Mircea Moscu

Ricardo Borsoi

Cédric Richard

Online Kernel-Based Graph Topology Identification with Partial-Derivative-Imposed Sparsity

Keywords: topology inference, partial derivative sparsity, brain connectivity, nonlinear interactions, kernels

In many applications, such as brain network connectivity or shopping recommendations, the underlying graph explaining the different interactions between participating agents is unknown. Moreover, many of these interactions may be based on nonlinear relationships, rendering the topology inference problem more complex. This paper presents a new topology inference method that estimates a possibly directed adjacency matrix in an online manner. In contrast to previous approaches which are based on additive models, the proposed model is able to explain general nonlinear interactions between the agents. Partial-derivative-imposed sparsity is implemented, while reproducing kernels are used to model nonlinearities. The impact of the increasing number of data points is alleviated by using dictionaries of kernel functions. A comparison with a previously developed method showcases the generality of the new model.

I. INTRODUCTION

In analyzing networks such as gene regulation systems [START_REF] Richard | A second generation human haplotype map of over 3.1 million SNPs[END_REF], socio-economical interactions [START_REF] Heiberger | Predicting economic growth with stock networks[END_REF], or brain activity [START_REF] Kramer | Emergent network topology at seizure onset in humans[END_REF], graphs have proven to be a useful tool, given their inherently distributed nature. Most graph signal processing algorithms, however, assume the graph topology as known beforehand. Recently, significant interest has been dedicated to the estimation of the graph topology from available data. Most of these works assume linear dependencies between the agents, e.g., brain regions, genes in a network, or sectors of a market economy. However, the presence of nonlinear interactions in real-world applications imposes the need of developing more general algorithms. As such, the ability of reproducing kernels to model nonlinear relationships between nodal signals makes them a powerful tool to the graph inference process.

In this article, we consider a setting where online nodal measurements are acquired and used in order to infer the topology of the underlying network. In the developed approach, it is desired to estimate a possibly directed adjacency matrix while accounting for general nonlinear dependencies between nodal signals, and to distribute it over the different agents. Since many real world examples, such as social graphs, show considerable edge sparsity, a sparsity-inducing framework based on partial derivatives is employed. Definitions: A graph G consists of a set N of (N + 1) nodes, and a set E of edges such that if nodes m and n are linked, The work of C. Richard was funded in part by the ANR under grant ANR-19-CE48-0002, and by the 3IA Côte d'Azur Senior Chair program. then (m, n) ∈ E. For undirected graphs, these node pairs are unordered. At node level, we collect a real-valued signal y(i)

[y 1 (i), . . . , y N +1 (i)] , where y n (i) is the sample of the signal y(i) at node n and time instant i. The adjacency matrix A [START_REF] Biggs | Algebraic Graph Theory[END_REF], [START_REF] Sandryhaila | Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure[END_REF], is defined as an (N + 1) × (N + 1) matrix whose entries a nm are zero if (m, n) / ∈ E and set to one otherwise. Throughout the paper we consider a nn = 0, ∀n. Prior works: Early advances in topology identification are put forward in [START_REF] Dempster | Covariance selection[END_REF], where a covariance estimation based method of inferring links is introduced. On the same line, in [START_REF] Friedman | Sparse inverse covariance estimation with the graphical Lasso[END_REF] the graphical Lasso is employed in order to estimate the inverse covariance matrix from data. In [8], the authors advocate that connectivity can be recovered from estimated spectral templates, while the authors in [START_REF] Vlaski | Online graph learning from sequential data[END_REF] propose an online adaptive algorithm for learning the topology from streaming graph signals driven by a diffusion process. Another online and adaptive algorithm is developed in [START_REF] Moscu | Learning causal networks topology from streaming graph signals[END_REF], where the authors assume a linear model. Linearity of interactions and signal stationarity are assumed in [START_REF] Shafipour | Online topology inference from streaming stationary graph signals[END_REF], while developing an ADMM algorithm.

In modeling non-linear phenomena, works such as [START_REF] Harring | A comparison of methods for estimating quadratic effects in nonlinear structural equation models[END_REF], [START_REF] Finch | Modeling nonlinear structural equation models: A comparison of the two-stage generalized additive models and the finite mixture structural equation model[END_REF] focus on polynomial structural equation models, while the authors of [START_REF] Lim | Operatorvalued kernel-based vector autoregressive models for network inference[END_REF] use their nonlinear counterparts. They, however, have some limitations, such as assuming knowledge of certain connections or the form of the non-linear functions. Reproducing kernels have seen widespread use in topology inference problems. One of these works is [START_REF] Shen | Topology inference of directed graphs using nonlinear structural vector autoregressive models[END_REF] where kernels model nonlinear relationships between nodes. In [START_REF] Moscu | Online graph topology inference with kernels for brain connectivity estimation[END_REF], a kernelbased approach was considered in an online algorithm to deal with dynamical settings. The multi-kernel approach in [START_REF] Zhang | Going beyond linear dependencies to unveil connectivity of meshed grids[END_REF] uses partial correlations to encode graph topology and p -norm regression to enhance performance. In [START_REF] Giannakis | Topology Identification and Learning over Graphs: Accounting for Nonlinearities and Dynamics[END_REF], a thorough analysis of the kernel-based topology inference problem is given, focusing on capturing non-linear and dynamic links. A review of state-of-the art methods in topology inference is in [START_REF] Dong | Learning graphs from data: A signal representation perspective[END_REF]. All these works, however, share a common limitation: the interactions between the different nodes are restricted to be additive.

Encoding the graph topology with the adjacency matrix A is ubiquitous in graph signal models. This operator describes the interactions between entities and, by extension, it can be considered as a tool for representing relationships between data. In this work, we propose an online approach able to estimate an adjacency matrix based on a general nonlinear model. This model ensures a better representativity of nonlinear interactions, without assuming a particular manner on how agents in a network influence each other. Moreover, due to the online nature of the method, the data acquisition process can be stopped exactly when an estimate is obtained, a property which can prove useful in domains such as medical research. Notations: Normal font letters denote scalars, while boldface lowercase and uppercase letters stand for column vectors and matrices, respectively. Operator • 0 returns the number of their non-zero entries. Uppercase calligraphic letters denote sets, of cardinality |•|. Finally, E{•} is the expectation operator.

II. LOCAL PROBLEM FORMULATION

Consider an (N + 1)-node graph with adjacency matrix A which models a system such as the brain network or a power grid. In this setting, the electrical activity of different brain-regions [START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF], [START_REF] Shen | Nonlinear structural vector autoregressive models for inferring effective brain network connectivity[END_REF], or the voltage angle per bus [START_REF] Zhang | Going beyond linear dependencies to unveil connectivity of meshed grids[END_REF], numbered from 1 through (N + 1), can be measured at different time instants i ∈ N, leading to a dynamic graph signal y(i). The signal at each node influences and is influenced by the signals at the other nodes, with nonlinear relationships being reported in many applications such as, e.g., in the case of brain connectivity [START_REF] Freeman | EEG analysis gives model of neuronal templatematching mechanism for sensory search with olfactory bulb[END_REF], [START_REF] De Zwart | Hemodynamic nonlinearities affect BOLD fMRI response timing and amplitude[END_REF].

The distributed nature of graphs allows for a local problem formulation. As such, we focus on a single node n, while keeping in mind that the following reasoning can be applied for any other particular node. For ease of notation, we assume that n ≡ (N +1) (i.e., we identify n with the (N +1) th node of the graph), which allows us to denote N \ {n} = {1, . . . , N }. Recent methods have considered models of the form [START_REF] Moscu | Online graph topology inference with kernels for brain connectivity estimation[END_REF], [START_REF] Shen | Kernel-based structural equation models for topology identification of directed networks[END_REF]:

y n (i) = N m=1 a nm g m (y Lm (i)) + v n (i) , (1)
where v n (i) represents innovation noise, and a nm is the (n, m) th entry of the graph adjacency matrix A. This matrix models how each function g m , m = 1, . . . , N influences the signal observed at node n. Let g m : R Lm → R be a nonlinear function whose possibly vector-valued argument is y

Lm (i) = [y m (i), . . . , y m (i -L m + 1)] , for L m ≥ 1.
Therefore, the signal at each node depends non-linearly on the signals at all the other nodes, up to the past L m samples. Given nodal measurements y(i) acquired online and model ( 1), the goal is to estimate the adjacency matrix A locally at each node:

argmin an,g1,...,g N 1 2 E y n (i) - N m=1 a nm g m (y Lm (i)) 2 + φ(a n ) subject to a nm ∈ {0, 1} , (2)
where a n is the n th row of A, and function φ is a sparsity promoting regularizer. However, models such as (1) do not consider the nonlinear interactions between multiple nodes, as they assume an additive model for y n (i) [START_REF] Buja | Linear smoothers and additive models[END_REF]. To overcome this issue, we propose to consider the following general nonlinear model:

y n (i) = f n (y L1 (i), . . . , y L N (i)) + v n (i) . (3)
Model (3), compared to (1), captures much more complex relationships between the different nodes without relying on an assumed additive model, therefore rendering it more general.

A. Nonparametric sparsity

Although kernel-based and other nonlinear regression frameworks can be applied to estimate the function f n described in equation ( 3), there remains a challenge to relate f n to the underlying graph topology A. Although the lack of an additive model precludes a straightforward relationship such as in [START_REF] Richard | A second generation human haplotype map of over 3.1 million SNPs[END_REF], the influence of a certain variable can be quantified by the norm of the corresponding partial derivative, i.e.: node m does not influence n ⇐⇒ ∂f n ∂y Lm = 0 , (4)

under the assumption that f n is continuously differentiable. If we do not assume the additive model, we can generalize problem (2) as:

argmin fn 1 2 E y n (i) -f n (y L1 (i), . . . , y L N (i)) 2 subject to col ∂f n ∂y Lm N m=1 being sparse . (5)
Let us denote

y(i) = [y L1 (i) , . . . , y L N (i) ]
. As thoroughly explained in [START_REF] Rosasco | Nonparametric sparsity and regularization[END_REF], in order to define a nonparametric notion of sparsity that leads to a convex optimization problem, one can define the sparsity through the following functional:

Ω E (f n ) = N m=1 ∂f n ∂y Lm E = N m=1 E y ∂f n ( y) ∂y Lm 2 . (6)
The expectation involving the derivatives in ( 6) can be approximated by the empirical average on all the data samples available up to instant i. Employing the 2 -norm, as proposed in the aforementioned paper, and approximating the expectation, we obtain the following sampled version functional:

Ω(f n ) = N m=1 ∂f n ∂y Lm i = N m=1 1 i i p=1 ∂f n ( y(p)) ∂y Lm 2 . (7)
By including Ω(f n ) as an additive term in the cost function of [START_REF] Sandryhaila | Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure[END_REF], we are able to obtain a convex optimization problem, which allow us to obtain more efficient algorithms, and cater to real-word graphs, which tend to be sparse.

B. Sparsity in Reproducing Kernel Hilbert Spaces

The penalty term proposed in the previous section allows us to promote sparsity in the estimated topology without the restrictive constraint of an additive model. However, there remains a fundamental step to constrain f n to an adequate class of functions that is flexible but allows for an efficient, finite dimensional implementation. Several solutions exist in the literature, including non-linear and polynomial Structural Equation Models [START_REF] Jöreskog | Nonlinear structural equation models: The Kenny-Judd model with interaction effects[END_REF] and function selection from function sets defined a priori [START_REF] Song | A bayesian modeling approach for generalized semiparametric structural equation models[END_REF]. In this work we will consider kernel methods, which address the presence of nonlinearities in classification or regression problems by applying linear algorithms to a highdimensional feature space obtained by mapping the input data to a Reproducing Kernel Hilbert Space (RKHS) H κ associated with a positive definite reproducing kernel κ(•, •). RKHS-based solutions have been applied in the context of nonlinear additive models for online topology estimation in, e.g., [START_REF] Moscu | Online graph topology inference with kernels for brain connectivity estimation[END_REF].

For simplicity, consider from now on L m = 1, ∀m ∈ N , which means that y Lm (i) = y m (i). Using the sparsity penalty in [START_REF] Friedman | Sparse inverse covariance estimation with the graphical Lasso[END_REF], constraining f n to belong to an RKHS H κ and approximating the expectation in ( 5) by an empirical average leads to the following optimization problem:

argmin fn∈Hκ 1 2i i =1 y n ( ) -f n ( y( )) 2 (8) +η n   N m=1 1 i i p=1 ∂f n ( y(p)) ∂y m 2 + ψ f n Hκ   .
In (8), parameter η n > 0 controls the relative importance of respecting the constraint on f n , and ψ : R → [0, ∞[ is a monotonically increasing function. Despite allowing us to introduce sparsity in f n without constraining it to an additive model, the cost function in (8) also contains a significant challenge to an RKHS-based solution: the sparsity-promoting penalty term precludes the direct application of traditional representation theorems to obtain a finite dimensional representation. Fortunately, if we suppose that the kernel κ(•, •) is at least twice differentiable, then the following relation holds [START_REF] Zhou | Derivative reproducing properties for kernel methods in learning theory[END_REF]:

H κ ∂f n ( y) ∂y m = f n n, κ ∂m (•, y) Hκ , (9)
where

κ ∂m (•, y(q)) = ∂κ(•, s) ∂s m s= y(q) . (10)
This means that for sufficiently smooth kernels, the derivative of functions in H κ also belong to H κ , and can be evaluated in the form of simple inner products. This makes it possible to obtain a finite dimensional representation of the solution of (8), as previously demonstrated in [START_REF] Rosasco | Nonparametric sparsity and regularization[END_REF]. As such, the representer theorem holds for our approach [START_REF] Rosasco | Nonparametric sparsity and regularization[END_REF], and a solution of (8) is:

f * n = i p=1 α p κ(•, y(p)) + N m=1 i q=1 β m,q κ ∂m (•, y(q)) , (11)
which can be substituted in (8) in order to obtain a finite dimensional optimization problem.

III. AN ONLINE ALGORITHM

An immediate observation concerning solution [START_REF] Shafipour | Online topology inference from streaming stationary graph signals[END_REF] is that the number of coefficients α p and β m,q can become prohibitive as i increases, since each acquired measurement increases the number of kernel functions. A solution to this problem is the use of kernel dictionaries which can be defined a priori [START_REF] Chen | Convergence analysis of kernel LMS algorithm with pre-tuned dictionary[END_REF] or which can admit a new candidate kernel function only if the candidate function passes a certain sparsification rule based on, e.g., the coherence criterion [START_REF] Richard | Online prediction of time series data with kernels[END_REF]. Under this framework, each node n in the network creates, updates, and stores a dictionary of kernel functions and their derivatives, D n = {{κ(•, y(ω j )), κ ∂1 (•, y(ω j )), . . . , κ ∂ N (•, y(ω j ))} : ω j ∈ I i n ⊂ {1, . . . , i -1}}, where I i n represents the set of time indices of elements selected for the dictionary, before instant i. This entails the fact that, after a sufficient number of samples i has been acquired, only a number |I i n | i of coefficient couples will be needed. A candidate kernel function κ(•, y(i)) is added to D n if the following sparsification condition holds [START_REF] Richard | Online prediction of time series data with kernels[END_REF]:

max ωj ∈I i n |κ( y(i), y(ω j ))| ≤ ξ n , (12)
where ξ n ∈ [0, 1[ determines the level of sparsity and coherence of the dictionary. The number of entries in the dictionary satisfies |I i n | < ∞ when i → ∞ [START_REF] Richard | Online prediction of time series data with kernels[END_REF]. We rewrite [START_REF] Shafipour | Online topology inference from streaming stationary graph signals[END_REF] as:

f * n = |I i n | p=1 α p κ(•, y(ω p )) + N m=1 |I i n | q=1 β m,q κ ∂m (•, y(ω q )) . ( 13
) Let vectors α = col{{α p } |I i n | p=1 }, β = col{{β m } N m=1 }, with β m = col{{β m,q } |I i n |
q=1 }, group the coefficients in [START_REF] Finch | Modeling nonlinear structural equation models: A comparison of the two-stage generalized additive models and the finite mixture structural equation model[END_REF]. Considering the online version of the batch cost function (8) with the instantaneous MSE estimate (measured only at instant i), setting ψ = 0, and using the dictionary-based representation of f * n in ( 13), we obtain the following finite-dimensional optimization problem:

argmin γ 1 2 y n (i) -γ s(i) 2 ( 14
)
+ η n N m=1 1 i i p=1 γ t m (p) 2 , with s(i) = z(i) k(i) , γ = β α , t m (p) = m (p) z m (p)
, and:

k(i) = col κ( y(i), y(ω q )) |I i n | q=1 , (15)
z(i) = z 1 (i), . . . , z N (i) , (16)
[z m (i)] q = ∂κ( y(i), y(ω q ))

∂y m (ω q ) q=1,...,|I i n | , m (i) = 1,m (i), . . . , N,m (i) , (17)
[ m1,m2 (i)] q = ∂ 2 κ( y(i), y(ω q )) ∂y m1 (ω q )∂y m2 (ω q ) q=1,...,|I i n |
.

The quantities ( 15), [START_REF] Moscu | Online graph topology inference with kernels for brain connectivity estimation[END_REF], and ( 17) can be computed in closed form when an explicit expression of the continuously differentiable kernel κ(•, •) is chosen.

One difficulty with problem ( 14) is that the summation of t m (p), p = 1, . . . , i in the regularization term grows linearly with i, and is thus not scalable in this form. Note that we can write the finite-dimensional regularizer in [START_REF] Lim | Operatorvalued kernel-based vector autoregressive models for network inference[END_REF] as:

Ω(f n ) = N m=1 1 i i p=1 γ t m (p) 2 (18) = N m=1 γ 1 i i p=1 T m (p) γ = N m=1 γ T m (i)γ , with T m (p) = t m (p)t m (p) and T m (i) = 1 i i p=1 T m (p), for
T m (i) = 1 i T m (i) + i -1 i T m (i -1) , ∀i ≥ 2 , (19)
we can compute T m (i) recursively for all i with a fixed complexity. Optimization problem ( 14) now becomes:

argmin γ 1 2 y n (i) -γ s(i) 2 + η n N m=1 γ T m (i)γ . ( 20
)
Cost function [START_REF] Rubinov | Complex network measures of brain connectivity: uses and interpretations[END_REF] can then be optimized iteratively using the subgradient descent algorithm as:

γ(i + 1) = γ(i) + µ n s(i) y n (i) -s (i)γ(i) (21)
-

µ n η n N m=1 T m (i)γ(i) Λm (i) , with Λm (i) = γ (i)T m (i)γ(i).
In ( 20) and ( 21), each Λ m represents the estimate of the partial derivative of f n with respect to y m . The proposed method, summarized in Algorithm 1, has a per-iteration complexity of O(N 2 ). Approximate strategies can be considered to obtain a scalable implementation. Parameter τ n acts as an edge identification threshold. It is used to identify the topology from the estimated coefficients Λm (i), determining whether there exist links from each node m ∈ N \ {n} towards n. When processing real data, τ n can be set as to obtain an estimated topology which realistically explains the studied process.

Algorithm 1: Kernel-based online graph inference

Inputs: For every node n: µn, ηn, κ(•, •), ξn, and τn Initialization: Set all entries of γ(0) to 0 Algorithm: At each time instant i ≥ 1 Update Dn if κ(•, y(i)) satisfies condition [START_REF] Harring | A comparison of methods for estimating quadratic effects in nonlinear structural equation models[END_REF] Compute s(i) and Tm(i) with ( 15), ( 16), ( 17), [START_REF] Dong | Learning graphs from data: A signal representation perspective[END_REF] Update γ(i) using [START_REF] Shen | Nonlinear structural vector autoregressive models for inferring effective brain network connectivity[END_REF] Set ânm(i) to 1 if Λm(i) ≥ τn, to 0 otherwise

IV. EXPERIMENTAL RESULTS

In this section, the performance of the proposed method is evaluated by two experiments: one considering synthetic data, and another considering real epilepsy data. The Gaussian kernel κ(a, b) = exp -ab 2 /2σ 2 n was used in both cases. Lorenz graph: Consider the first experiment in [START_REF] Moscu | Online graph topology inference with kernels for brain connectivity estimation[END_REF] (hereafter called RA for reference algorithm), based on the discretized version of the Lorenz attractor [START_REF] Lorenz | Computational chaos-a prelude to computational instability[END_REF]. The system parameters were set to the same values as in [START_REF] Tobar | Multikernel least mean square algorithm[END_REF], with initial conditions y n (0) = 10 -10 , ∀n, for both RA and the proposed algorithm (PA). This system contains nonlinear interactions which cannot be completely characterized by additive models. We set n and τ n as to achieve the fastest convergence. Parameters σ n and ξ n were set as to obtain the same number of dictionary entries per node, nine, in order to achieve a meaningful comparison. Finally, τ n were set as to achieve the best performance in terms of edge identification error rate, EIER A gt -Â 0 /(N (N -1)) • 100%, where A gt is the ground truth depicted in Fig. 1(b). Fig. 1(a) depicts the EIER of both algorithms as a function of the iterations. In Fig. 1(c) and(d), each entry represents the mean of the normalized Λm (i), per n, over i = 5000 iterations, which encodes the strength of a link from node m towards n before thresholding. It is desired that the amplitude of the elements for which a nm = 1 be larger and as separated as possible from the amplitude of the elements for which a nm = 0, since this makes distinguishing the active and inactive links easier. On the first row, the more general model of the current method is able to better differentiate between the absence and presence of a link. This is seen through the larger difference between Λ2 and Λ3 corresponding to â12 = 1 and â13 = 0, respectively, in the case of PA, while for the RA the strength of the link corresponding to â13 is much closer to that of the active link. The lower performance of RA was expected since it constrains the interactions to obey an additive model. Real data: The data for this experiment come from a 39-yearold female subject suffering from intractable epilepsy. The data acquisition and pre-processing information is provided in [START_REF] Kramer | Emergent network topology at seizure onset in humans[END_REF]. The data-set contains 8 instances of electrocorticography (ECoG) time series, each instance representing one seizure and contains voltage measurements from 76 different regions on and inside the brain, during the 10 seconds before the epilepsy seizure (preictal interval) and the first 10 seconds during the seizure (ictal interval). We set µ n = 10 -3 , η n = 10 2 , ξ n = 0.8 for each n. Since there was no ground truth, parameters σ n and τ n were set as to obtain coherent results.

Fig. 2(a) and (b) show the estimated connectivity of the brain, for each interval, averaged over the 8 instances. Fig. 2(c) depicts the degree (sum of in-and out-degree), encoded in the radii of the circles, relative to each interval. Interestingly, our online estimate reveals roughly the same behavior before and during the seizure as the estimate obtained using the method batch developed in [START_REF] Shen | Nonlinear structural vector autoregressive models for inferring effective brain network connectivity[END_REF] and the online method in [START_REF] Moscu | Online graph topology inference with kernels for brain connectivity estimation[END_REF]. More precisely, the number of total connections decreases from one interval to the other, especially due to the variation of indegrees for nodes 30 to 50. Further analyzing the connections, nodes 75 and 76 have a small in-degree, however they present a more important out-degree. Observe the decrease of the degree of node 26 or the major increase for node 73. This behavior is consistent with the findings of the aforementioned papers.

The algorithm is therefore able to obtain results similar to those obtained in previous works, based on a general data model. For reference, the number of kernel functions inserted in the dictionaries was at most 9, after 4000 samples. These results show how a small number of kernel functions are actually needed in order to obtain a satisfactory topology estimate. This fact, alongside the online approach and general model, can translate in reduced computational complexity due to the drastically reduced number of needed kernel functions.

V. CONCLUSION

In this paper, a new kernel-based online topology estimation method was proposed accounting for general nonlinear interactions between the agents in a network. Previous work only considered models based on additive interactions between the signals at the different nodes. However, such a simplifying model is not entirely justified in many practical applications. Following a more general approach, we consider arbitrary interactions between the nodes, which render our model much more general. By encoding links as partial derivatives of the nonlinear functions, we are able to benefit from the kernel machinery framework to estimate a possibly directed, sparse adjacency matrix. An online algorithm is proposed, using kernel dictionaries and recursive computations of the regularization terms to operate with bounded complexity. Preliminary results indicate the proposed method can lead to more accurate estimates for more general nonlinear systems.

Fig. 1 .

 1 Fig. 1. EIER, ground truth and estimates. White represents 0. m = 1, . . . , N . Since the following relation is satisfied:

  Ictal (c) Left: Preictal. Right: Ictal.

Fig. 2 .

 2 Fig. 2. (a) and (b): Estimated adjacency matrices. (c): Summed in-and out-degrees for the estimated graphs. The larger the radius corresponding to node n, the larger the summed degree of node n.


Ruppert Vimal -

The impact of the Covid-19 lockdown on the human experience of nature

Keywords:

The Covid-19 pandemic has resulted in extensive lockdowns implemented all around the world and billion of people have been asked to stay at home for several weeks. Although this global confinement has had potentially huge unintended consequences on the environment and on its associated wildlife, this study shows that it has also impacted the human experience of nature. Based on an online questionnaire, this study aims to assess how the significant changes in people's everyday lives induced by the French lockdown impacted their relationship with other species. Participants did not only observe and interact more with non human species, but also discovered new traits characterizing them, and felt less lonely thanks to them. The impact of the lockdown was stronger on people's relationship with their pets, farm animals, home plants and with birds than with other plants and animals. This study further demonstrates that participants with different profiles have been affected differently. In particular, women and people with better access to nature were clearly more sensitive to changes and have been more positively impacted in their relationships with other species. Acting as a real world experiment, the lockdown reveals to which extent our experience of nature is embedded in social, cultural and political contexts.

INTRODUCTION

In December 2019, the Coronavirus SARS-CoV-2 appeared in China and spread worldwide before to being declared on March 2019 by the World Health Organization as a pandemic. In order to face the Covid-19 outbreak, governments around the world imposed strong restrictions to their populations such as keeping physical distance, stopping non-essential activities and limiting the movements of people. This global "lockdown" resulted in almost two thirds of the world population being asked to stay at home and placed under a confinement by April 2020. Since then, in many parts of the world, such drastic measures have also been reactivated to face the second and third waves of the pandemic.

Besides investigating the outbreak dynamics, researchers around the world studied the diverse unintended consequences of the lockdown such as the impact on economy and finances (e.g Huo & Qiu 2020; Kanu 2020), mental health and psychology (e.g Guessoum et al. 2020;Singh et al. 2020), education (e.g Mishra et al. 2020;Alvi & Gupta 2020) or social relationships (e.g Dewitte et al. 2020;Sommerlad et al. 2020).

As suggested by Bates et al. (2020), the global confinement also represented a unique opportunity, a world scale lab-in-the-field experiment, for environmental researches. Scholars have documented the consequences of the lockdown on carbon emission (Evangeliou et al. 2020;Han et al. 2021), air and water quality (Menut et al. 2020;Dutta et al. 2020;Chen et al. 2020), noise disturbances (Aletta et al. 2020;Basu et al. 2020) or light pollution (Bustamante-Calabria et al. 2020). They also assessed how the Covid-19 crisis impacted biodiversity. In this perspective, their main interest was in understanding of how non human species can adapt to rapid transformations of their environment and whether or not nature has recovered from this unprecedented reduction of human activities. As for examples, Derryberry et al. (2020) showed that the reduction in traffic noise led to a shift in song frequencies in a bird species. Gordo et al. (2020) revealed that although birds detectability increased in Spanish urban areas, the probability of their occurrence remained stable.

Overall, with the emergence of the Covid-19 zoonosis, the relationship between human and wildlife has been placed under a global spotlight (Gaynor et al. 2020). The pandemic reveals the complex and interacting pathways in which animal populations and habitats are intrinsically connected to human health and livelihoods. Little has been said, however, regarding the impact of the lockdown on people's relationship with nature, especially in terms of their daily interactions with the environment and living creatures. How can this "anthropause", as a dramatic slow down in human activities (Stokstad 2020), affect human experience of nature?

It is acknowledged that people in western societies have fewer and fewer direct contacts with natural environments and their associated wildlife in their everyday lives (Turner et al. 2004;Soga & Gaston 2016). For instance, it has been shown that in Japan, older people had more frequent experiences during their childhood with flowering plants and with more diverse species than younger ones (Soga et al. 2018). Similarly, the proportion of US children participating in outdoor activities as well as the time spent outdoors dramatically decreased over the last decades (Hofferth 2009). Some authors recently claimed that the ecological crisis could be first of all related to a loss of sensitivity of modern societies toward the living (Morizot 2020). Indeed, it is recognized that this "extinction of experience" of nature (Pyle 2011) can have deleterious consequences not only on human well-being but also on people's emotional, attitudinal, and behavioral relations to nature and biodiversity (Turner et al. 2004;Chan et al. 2016;Prévot et al. 2018).

In that perspective, the Covid-19 crisis and its associated lockdowns provide an opportunity to understand how a major change in society can impact human experience of nature. Only a few studies have documented the effect of the lockdown on people interactions with their environments. Part of them have been dedicated to demonstrate the importance of greenspace access for people living in urban areas (Biswas & Sen 2020;Kleinschroth & Kowarik 2020;Derks et al. 2020;Venter et al. 2020;Ugolini et al. 2020). Other studies have shown how the lockdown impacted birders behaviors and activities (Randler et al. 2020), explored how digital instruments and virtual portals helped people to connect to nature and reduce anxiety [START_REF] Zabini | Comparative Study of the Restorative Effects of Forest and Urban Videos during COVID-19 Lockdown: Intrinsic and Benchmark Values[END_REF]Jarratt 2020) or analyzed the impact of animal ownership on mental health and loneliness (Bowen et al. 2020;Ratschen et al. 2020).

This study aims at understanding how the significant changes of people's lifestyle generated by the french confinement impacted their experience of nature in terms of their relationship with other beings.

Based on an online questionnaire, I first assessed the impact of the Covid-19 lockdown on the way people observed, interacted, learned from different species and how the latter helped them to feel less lonely. I then evaluated whether people of different profiles were impacted differently.

METHODS

Context of the study

From March 17 th 2020 to May 11 th 2020, the French authorities established a state-wide lockdown stopping all « non-essential activities ». Apart from food retail sector and healthcare institutions, most of the other activities were brought to a halt. People were asked to stay home as much as possible and were allowed out only to meet their basic needs. A one hour per day permission was also granted to exercise, walk pets, get some fresh air within 1km maximum of one's place of residence.

Questionnaire implementation

Using the Limesurvey software (Limesurvey GmbH), I proposed an online questionnaire running from the April 8 th 2020 to May 11 th 2020 (end of the first French lockdown). The questionnaire was written in French and was available to anyone over 17 years old and living in France (mainland). The study was fully independent and has not been commissioned by any public or private organisms.

The questionnaire was composed of a short preamble describing the aim of the study, some explanations on how to answer it, and 24 questions. It has been released via social networks, emails, and local media. The estimated time of answer was about 15 minutes.

The first part of the questionnaire (9 questions) focused on the participants' profiles. I asked people for their gender, their age (between 18 and 40 years of age, between 40 and 60, above 60), their education (higher than high-school degree or not), their geographic location (city, outskirts, countryside), their access to nature (no access, access to parks, access to natural areas), whether they had a garden or not, whether they were students/workers or not (unemployed, retired), whether they were confined alone or not and since when (number of weeks) and whether they were familiar, through education, profession or pleasure, with biodiversity study, management, or protection (note however, that, due to a mistake in implementing the questionnaire online, only 575 participants answered this last question).

The second part of the questionnaire (15 questions) was dedicated to the effect of the lockdown on the respondents' relationships with the species of 5 different groups: pets and farm animals of the respondents ("Own animals"), indoor and outdoor plants and trees of the respondents ("Home plants"), birds ("Birds"), plants and trees outside the respondents home ("Other plants"), animals which are not considered as pets, farm animals or birds ("Other animals").

I assessed the impact of the lockdown on the respondent observation/interaction, learning, and loneliness following three questions respectively: i) since the beginning of the lockdown, have you been spending more time observing or interacting with [species group]? If yes, please specify where and how. ii) since the beginning of the lockdown, have you discovered new behavioral, biological or morphological traits from [species group]? If yes, please specify which ones. iii) since the beginning of the lockdown, have you felt less lonely thanks to [species group]? For the first two questions, people had four choices: "No, less than before", "No", "I don't know", "Yes". For the third question, people had three choices: "No", "I don't know", "Yes".

Participant profiles

I obtained a total of 1292 complete answers (see appendix 1). Most respondents were women (70%), did not live alone (83,2%) and had at least a high-school degree (81%). Forty one percent lived in the countryside, 39% said they had a direct access to nature when going out, and 70% had a garden. When they answered the questionnaire, more than half of the respondents had been confined for 4 to 5 weeks whereas 11% had been confined for 1 to 3 weeks and 30% for 6 to 8 weeks. Fifty-four percent were aged 18 to 40, 32% 40 to 60 and 14% above 60. Finally, 32% declared they were not familiar with biodiversity study, management, or protection (estimation based on the 575 respondents who had access to this question).

Analyses

All the analysis have been conducted for each group of species and function of each of the considered impacts (observation/interaction, learning and loneliness). For the sake of simplicity, answers "I don't know" and "No, less than before" have been considered as "Not Available" and "No" respectively. Indeed, only a small number of both types of answers were found (see appendix 1).

I first analyzed the proportion of the participants who declared they were positively impacted by the lockdown as far as their relationship with other species were concerned (Figure 1). I also used the qualitative information given by the participants ("if yes, please specify where and how", "if yes, please specify which ones") to provide a complementary assessment of the lockdown impact.

I then analyzed the effect of the participants profiles (variables described in the previous section) concerning the lockdown impact on their relationship with other species. For this part of the study, I discarded the answers of people whose gender and nature access was "other" (n=6 and n=93 respectively). For each species group and each considered impact (i.e. observation/interaction, learning, loneliness), I ran generalized linear models (GLM) in order to test for the effect of each variable, holding the effects of the other variables constant. In all 15 models, as the categories of answers (i.e. the dependent variables) were binary ("Yes" or "No"), I fitted the models using a binomial error structure and logit link function (McCullagh & Nelder 1989). The models were fitted in R (version 3.6.3; R Core Team 2020) using the "glm" function.

For each model, the selection of the used variables was made according to a step by step descending process. In such an approach, the performance of the model is iteratively increased by removing at each step the least important variable. The selected model is the one for which the suppression of a new variable does not increase its performance. The performance was assessed using the Akaike Information Criterion (AIC). Such selection was made using the "step" function in R.

The detailed results of the different models are presented in appendix 3. For the sake of simplicity, within the main manuscript (Table 2), I present only the significance of each variable (p>0,001) for each of the considered models.

RESULTS

The impact of the lockdown

Overall, the lockdown had a positive effect on people relationships with the other species (Fig. 1). The most important effect was on the respondents observation and interaction with other species. More than 65% declared they observed and interacted more with their animals and plants as well as with birds and more than 40% with other animals and plants. Fewer declared that they learned something new (from 20% and above for home plants, own animals and birds to 15% and less for other plants and animals).

While 30 to 40 % of the participants declared that other animals and plants, birds and home plants helped them to feel less lonely, the effect was more significant concerning their own animals. Indeed, 60% of the respondents having animals felt less lonely thanks to them.

Of course such results should be used with caution since the pool of participants is biased toward women, wealthy and educated people having a garden and familiar with biodiversity. Nonetheless, a detailed analysis of the results (Appendix 1) reveals that positive impacts of the lockdown, although with various intensity, were found for each profile.People declared they spent much more time taking care of their animals and plants. They groomed their pets more, played more with their dogs, took time to prune their trees, sow seeds, and water the garden. The participants mentioned that they got to know their animals better : what do cats eat, when do dogs sleep during the day and for how long, how do hamsters react to a new environment, how to recognize different types of hens. Some of them also explained the impact of the lockdown on their pet behavior. Lot of people declared they observed their plants and trees more. Thanks to the spring season, they mentioned for instance the growing of the leaves and the appearance of the first flowers. Through their observation, they learn new things about them: "... it's the first time I have the opportunity to observe the seedlings grow and to realize that there are similarities and small differences between tomatoes and egg plants" (Answer n°786). Overall, the respondents do not only say they interact and observe more but also that they do it better and with much more precision: "I observe the slightest changes there can be, several times a day. I water them regularly, I realize their speed of growth, their needs…"(Answer n°639).

People spend also more time observing and listening to birds, from their balcony, in their garden or outside during their walk. Several people mentioned that they tried to communicate with them by whistling: "Once or twice, when walking in the street, I answered to a bird. I think it was because there were fewer people there and no cars" (Answer n°120). "I'm whistling and sometimes I got the impression that they are answering" (Answer n°357). Participants also said they took more care of birds by for instance protecting them from their cats, giving them food, water or building them a nest or a feeder. People declaring that they learned new things about birds mentioned their feeding habits, the variety of singing, behavioral differences and interactions between bird species. Interestingly, people often mentioned how they observe, interact with and get to know specific, well identified individuals: "One bird specifically comes back every day and several times a day to stand on close by the house.

When I'm reading, I often stop to observe and listen to him/her" (Answer n°830). "There is one pair of birds of prey that fly always at the same time above the house and one pair of red-tailed birds that have nested between the roof and the chimney" (Answer n°825).

People observe more the plants and trees in the greenspaces around their home or in their neighboring gardens. Several respondents declared that they sketched them or took pictures of them and that they learned more about their phenology: plant growth speed, leaf color changes and flowering. Although some of the participants regretted the fact that they have only poor access to plants and trees within their surrounding, others mentioned it as a reminder to pay more attention, to be more mindful: "there is only a little greenery around my place. I walk there and understand how lucky I am as well as the scarcity of these spaces. I take full advantage of it, with "gratitude""(Answer n°819). They declared how access to such greenspaces and interaction with plants and trees comforts them: "On a walk, I touch the trees, hug them and sit against them. It regenerates me. I felt a strong lack of nature, especially at the beginning of the confinement. My walks were life-saving" (Answer n°1062).

People who declared observing and interacting more with other animals mentioned species such as snakes, different insects, rats, lizards, or squirrels. Observations focus mainly on behavior: I've spent two weeks observing a bee in the holes of the veranda door. I really spent several hours observing it. I think I would have never taken this time before" (Answer n°120). Although several people mentioned they should leave such animals alone ("I don't want to encroach on their territory and their lives" (Answer n°975)), others mentioned how they took care of them by protecting one animal from the other, by gently releasing insects which were trapped indoors. As for plants and trees, several participants also declared they did not spend more time interacting with or observing such animals because they have poor access to nature.

The impact of the participant profiles

My results reveal that the profile of the participants can play a part on how the lockdown influenced their relationships with other species (Table 2). The gender effect is clearly the most important one. For all species groups, the lockdown had more impact on women than on men regarding their observation/interaction with non-humans as well as how the latter helped them fight loneliness. Women also declared more than men that they learned new things from plants and from their animals.

Additionally, the degree of access to nature has also a strong positive effect on the respondents observation/interaction with other animals and other plants, on their leaning from animals as well on the way home plants, birds, other plants and animals helped participants feel less lonely. People who have their own garden are also more positively impacted concerning their observation/interaction with their home plants and birds as well as regarding their learning from birds and other animals. They also declared fewer times than others that they were positively impacted regarding how their animals helped them feel less lonely. People living in the countryside have been less impacted than others in their observation/interaction with their animals. While they were also less impacted regarding what they learn from the latter, they declared more than others that they learned from plants. Participants with experience in biodiversity conservation, study or management overall have been more impacted by the lockdown in terms of their observation and interaction with other species except concerning their animals. Interestingly, they learned more than others from birds and said more often than others that they felt less lonely thanks to them. Educated people as well have been more impacted regarding their observation/interaction with their plants, birds and other plants but declared less than others that their animals and the other animals helped them feel less lonely. Being active has only a positive effect on the respondents observation and interaction with animals. Older people have been overall less impacted than others regarding their observation/interaction with other species and declared less than others that they learned something from plants or that they felt less lonely thanks to their animals. People living alone have been less impacted in their observation/interaction with birds and other plants and their learning from birds. Their animals and their plants helped them more to fight loneliness. Finally, people being confined for a longer period declared more than others that the lockdown had an effect on what they learned from their animals and from other plants. Compared to other participants, the other plants helped them more overcome loneliness while their animals helped them less.

DISCUSSION

This study shows that the Covid-19 lockdown affected the human relationship with other species. The questionnaire answers provide diverse stories about how people took more care of their pets, observed the daily growth of plants, fed wild animals, rescued insects, got to know specific birds individually, etc. People did not only declare that they observed and interacted more with non human species but also that they learned new things about them and that they felt less lonely thanks to them.

If the pandemic has been considered as an opportunity to take a step back, to understand the "grave consequences of causing an imbalance in natural processes shaped over millennia" (Bang & Khadakkar 2020), this study demonstrates that the zoonosis was first of all an opportunity in itself to concretely reconnect with nature. Whereas several studies have shown how biodiversity clawed back spaces in human-dominated landscape (e.g Pearson et al. 2020;Gordo et al. 2020;Silva-Rodríguez et al. 2020), my work rather shows how the lockdown modified people attention towards other beings. Non-humans species did not only recover spatially but as well in the everyday life of humans and this can also lead to a positive impact on biodiversity. Indeed, it is recognized that an increased experience of nature is likely to improve people commitment to protect wildlife (Chan et al. 2016;Prévot et al. 2018;Cazalis & Prévot 2019).

This study demonstrates how new experiences can emerge from societal changes. Due to the lockdown, movements and social interactions have been drastically reduced whereas time spent at home has consistently increased. This transformation of the everyday life of people induced significant changes in their relationship with other species. Nonetheless, this study further demonstrates that such major changes do not equally impact people. Indeed, participants with different profiles have been affected differently regarding their relationship with other beings during the lockdown.

In particular, there was a clear insight that women have been more positively impacted than men in terms of their observation, interaction, learning regarding different species as well as how the latter influences their loneliness. Although it is known that women have a different sensitivity for nature (Merchant 1996), this study contributes to show that they are also more sensitive to changes. The environment in which people live also appears to be determinant regarding how a major change can impact their relationships towards nature. Although living in a city versus living in the countryside does not seem to affect that much the respondents' sensitivity, people having a garden, access to a park or living close to natural environments were more likely to change their relationship with other species during the lockdown.

Other characteristics of the participants had an impact on their answers. For instance, older people were overall less affected than others. Educated people seemed to be more sensitive in their observation/interaction with plants and birds but less than others regarding how animals helped them to feel less lonely. Participants being already familiar with biodiversity are more likely to be impacted and have stronger sensitivity to birds. People living alone have been more affected regarding how their plants and animals helped them fight against loneliness, but have been less sensitive in terms of observation, interaction and learning from other species. People being confined for longer periods of time seem to have developed a higher sensitivity to a priori less accessible species namely the plants.

Such observations generate a number of hypotheses and further studies would be valuable to fully address the involved mechanisms. For instance, it would be interesting to understand whether educated people are more rational toward nature whereas less educated people would be more emotional and how being in a group and living collectively could facilitate the observation, interaction and learning from wild species.

Of course, it is very unlikely that a 2 month confinement or even a 1 or 2 year pandemic will transform human relation to other beings in the long term. But, while some authors claimed that the "concerted and urgent global response to Covid-19 should pave the way for similar responses to global ecological crises (Bang & Khadakkar 2020)", the main interest of my study is to demonstrate how such responses can also transform people experience of nature. Acting as a real world experiment, the lockdown does not only reveal how a major change in society can trigger this experience but also who may be more impacted. Experiences of nature are complex and diverse and are embedded in social, cultural and political contexts (Clayton et al. 2017). One of the main conclusions of this study is that people are not equal regarding changes such as the ones induced by the lockdown. Although it is globally accepted that mankind needs to reconnect with nature and to develop a new sensitivity towards other beings, this study contributes to demonstrate the importance of considering such issues in light of the social and political organizations of human societies. In that respect, I identified key profiles that should be considered when facing deep and global societal changes.

Table 1. The effects of the participants profiles on their answers. The results report the positive (black) or negative (grey) significant effects (p>0.001) of the different variables on the three categories of answers (observing/interacting, learning, feeling less lonely) for each of the considered species groups. Results are extracted from the logistic binomial regressions which detailed results are presented in Appendix 2.

Yamamoto T, Uchiumi C, Suzuki N, Yoshimoto J, Murillo-Rodriguez E. 2020. The psychological impact of "mild lockdown" in Japan during the COVID-19 pandemic: a nationwide survey under a declared state of emergency. medRxiv:2020.07.17.20156125. Cold Spring Harbor Laboratory Press. Zabini F et al. 2020. Comparative Study of the Restorative Effects of Forest and Urban Videos during COVID-19 Lockdown: Intrinsic and Benchmark Values. International Journal of Environmental Research and Public Health 17:8011. Multidisciplinary Digital Publishing Institute.

FIGURES AND TABLES