settings with strong inter-modality correlations. Furthermore, the learned representations allow for accurate downstream cross-modal predictions. While performance may vary with dataset complexity, MoRE-GNN offers an adaptive, scalable and interpretable framework for advancing multi-omics integration.

Graph clustering, which aims to divide a graph into several homogeneous groups, is a critical area of study with applications that span various fields such as social network analysis, bioinformatics, and image segmentation. This paper explores both traditional and more recent approaches to graph clustering. Firstly, key concepts and definitions in graph theory are introduced. The background section covers essential topics, including graph Laplacians and the integration of Deep Learning in graph analysis. The paper then delves into traditional clustering methods, including Spectral Clustering and the Leiden algorithm. Following this, state-of-the-art clustering techniques that leverage deep learning are examined. A comprehensive comparison of these methods is made through experiments. The paper concludes with a discussion of the practical applications of graph clustering and potential future research directions.

Scene generation is crucial to many computer graphics applications. Recent advances in generative AI have streamlined sketch-to-image workflows, easing the workload for artists and designers in creating scene concept art. However, these methods often struggle for complex scenes with multiple detailed objects, sometimes missing small or uncommon instances. In this paper, we propose a Training-free Triplet Tuning for Sketch-to-Scene (T3-S2S) generation after reviewing the entire cross-attention mechanism. This scheme revitalizes the existing ControlNet model, enabling effective handling of multi-instance generations, involving prompt balance, characteristics prominence, and dense tuning. Specifically, this approach enhances keyword representation via the prompt balance module, reducing the risk of missing critical instances. It also includes a characteristics prominence module that highlights TopK indices in each channel, ensuring essential features are better represented based on token sketches. Additionally, it employs dense tuning to refine contour details in the attention map, compensating for instance-related regions. Experiments validate that our triplet tuning approach substantially improves the performance of existing sketch-to-image models. It consistently generates detailed, multi-instance 2D images, closely adhering to the input prompts and enhancing visual quality in complex multi-instance scenes. Code is available at https://github.com/chaos-sun/t3s2s.git.

We introduce a method for the generation of images from an input scene graph. The method separates between a layout embedding and an appearance embedding. The dual embedding leads to generated images that better match the scene graph, have higher visual quality, and support more complex scene graphs. In addition, the embedding scheme supports multiple and diverse output images per scene graph, which can be further controlled by the user. We demonstrate two modes of per-object control: (i) importing elements from other images, and (ii) navigation in the object space, by selecting an appearance archetype. Our code is publicly available at https://www.github.com/ashual/scene_generation

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

Since the pioneering work on the lottery ticket hypothesis for graph neural networks (GNNs) was proposed in Chen et al. (2021), the study on finding graph lottery tickets (GLT) has become one of the pivotal focus in the GNN community, inspiring researchers to discover sparser GLT while achieving comparable performance to original dense networks. In parallel, the graph structure has gained substantial attention as a crucial factor in GNN training dynamics, also elucidated by several recent studies. Despite this, contemporary studies on GLT, in general, have not fully exploited inherent pathways in the graph structure and identified tickets in an iterative manner, which is time-consuming and inefficient. To address these limitations, we introduce TEDDY, a one-shot edge sparsification framework that leverages structural information by incorporating edge-degree information. Following edge sparsification, we encourage the parameter sparsity during training via simple projected gradient descent on the ell_0 ball. Given the target sparsity levels for both the graph structure and the model parameters, our TEDDY facilitates efficient and rapid realization of GLT within a single training. Remarkably, our experimental results demonstrate that TEDDY significantly surpasses conventional iterative approaches in generalization, even when conducting one-shot sparsification that solely utilizes graph structures, without taking feature information into account.

Recent mesh generation approaches typically tokenize triangle meshes into sequences of tokens and train autoregressive models to generate these tokens sequentially. Despite substantial progress, such token sequences inevitably reuse vertices multiple times to fully represent manifold meshes, as each vertex is shared by multiple faces. This redundancy leads to excessively long token sequences and inefficient generation processes. In this paper, we propose an efficient framework that generates artistic meshes by treating vertices and faces separately, significantly reducing redundancy. We employ an autoregressive model solely for vertex generation, decreasing the token count to approximately 23\% of that required by the most compact existing tokenizer. Next, we leverage a bidirectional transformer to complete the mesh in a single step by capturing inter-vertex relationships and constructing the adjacency matrix that defines the mesh faces. To further improve the generation quality, we introduce a fidelity enhancer to refine vertex positioning into more natural arrangements and propose a post-processing framework to remove undesirable edge connections. Experimental results show that our method achieves more than 8times faster speed on mesh generation compared to state-of-the-art approaches, while producing higher mesh quality.

Semi-transitive graphs, defined in hps98 as examples where ``uniform percolation" holds whenever p>p_c, are a large class of graphs more general than quasi-transitive graphs. Let G be a semi-transitive graph with one end which can be properly embedded into the plane with uniformly bounded face degree for finite faces and minimal vertex degree at least 7. We show that p_u^{site}(G) +p_c^{site}(G_*)=1, where G_* denotes the matching graph of G. This fulfils and extends an observation of Sykes and Essam in 1964 (SE64) to semi-transitive graphs.

Let S be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps f,g,h:A^2 --to Ssubset P^n such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps f,g:A^2 --to S, such that the union of their images covers the affine surface Scap A^n. In the affine case, the number of rational maps involved in the cover is in general optimal.

This paper has two aims, first one is to introduce special kind of proximal contractions guaranteeing a finite number of best proximity points, and second one is to derive best proximity point results for perimetric contractions. To meet these two aims, we introduce two new proximal contractions: perimetric proximal contractions of the first and the second kind, and derive best proximity point results for these mappings. We establish that for these particular mappings, best proximity points are not necessarily unique; however, we provide an upper bound, proving that at most two such points can exist. To establish the validity of our results, we provide illustrative examples demonstrating that these newly defined mappings can possess unique or exactly two best proximity points.

An open-source C++ framework for discovering fast matrix multiplication schemes using the flip graph approach is presented. The framework supports multiple coefficient rings -- binary (Z_2), modular ternary (Z_3) and integer ternary (Z_T = {-1,0,1}) -- and implements both fixed-dimension and meta-dimensional search operators. Using efficient bit-level encoding of coefficient vectors and OpenMP parallelism, the tools enable large-scale exploration on commodity hardware. The study covers 680 schemes ranging from (2 times 2 times 2) to (16 times 16 times 16), with 276 schemes now in Z_T coefficients and 117 in integer coefficients. With this framework, the multiplicative complexity (rank) is improved for 79 matrix multiplication schemes. Notably, a new 4 times 4 times 10 scheme requiring only 115 multiplications is discovered, achieving ωapprox 2.80478 and beating Strassen's exponent for this specific size. Additionally, 93 schemes are rediscovered in ternary coefficients that were previously known only over rationals or integers, and 68 schemes in integer coefficients that previously required fractions. All tools and discovered schemes are made publicly available to enable reproducible research.

This is the third paper of the CayleyPy project applying artificial intelligence to problems in group theory. We announce the first public release of CayleyPy, an open source Python library for computations with Cayley and Schreier graphs. Compared with systems such as GAP and Sage, CayleyPy handles much larger graphs and performs several orders of magnitude faster. Using CayleyPy we obtained about 200 new conjectures on Cayley and Schreier graphs, focused on diameters and growth. For many Cayley graphs of symmetric groups Sn we observe quasi polynomial diameter formulas: a small set of quadratic or linear polynomials indexed by n mod s. We conjecture that this is a general phenomenon, giving efficient diameter computation despite the problem being NP hard. We propose a refinement of the Babai type conjecture on diameters of Sn: n^2/2 + 4n upper bounds in the undirected case, compared to previous O(n^2) bounds. We also provide explicit generator families, related to involutions in a square with whiskers pattern, conjectured to maximize the diameter; search confirms this for all n up to 15. We further conjecture an answer to a question posed by V M Glushkov in 1968 on directed Cayley graphs generated by a cyclic shift and a transposition. For nilpotent groups we conjecture an improvement of J S Ellenberg's results on upper unitriangular matrices over Z/pZ, showing linear dependence of diameter on p. Moreover. Some conjectures are LLM friendly, naturally stated as sorting problems verifiable by algorithms or Python code. To benchmark path finding we created more than 10 Kaggle datasets. CayleyPy works with arbitrary permutation or matrix groups and includes over 100 predefined generators. Our growth computation code outperforms GAP and Sage up to 1000 times in speed and size.

Multimodal Attributed Graphs (MMAGs) are an expressive data model for representing the complex interconnections among entities that associate attributes from multiple data modalities (text, images, etc.). Clustering over such data finds numerous practical applications in real scenarios, including social community detection, medical data analytics, etc. However, as revealed by our empirical studies, existing multi-view clustering solutions largely rely on the high correlation between attributes across various views and overlook the unique characteristics (e.g., low modality-wise correlation and intense feature-wise noise) of multimodal attributes output by large pre-trained language and vision models in MMAGs, leading to suboptimal clustering performance. Inspired by foregoing empirical observations and our theoretical analyses with graph signal processing, we propose the Dual Graph Filtering (DGF) scheme, which innovatively incorporates a feature-wise denoising component into node representation learning, thereby effectively overcoming the limitations of traditional graph filters adopted in the extant multi-view graph clustering approaches. On top of that, DGF includes a tri-cross contrastive training strategy that employs instance-level contrastive learning across modalities, neighborhoods, and communities for learning robust and discriminative node representations. Our comprehensive experiments on eight benchmark MMAG datasets exhibit that DGF is able to outperform a wide range of state-of-the-art baselines consistently and significantly in terms of clustering quality measured against ground-truth labels.

We study the H-chromatic symmetric functions X_G^H (introduced in (arXiv:2011.06063) as a generalization of the chromatic symmetric function (CSF) X_G), which track homomorphisms from the graph G to the graph H. We focus first on the case of self-chromatic symmetric functions (self-CSFs) X_G^G, making some progress toward a conjecture from (arXiv:2011.06063) that the self-CSF, like the normal CSF, is always different for different trees. In particular, we show that the self-CSF distinguishes trees from non-trees with just one exception, we check using Sage that it distinguishes all trees on up to 12 vertices, and we show that it determines the number of legs of a spider and the degree sequence of a caterpillar given its spine length. We also show that the self-CSF detects the number of connected components of a forest, again with just one exception. Then we prove some results about the power sum expansions for H-CSFs when H is a complete bipartite graph, in particular proving that the conjecture from (arXiv:2011.06063) about p-monotonicity of ω(X_G^H) for H a star holds as long as H is sufficiently large compared to G. We also show that the self-CSFs of complete multipartite graphs form a basis for the ring Λ of symmetric functions, and we give some construction of bases for the vector space Λ^n of degree n symmetric functions using H-CSFs X_G^H where H is a fixed graph that is not a complete graph, answering a question from (arXiv:2011.06063) about whether such bases exist. However, we show that there generally do not exist such bases with G fixed, even with loops, answering another question from (arXiv:2011.06063). We also define the H-chromatic polynomial as an analogue of the chromatic polynomial, and ask when it is the same for different graphs.

Unsupervised Multiplex Graph Learning (UMGL) aims to learn node representations on various edge types without manual labeling. However, existing research overlooks a key factor: the reliability of the graph structure. Real-world data often exhibit a complex nature and contain abundant task-irrelevant noise, severely compromising UMGL's performance. Moreover, existing methods primarily rely on contrastive learning to maximize mutual information across different graphs, limiting them to multiplex graph redundant scenarios and failing to capture view-unique task-relevant information. In this paper, we focus on a more realistic and challenging task: to unsupervisedly learn a fused graph from multiple graphs that preserve sufficient task-relevant information while removing task-irrelevant noise. Specifically, our proposed Information-aware Unsupervised Multiplex Graph Fusion framework (InfoMGF) uses graph structure refinement to eliminate irrelevant noise and simultaneously maximizes view-shared and view-unique task-relevant information, thereby tackling the frontier of non-redundant multiplex graph. Theoretical analyses further guarantee the effectiveness of InfoMGF. Comprehensive experiments against various baselines on different downstream tasks demonstrate its superior performance and robustness. Surprisingly, our unsupervised method even beats the sophisticated supervised approaches. The source code and datasets are available at https://github.com/zxlearningdeep/InfoMGF.

We address the problem of determining the Hausdorff dimension of sets consisting of complex irrationals whose complex continued fraction digits satisfy prescribed restrictions and growth conditions. For the Hurwitz continued fraction, we confirm Hirst's conjecture, as a complex analogue of the result of Wang and Wu [Bull. Lond. Math. Soc. {\bf 40} (2008), no. 1, 18--22] for the regular continued fraction. We also prove a complex analogue of the second-named author's result on the Hausdorff dimension of sets with restricted slowly growing digits [Proc. Amer. Math. Soc. {\bf 151} (2023), no. 9, 3645--3653]. To these ends, we exploit an infinite conformal iterated function system associated with the Hurwitz continued fraction.

We introduce MeshAnything V2, an autoregressive transformer that generates Artist-Created Meshes (AM) aligned to given shapes. It can be integrated with various 3D asset production pipelines to achieve high-quality, highly controllable AM generation. MeshAnything V2 surpasses previous methods in both efficiency and performance using models of the same size. These improvements are due to our newly proposed mesh tokenization method: Adjacent Mesh Tokenization (AMT). Different from previous methods that represent each face with three vertices, AMT uses a single vertex whenever possible. Compared to previous methods, AMT requires about half the token sequence length to represent the same mesh in average. Furthermore, the token sequences from AMT are more compact and well-structured, fundamentally benefiting AM generation. Our extensive experiments show that AMT significantly improves the efficiency and performance of AM generation. Project Page: https://buaacyw.github.io/meshanything-v2/

We look at the eigenvalues of the Hessian of a loss function before and after training. The eigenvalue distribution is seen to be composed of two parts, the bulk which is concentrated around zero, and the edges which are scattered away from zero. We present empirical evidence for the bulk indicating how over-parametrized the system is, and for the edges that depend on the input data.

Multi-view image generation holds significant application value in computer vision, particularly in domains like 3D reconstruction, virtual reality, and augmented reality. Most existing methods, which rely on extending single images, face notable computational challenges in maintaining cross-view consistency and generating high-resolution outputs. To address these issues, we propose the Geometry-guided Multi-View Diffusion Model, which incorporates mechanisms for extracting multi-view geometric information and adjusting the intensity of geometric features to generate images that are both consistent across views and rich in detail. Specifically, we design a multi-view geometry information extraction module that leverages depth maps, normal maps, and foreground segmentation masks to construct a shared geometric structure, ensuring shape and structural consistency across different views. To enhance consistency and detail restoration during generation, we develop a decoupled geometry-enhanced attention mechanism that strengthens feature focus on key geometric details, thereby improving overall image quality and detail preservation. Furthermore, we apply an adaptive learning strategy that fine-tunes the model to better capture spatial relationships and visual coherence between the generated views, ensuring realistic results. Our model also incorporates an iterative refinement process that progressively improves the output quality through multiple stages of image generation. Finally, a dynamic geometry information intensity adjustment mechanism is proposed to adaptively regulate the influence of geometric data, optimizing overall quality while ensuring the naturalness of generated images. More details can be found on the project page: https://sobeymil.github.io/GeoMVD.com.

Modern VLSI designs comprise tens of billions of components, making scalable hypergraph partitioning critical for parallel and hierarchical optimization. Although multilevel partitioning remains the dominant paradigm, its coarsening stage can distort structural information, especially in hypergraphs with many high-degree hyperedges, leading to increased refinement overhead and limited scalability. Recent approaches incorporate spectral information to guide coarsening, but only in a heuristic manner, without directly optimizing the partitioning objectives. We introduce HySpecPro, a single-level hypergraph partitioner that performs end-to-end optimization in a spectral embedding space. HySpecPro constructs embeddings from a bipartite Laplacian and performs efficient projection-based search, supported by a fully GPU-accelerated implementation. Experiments show that HySpecPro delivers cut quality comparable to state-of-the-art multilevel methods while scaling linearly with the total hyperedge degree.

We introduce the Multi-Instance Generation (MIG) task, which focuses on generating multiple instances within a single image, each accurately placed at predefined positions with attributes such as category, color, and shape, strictly following user specifications. MIG faces three main challenges: avoiding attribute leakage between instances, supporting diverse instance descriptions, and maintaining consistency in iterative generation. To address attribute leakage, we propose the Multi-Instance Generation Controller (MIGC). MIGC generates multiple instances through a divide-and-conquer strategy, breaking down multi-instance shading into single-instance tasks with singular attributes, later integrated. To provide more types of instance descriptions, we developed MIGC++. MIGC++ allows attribute control through text \& images and position control through boxes \& masks. Lastly, we introduced the Consistent-MIG algorithm to enhance the iterative MIG ability of MIGC and MIGC++. This algorithm ensures consistency in unmodified regions during the addition, deletion, or modification of instances, and preserves the identity of instances when their attributes are changed. We introduce the COCO-MIG and Multimodal-MIG benchmarks to evaluate these methods. Extensive experiments on these benchmarks, along with the COCO-Position benchmark and DrawBench, demonstrate that our methods substantially outperform existing techniques, maintaining precise control over aspects including position, attribute, and quantity. Project page: https://github.com/limuloo/MIGC.

This paper studies the problem of recovering cameras from a set of fundamental matrices. A set of fundamental matrices is said to be compatible if a set of cameras exists for which they are the fundamental matrices. We focus on the complete graph, where fundamental matrices for each pair of cameras are given. Previous work has established necessary and sufficient conditions for compatibility as rank and eigenvalue conditions on the n-view fundamental matrix obtained by concatenating the individual fundamental matrices. In this work, we show that the eigenvalue condition is redundant. We provide explicit homogeneous polynomials that describe necessary and sufficient conditions for compatibility in terms of the fundamental matrices and their epipoles. In this direction, we find that quadruple-wise compatibility is enough to ensure global compatibility for any number of cameras. We demonstrate that for four cameras, compatibility is generically described by triple-wise conditions and one additional equation involving all fundamental matrices.

Despite the recent successes of vanilla Graph Neural Networks (GNNs) on many tasks, their foundation on pairwise interaction networks inherently limits their capacity to discern latent higher-order interactions in complex systems. To bridge this capability gap, we propose a novel approach exploiting the rich mathematical theory of simplicial complexes (SCs) - a robust tool for modeling higher-order interactions. Current SC-based GNNs are burdened by high complexity and rigidity, and quantifying higher-order interaction strengths remains challenging. Innovatively, we present a higher-order Flower-Petals (FP) model, incorporating FP Laplacians into SCs. Further, we introduce a Higher-order Graph Convolutional Network (HiGCN) grounded in FP Laplacians, capable of discerning intrinsic features across varying topological scales. By employing learnable graph filters, a parameter group within each FP Laplacian domain, we can identify diverse patterns where the filters' weights serve as a quantifiable measure of higher-order interaction strengths. The theoretical underpinnings of HiGCN's advanced expressiveness are rigorously demonstrated. Additionally, our empirical investigations reveal that the proposed model accomplishes state-of-the-art (SOTA) performance on a range of graph tasks and provides a scalable and flexible solution to explore higher-order interactions in graphs.

Subgraph matching is vital in knowledge graph (KG) question answering, molecule design, scene graph, code and circuit search, etc. Neural methods have shown promising results for subgraph matching. Our study of recent systems suggests refactoring them into a unified design space for graph matching networks. Existing methods occupy only a few isolated patches in this space, which remains largely uncharted. We undertake the first comprehensive exploration of this space, featuring such axes as attention-based vs. soft permutation-based interaction between query and corpus graphs, aligning nodes vs. edges, and the form of the final scoring network that integrates neural representations of the graphs. Our extensive experiments reveal that judicious and hitherto-unexplored combinations of choices in this space lead to large performance benefits. Beyond better performance, our study uncovers valuable insights and establishes general design principles for neural graph representation and interaction, which may be of wider interest.

Multivariate time series is prevalent in many scientific and industrial domains. Modeling multivariate signals is challenging due to their long-range temporal dependencies and intricate interactions--both direct and indirect. To confront these complexities, we introduce a method of representing multivariate signals as nodes in a graph with edges indicating interdependency between them. Specifically, we leverage graph neural networks (GNN) and attention mechanisms to efficiently learn the underlying relationships within the time series data. Moreover, we suggest employing hierarchical signal decompositions running over the graphs to capture multiple spatial dependencies. The effectiveness of our proposed model is evaluated across various real-world benchmark datasets designed for long-term forecasting tasks. The results consistently showcase the superiority of our model, achieving an average 23\% reduction in mean squared error (MSE) compared to existing models.

Tangles were originally introduced as a concept to formalize regions of high connectivity in graphs. In recent years, they have also been discovered as a link between structural graph theory and data science: when interpreting similarity in data sets as connectivity between points, finding clusters in the data essentially amounts to finding tangles in the underlying graphs. This paper further explores the potential of tangles in data sets as a means for a formal study of clusters. Real-world data often follow a normal distribution. Accounting for this, we develop a quantitative theory of tangles in data sets drawn from Gaussian mixtures. To this end, we equip the data with a graph structure that models similarity between the points and allows us to apply tangle theory to the data. We provide explicit conditions under which tangles associated with the marginal Gaussian distributions exist asymptotically almost surely. This can be considered as a sufficient formal criterion for the separabability of clusters in the data.

This tutorial serves as a comprehensive guide for understanding graph databases, focusing on the fundamentals of graph theory while showcasing practical applications across various fields. It starts by introducing foundational concepts and delves into the structure of graphs through nodes and edges, covering different types such as undirected, directed, weighted, and unweighted graphs. Key graph properties, terminologies, and essential algorithms for network analysis are outlined, including Dijkstras shortest path algorithm and methods for calculating node centrality and graph connectivity. The tutorial highlights the advantages of graph databases over traditional relational databases, particularly in efficiently managing complex, interconnected data. It examines leading graph database systems such as Neo4j, Amazon Neptune, and ArangoDB, emphasizing their unique features for handling large datasets. Practical instructions on graph operations using NetworkX and Neo4j are provided, covering node and edge creation, attribute assignment, and advanced queries with Cypher. Additionally, the tutorial explores common graph visualization techniques using tools like Plotly and Neo4j Bloom, which enhance the interpretation and usability of graph data. It also delves into community detection algorithms, including the Louvain method, which facilitates clustering in large networks. Finally, the paper concludes with recommendations for researchers interested in exploring the vast potential of graph technologies.

Text-driven image synthesis has made significant advancements with the development of diffusion models, transforming how visual content is generated from text prompts. Despite these advances, text-driven image editing, a key area in computer graphics, faces unique challenges. A major challenge is making simultaneous edits across multiple objects or attributes. Applying these methods sequentially for multi-aspect edits increases computational demands and efficiency losses. In this paper, we address these challenges with significant contributions. Our main contribution is the development of MultiEdits, a method that seamlessly manages simultaneous edits across multiple attributes. In contrast to previous approaches, MultiEdits not only preserves the quality of single attribute edits but also significantly improves the performance of multitasking edits. This is achieved through an innovative attention distribution mechanism and a multi-branch design that operates across several processing heads. Additionally, we introduce the PIE-Bench++ dataset, an expansion of the original PIE-Bench dataset, to better support evaluating image-editing tasks involving multiple objects and attributes simultaneously. This dataset is a benchmark for evaluating text-driven image editing methods in multifaceted scenarios. Dataset and code are available at https://mingzhenhuang.com/projects/MultiEdits.html.

We study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a very explicit simplicial model of the injective hull of a Helly graphs, in terms of cliques which are intersections of balls. We use these subdivisions to prove that any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic. Moreover, every such hyperbolic automorphism has an axis in an appropriate Helly subdivision, and its translation length is rational with uniformly bounded denominator.

In this paper we provide an O(m (log log n)^{O(1)} log(1/ε))-expected time algorithm for solving Laplacian systems on n-node m-edge graphs, improving improving upon the previous best expected runtime of O(m log n (log log n)^{O(1)} log(1/ε)) achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of ell_p-stretch graph approximations with improved stretch and sparsity bounds. Additionally, as motivation for this work, we show that for every set of vectors in R^d (not just those induced by graphs) and all k > 1 there exist ultrasparsifiers with d-1 + O(d/k) re-weighted vectors of relative condition number at most k. For small k, this improves upon the previous best known relative condition number of O(k log d), which is only known for the graph case.

Sparse Mixture-of-Experts (MoE) layers route tokens through a handful of experts, and learning-free compression of these layers reduces inference cost without retraining. A subtle obstruction blocks every existing compressor in this family: three experts can each be pairwise compatible yet form an irreducible cycle when merged together, so any score that ranks experts on pairwise signals is structurally blind to which triples are jointly mergeable. We show the obstruction is a precise mathematical object, the harmonic kernel of the simplicial Laplacian on a 2-complex whose vertices are experts, whose edges carry KL merge barriers, and whose faces carry triplet barriers; Hodge-decomposing the edge-barrier signal isolates the kernel exactly. We turn the diagnostic into a selection objective: HodgeCover greedily covers the harmonic-critical edges and triplet-critical triangles, and a hybrid variant of HodgeCover pairs it with off-the-shelf weight pruning on survivors. On three open-weight Sparse MoE backbones under aggressive expert reduction, HodgeCover matches state-of-the-art learning-free baselines on the expert-reduction axis, leads on the aggressive-compression frontier of the hybrid axis, and uniquely balances retained mass across all four Hodge components. These results show that exposing the harmonic kernel of a learned MoE structure changes which compressor wins at the regime that matters most.

Most real-world graphs exhibit a hierarchical structure, which is often overlooked by existing graph generation methods. To address this limitation, we propose a novel graph generative network that captures the hierarchical nature of graphs and successively generates the graph sub-structures in a coarse-to-fine fashion. At each level of hierarchy, this model generates communities in parallel, followed by the prediction of cross-edges between communities using separate neural networks. This modular approach enables scalable graph generation for large and complex graphs. Moreover, we model the output distribution of edges in the hierarchical graph with a multinomial distribution and derive a recursive factorization for this distribution. This enables us to generate community graphs with integer-valued edge weights in an autoregressive manner. Empirical studies demonstrate the effectiveness and scalability of our proposed generative model, achieving state-of-the-art performance in terms of graph quality across various benchmark datasets. The code is available at https://github.com/Karami-m/HiGen_main.

In this study, we present a technique that spans multi-scale views (global scale -- meaning brain network-level and local scale -- examining each individual ROI that constitutes the network) applied to resting-state fMRI volumes. Deep learning based classification is utilized in understanding neurodegeneration. The novelty of the proposed approach lies in utilizing two extreme scales of analysis. One branch considers the entire network within graph-analysis framework. Concurrently, the second branch scrutinizes each ROI within a network independently, focusing on evolution of dynamics. For each subject, graph-based approach employs partial correlation to profile the subject in a single graph where each ROI is a node, providing insights into differences in levels of participation. In contrast, non-linear analysis employs recurrence plots to profile a subject as a multichannel 2D image, revealing distinctions in underlying dynamics. The proposed approach is employed for classification of a cohort of 50 healthy control (HC) and 50 Mild Cognitive Impairment (MCI), sourced from ADNI dataset. Results point to: (1) reduced activity in ROIs such as PCC in MCI (2) greater activity in occipital in MCI, which is not seen in HC (3) when analysed for dynamics, all ROIs in MCI show greater predictability in time-series.

Message Passing Neural Networks (MPNNs) are instances of Graph Neural Networks that leverage the graph to send messages over the edges. This inductive bias leads to a phenomenon known as over-squashing, where a node feature is insensitive to information contained at distant nodes. Despite recent methods introduced to mitigate this issue, an understanding of the causes for over-squashing and of possible solutions are lacking. In this theoretical work, we prove that: (i) Neural network width can mitigate over-squashing, but at the cost of making the whole network more sensitive; (ii) Conversely, depth cannot help mitigate over-squashing: increasing the number of layers leads to over-squashing being dominated by vanishing gradients; (iii) The graph topology plays the greatest role, since over-squashing occurs between nodes at high commute (access) time. Our analysis provides a unified framework to study different recent methods introduced to cope with over-squashing and serves as a justification for a class of methods that fall under graph rewiring.

We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of 6-dimensional (stable minimal) immersions. The key step is an ε-regularity theorem, that assumes smallness of the scale-invariant L^2 norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension ngeq 2, that may have a singular set Σ of locally finite H^{n-2}-measure, and that is weakly close to a hyperplane. (In fact, if H^{n-2}(Σ)=0, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an ε-regularity theorem, that assumes smallness of the scale-invariant L^2 tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen--Simon theorem (1981). In both ε-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method à la De Giorgi (from the linear De Giorgi--Nash--Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both ε-regularity theorems the method gives explicit constants that quantify the required smallness.)

Given a 3-dimensional (para-)CR structure, its family of chains define a 3-dimensional path geometry. We provide necessary and sufficient conditions that determine whether a path geometry in dimension three arises from chains of a CR or para-CR 3-manifold. We demonstrate how our characterization can be verified computationally for a given 3-dimensional path geometry and discuss a few examples.

Representational analysis explores how input data of a neural system are encoded in high dimensional spaces of its distributed neural activations, and how we can compare different systems, for instance, artificial neural networks and brains, on those grounds. While existing methods offer important insights, they typically do not account for local intrinsic geometrical properties within the high-dimensional representation spaces. To go beyond these limitations, we explore Ollivier-Ricci curvature and Ricci flow as tools to study the alignment of representations between humans and artificial neural systems on a geometric level. As a proof-of-principle study, we compared the representations of face stimuli between VGG-Face, a human-aligned version of VGG-Face, and corresponding human similarity judgments from a large online study. Using this discrete geometric framework, we were able to identify local structural similarities and differences by examining the distributions of node and edge curvature and higher-level properties by detecting and comparing community structure in the representational graphs.

In computer-aided design (CAD) systems, 2D line drawings are commonly used to illustrate 3D object designs. To reconstruct the 3D models depicted by a single 2D line drawing, an important key is finding the edge loops in the line drawing which correspond to the actual faces of the 3D object. In this paper, we approach the classical problem of face identification from a novel data-driven point of view. We cast it as a sequence generation problem: starting from an arbitrary edge, we adopt a variant of the popular Transformer model to predict the edges associated with the same face in a natural order. This allows us to avoid searching the space of all possible edge loops with various hand-crafted rules and heuristics as most existing methods do, deal with challenging cases such as curved surfaces and nested edge loops, and leverage additional cues such as face types. We further discuss how possibly imperfect predictions can be used for 3D object reconstruction.

Recent advances in large language model-powered multi-agent systems have demonstrated remarkable collective intelligence through effective communication. However, existing approaches face two primary challenges: (i) Ineffective group collaboration modeling, as they rely on pairwise edge representations in graph structures, limiting their ability to capture relationships among multiple agents; and (ii) Limited task-adaptiveness in communication topology design, leading to excessive communication cost for simple tasks and insufficient coordination for complex scenarios. These issues restrict the scalability and practical deployment of adaptive collaboration frameworks. To address these challenges, we propose HyperAgent, a hypergraph-based framework that optimizes communication topologies and effectively captures group collaboration patterns using direct hyperedge representations. Unlike edge-based approaches, HyperAgent uses hyperedges to link multiple agents within the same subtask and employs hypergraph convolutional layers to achieve one-step information aggregation in collaboration groups. Additionally, it incorporates a variational autoencoder framework with sparsity regularization to dynamically adjust hypergraph topologies based on task complexity. Experiments highlight the superiority of HyperAgent in both performance and efficiency. For instance, on GSM8K, HyperAgent achieves 95.07\% accuracy while reducing token consumption by 25.33\%, demonstrating the potential of hypergraph-based optimization for multi-agent communication.

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.

Combinatorial complexes have unified set-based (e.g., graphs, hypergraphs) and part-whole (e.g., simplicial, cellular complexes) structures into a common topological framework. Existing topological neural networks and Weisfeiler-Lehman variants remain fragmented, lacking a unified theoretical foundation for topological deep learning. In this work, we introduce the Combinatorial Complex Weisfeiler-Lehman (CCWL) test, an axiomatic-style extension of the WL test to combinatorial complexes. CCWL formalizes topological message passing through four types of neighborhood relation and provides a unified perspective on the expressive power of higher-order variants. We further prove that upper and lower neighborhoods are sufficient among the four adjacent WL tests to reach the expressivity of the full CCWL framework across topological structures of combinatorial complexes. Building on this framework, we also propose the Combinatorial Complex Isomorphism Network (CCIN) and evaluate it on synthetic and real-world benchmarks. Experimental results indicate CCIN outperforms baseline methods and offers a generalized expressive framework for topological deep learning.

Finite unions of convex sets are a central object of study in discrete and computational geometry. In this paper we initiate a systematic study of complements of such unions -- i.e., sets of the form S=R^d setminus (cup_{i=1}^n K_i), where K_i are convex sets. In the first part of the paper we study isolated points in S, whose number is related to the Betti numbers of cup_{i=1}^n K_i and to its non-convexity properties. We obtain upper bounds on the number of such points, which are sharp for n=3 and significantly improve previous bounds of Lawrence and Morris (2009) for all n ll 2^d{d}. In the second part of the paper we study coverings of S by well-behaved sets. We show that S can be covered by at most g(d,n) flats of different dimensions, in such a way that each x in S is covered by a flat whose dimension equals the `local dimension' of S in the neighborhood of x. Furthermore, we determine the structure of a minimum cover that satisfies this property. Then, we study quantitative aspects of this minimum cover and obtain sharp upper bounds on its size in various settings.

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions R^3, position and orientations R^3 {times} S^2, and the group SE(3) itself. Among these, R^3 {times} S^2 is an optimal choice due to the ability to represent directional information, which R^3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

The growing importance of understanding and addressing algorithmic bias in artificial intelligence (AI) has led to a surge in research on AI fairness, which often assumes that the underlying data is independent and identically distributed (IID). However, real-world data frequently exists in non-IID graph structures that capture connections among individual units. To effectively mitigate bias in AI systems, it is essential to bridge the gap between traditional fairness literature, designed for IID data, and the prevalence of non-IID graph data. This survey reviews recent advancements in fairness amidst non-IID graph data, including the newly introduced fair graph generation and the commonly studied fair graph classification. In addition, available datasets and evaluation metrics for future research are identified, the limitations of existing work are highlighted, and promising future directions are proposed.

Recent advances in learning multi-modal representation have witnessed the success in biomedical domains. While established techniques enable handling multi-modal information, the challenges are posed when extended to various clinical modalities and practical modalitymissing setting due to the inherent modality gaps. To tackle these, we propose an innovative Modality-prompted Heterogeneous Graph for Omnimodal Learning (GTP-4o), which embeds the numerous disparate clinical modalities into a unified representation, completes the deficient embedding of missing modality and reformulates the cross-modal learning with a graph-based aggregation. Specially, we establish a heterogeneous graph embedding to explicitly capture the diverse semantic properties on both the modality-specific features (nodes) and the cross-modal relations (edges). Then, we design a modality-prompted completion that enables completing the inadequate graph representation of missing modality through a graph prompting mechanism, which generates hallucination graphic topologies to steer the missing embedding towards the intact representation. Through the completed graph, we meticulously develop a knowledge-guided hierarchical cross-modal aggregation consisting of a global meta-path neighbouring to uncover the potential heterogeneous neighbors along the pathways driven by domain knowledge, and a local multi-relation aggregation module for the comprehensive cross-modal interaction across various heterogeneous relations. We assess the efficacy of our methodology on rigorous benchmarking experiments against prior state-of-the-arts. In a nutshell, GTP-4o presents an initial foray into the intriguing realm of embedding, relating and perceiving the heterogeneous patterns from various clinical modalities holistically via a graph theory. Project page: https://gtp-4-o.github.io/.