Antoine Huchet
email: antoine.huchet@univ-lr.fr

Jean-Loup Guillaume
email: jean-loup.guillaume.yacine.ghamri@univ-lr.fr

Yacine Ghamri-Doudane

On filtering the noise in consensual communities ⋆

Keywords: Consensual Community Detection, Noise, Complex Networks

Community detection is a tool to understand how networks are organised. Ranging from social, technological, information or biological networks, many real-world networks exhibit a community structure. Consensual community detection fixes some of the issues of classical community detection like non-determinism. This is often done through what is called a consensus matrix. We show that this consensus matrix is not filled with relevant information only, it is noisy. We then show how to filter out some of the noise and how it could benefit existing algorithms.

Introduction

Biological, social, technological and information networks can be represented by graphs [START_REF] Fortunato | Community detection in graphs[END_REF]. Thus, graph analysis has become an important tool to understand such networks. Graphs representing real-world data exhibit particular features that make them far from regular. Few nodes tend to have a lot of neighbours while many nodes tend to have few neighbours. The distribution of edges is not homogeneous: parts of the graph are densely connected, while between such dense parts, there tend to be only a few edges. Such feature of real-world graphs is called community structure [START_REF] Girvan | Community structure in social and biological networks[END_REF] and finding such densely connected parts of a graph is called community detection. In a network of purchase relationship between customers and products, community detection can identify communities of customers with similar interests, thus improve recommendation systems. In social graphs, it could help identify group of people: families, friends or co-workers.

Many community detection algorithms exists like Walktrap [START_REF] Pons | Computing communities in large networks using random walks[END_REF], Infomap [START_REF] Rosvall | The map equation[END_REF] or Louvain [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF]. Some algorithms are non-deterministic like Louvain, in which the communities produced are determined by the order in which the nodes are visited. Since the nodes may be visited in any order, such algorithm may produce different partitions of communities. Therefore, we need a way to pick a "good" partition. In order to do so, we would need a criterion to sort the partitions and pick the one that is the most representative of the actual community structure of the network.

In the absence of such criterion, we combine the information of different partitions of communities into consensual communities [START_REF] Lancichinetti | Consensus clustering in complex networks[END_REF] (also known in the literature as community cores or constant communities). Such communities allow working with dynamic graphs as it becomes easier to follow communities in a timestamped network when the computation is deterministic [START_REF] Seifi | Community cores in evolving networks[END_REF].

Our contribution is twofold: we show that the information from the different partitions of communities is noisy. Then when combining the partitions into consensual communities, we show that some of the noise can be avoided.

This article is organised as follows: Section 2 formally defines some concepts needed in Section 3, which gives an overview of related works. Then, Section 4 and 5 presents our contribution, followed by Section 6 in which our contribution is used to improve the quality of some state of the art algorithms. The last section concludes the article.

Definitions

A graph G = (V, E) is made of a set of nodes V and a set of edges E ∈ V × V , where |V | = n and |E| = m. Within a graph, communities are defined as a partition of the nodes. That is, each node belongs to exactly one community.

To evaluate the quality of a set of communities, the Modularity [START_REF] Newman | Modularity and community structure in networks[END_REF] compares the number of intra-community edges to the expected number of intra-community edges in a random graph with the same number of nodes, same number of edges and same degree distribution as the starting graph. It is defined as

Q = 1 2m i,j A ij - k i k j 2m δ(c i , c j ),
where i and j are nodes of the graph, and A ij is 1 if there is an edge between i and j, 0 otherwise. m is the number of edges of the graph, k i is the degree of i, and δ(c i , c j ) is 1 if nodes i and j are in the same community, 0 otherwise.

If we have ground-truth communities, we can directly compare a partition given by any community detection algorithm to this ground-truth. In this case we can use the Normalised Mutual Information (NMI ) [START_REF] Danon | Comparing community structure identification[END_REF]. It is based on a confusion matrix N where N ij is the number of nodes of the ground-truth community i that also belong to the computed community j. It if formally defined as

I(A, B) = -2 c A i=1 c B j=1 N ij log Nij N Ni.N.j c A i=1 N i. log Ni. N + c B j=1 N .j log N.j N
, where c A (resp. c B ) is the number of ground-truth (resp. computed) communities. The sum over row i of matrix N is denoted N i. and the sum over column j is denoted N .j . The NMI takes values from 0 (independent partitions) to 1 (same partitions).

Finally it is also possible to use synthetic graphs generated using the Lancichinetti, Fortunato and Radicchi (LFR) model [START_REF] Lancichinetti | Benchmark graphs for testing community detection algorithms[END_REF]. This model creates synthetic graphs with a known community structure. They are generated with a mixing parameter µ that ranges from 0 to 1, and denotes the fraction of edges that a node shares with other communities of the graph. Thus, the smaller µ, the stronger the communities are and the easier it will be to detect them.

Related Work

Most community detection algorithms are non-deterministic. Thus, running any such algorithm A multiple times on the same graph G may result in different partitions. Moreover, these partitions cannot be discriminated on the basis of their modularity alone, as it has been shown that there are significantly different partitions with similar modularity [START_REF] Good | Performance of modularity maximization in practical contexts[END_REF]. To address this issue, a solution is to search similarities between different partitions to obtain consensual communities.

These similarities has also been studied for random graphs. It has been shown that random graphs contain good communities (or at least partitions of high modularity) which is very unnatural. However, different partitions are very dissimilar from each other. The absence of similarities might therefore indicate that the communities are not meaningful [START_REF] Campigotto | The power of consensus: Random graphs have no communities[END_REF]. Peixoto et al. showed that when there is too much diversity in several partitions, it is in general not possible to obtain a consistent answer. Therefore, consensual communities are expected to yield good results only when there is a community structure in the graph.

The consensus matrix

To record similarities between partitions, most algorithms use, in one way or another, on a consensus matrix P . First, a (classical) community detection algorithm A is ran n p times on a graph G. Since A is non-deterministic, the n p results will most likely be different. Then, the consensus matrix P is built as follows: the consensus coefficient P ij is the number of times that nodes i and j were placed in the same community by A during the n p executions. P ij is thus between 0 and n p . Then, the weighted consensus graph G P whose adjacency matrix is P is built: nodes i and j are linked in G P by an edge whose weight is P ij . If P ij = 0, then nodes i and j are not connected [START_REF] Lancichinetti | Consensus clustering in complex networks[END_REF][START_REF] Seifi | Community cores in evolving networks[END_REF].

Rather than using A multiple times, some authors obtain diversity by using different algorithms to generate the first n p . The Azar method [START_REF] Kheirkhahzadeh | A consensus clustering method for clustering social networks[END_REF] picks n p different algorithms and each of them is run only once on G. Then the information from the n p partitions is aggregated in a consensus matrix as described before, from which they build the associated consensus graph.

Liu et al. [START_REF] Liu | Detecting spatial communities in vehicle movements by combining multi-level merging and consensus clustering[END_REF] chose to avoid local optimum of modularity by applying perturbations to the initial graph, then used a modified version of Louvain [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF] that may explores a wider range of possible solutions. Finally, they aggregate the result in a consensus matrix. Burgess et al. [START_REF] Burgess | Link-prediction enhanced consensus clustering for complex networks[END_REF] focuses on adding intracommunity edges to the starting graph G, hoping to increase the efficiency of community detection algorithm. They do so by computing the similarity between pairs of nodes, based on metrics such as Jaccard [START_REF] Jaccard | Distribution de la flore alpine dans le bassin des dranses et dans quelques régions voisines[END_REF] or the number of common neighbours. The edges are then added in a non-deterministic way to G, and they execute A on G. The process is repeated n p times to obtain a consensus matrix.

Rasero et al. [START_REF] Rasero | Consensus clustering approach to group brain connectivity matrices[END_REF] use another version of consensus matrix that is not obtained through community detection but from the original data. They study brain connectivity graphs from different persons and aggregate the different graphs into a consensus matrix.

From consensus matrix to consensual communities

The first type of approach developed by Seifi et al. [START_REF] Seifi | Community cores in evolving networks[END_REF] does not require additional community detection. It considers that low values of P ij are not significant and therefore keep only the values P ij higher than a given threshold τ and set all the others to 0. This gives a thresholded matrix P τ and the associated graph G Pτ . The connected components of G Pτ are then directly the consensual communities. Work has been done to get rid of τ and decide whether the P ij entry is statistically significant before keeping it or not [START_REF] Mandaglio | Consensus community detection in multilayer networks using parameter-free graph pruning[END_REF]. In the same vein, Chakraborty et al. [START_REF] Chakraborty | Constant communities in complex networks[END_REF] consider that constant communities are groups of nodes that are always in the same community in several executions (i.e., P ij = 1).

Another approach consists in executing one final community detection algorithm on the consensus graph. Liang et al. [START_REF] Liang | Detecting community structure using label propagation with consensus weight in complex network[END_REF] use the Label Propagation algorithm (LPA) [START_REF] Raghavan | Near linear time algorithm to detect community structures in large-scale networks[END_REF] n p times to build the consensus matrix, then perform a final single execution of a weighted version of LPA on the consensus graph. This has been generalized for instance in the LF procedure [START_REF] Lancichinetti | Consensus clustering in complex networks[END_REF] that runs A n p times again on G P . At this point, if the n p partition just computed are all the same, they are considered consensual and the algorithm stops. Otherwise, a new consensus matrix is built, and the process is repeated until all n p partitions are the same.

Wang and Fleury developed an algorithm for overlapping communities that builds a consensus graph, then merges nodes when their consensus coefficient is greater than a threshold α and repeat the process [START_REF] Wang | Detecting overlapping communities in graphs[END_REF]. Yang and Leskovec developed an algorithm for overlapping consensual community detection (a context in which a node may belong to several communities), with a consensus matrix [START_REF] Yang | Overlapping community detection using weighted consensus clustering[END_REF].

Complexity issues

The consensus matrix P is an n × n matrix. The computation and storage of such a matrix quickly becomes infeasible on large graphs. This is unfortunately regularly the case with complex networks with up to several billions of nodes [START_REF] Fortunato | Community detection in graphs[END_REF]. Several studies have therefore worked on improving the computation of the consensus matrix, as in the algorithms of Tandon [START_REF] Tandon | Fast consensus clustering in complex networks[END_REF] and ECG [START_REF] Poulin | Ensemble clustering for graphs[END_REF] where they only compute the entries P ij for the edges {i, j} of G. Instead of having n 2 entries in P , they only have m, the number of edges in the graph and all other entries are 0. Tandon chooses to construct a consensus graph G P on which A is run n p times until convergence. ECG, on the other hand, runs A one last time on the consensus graph G P and considers the result as final.

The presence of edges between communities limits the efficiency of community detection algorithms. If there were only intra-community edges, the communities would be the connected components of the graph and identifying them would be trivial. Therefore, adding all the entries that correspond to the edges of the graph into the consensus matrix adds noise. The two main contributions of this article are to highlight the presence of noise and to find another criterion that would decrease further the number of entries, hence limit this noise. In addition, decreasing the number of entries in P could also reduce the temporal and spatial complexity of the consensus matrix computation.

Information is noisy

This section shows that the consensus matrix P is noisy. We do this by showing that the quality of the community is maximal when the consensus matrix is only partially filled. This means that there is some noise in P when it is full.

First, we generate a synthetic LFR graph G (along with its ground-truth communities). We use a distance dist between nodes that will be described in more details in the following subsections. Next, we execute n p times a community detection algorithm A on G. Finally, we build a consensus matrix P , but we fill it one entry P ij at a time in increasing order of dist(i, j). In case of a tie, we break it by selecting an arbitrary pair of nodes among the tie. After each addition in the incomplete consensus matrix P , we build the associated partial consensus graph G P and execute A on G P . We then compute the NMI between the LFR ground-truth communities and the communities we just computed. We iterate this way until P is completely filled. This method allows us to study how the NMI varies based on the number of entries in P .

Our methodology (see Algorithm 1) uses several parameters: the LFR graph and its mixing parameter µ that defines the proportion of links between communities; the community detection algorithm A , the number n p of executions of A and the distance dist. In the rest of the paper we use A = Louvain as it is very fast, n p = 12 as it has been shown that the consensus matrix rapidly converges as n p grows [START_REF] Poulin | Ensemble clustering for graphs[END_REF] and we will vary µ and use several dist.

Graph distance

First we consider the graph distance (smallest number of consecutive edges) between two nodes as dist, and apply Algorithm 1 for different values of µ. We use the graph distance since in Tandon [START_REF] Tandon | Fast consensus clustering in complex networks[END_REF] and ECG [START_REF] Poulin | Ensemble clustering for graphs[END_REF] the authors restrict themselves to connected pairs of nodes, i.e., nodes at distance 1.

Figure 1 shows the NMI as a function of the number of entries in P , for LFR graphs with 10 000 nodes and 3 different values of the mixing parameter µ. The graphs were generated with the implementation available online. The parameters used for LFR are: number of nodes n = 10.000, average degree k = 20, maximum Algorithm 1 Ordering (an LFR graph G, A , n p , dist) 1: Build an ordered list L of the pairs of nodes (i, j), following the order given by dist 2: Execute np times algorithm A on G 3: while L is not empty do 4:

Pick the pair (i, j) whose dist(i, j) is the smallest among L 5:

Compute the consensus coefficient Pij and add it to P 6:

Execute A (GP ) 7:

Compute the NMI between the ground truth communities, and those found at the previous step 8: end while 9: return The plot with the NMI as a function of the number of entries in P degree max k = 50, degree sequence exponent t 1 = 2, community size distribution exponent t 2 = 3, minimum community size min c = 10, maximum community size max c = 50, weight mixing parameter µ w = 0.6. Then, the shape of the curves indicates that entries need to be filled. Indeed, after a certain number of entries, the NMI decreases. Intuitively, community detection algorithms would work better if there were no inter-community edges, and if all intra-community edges were present. The non-determinism of A makes the n p partitions potentially different and thus, some non-zero consensus coefficients correspond to pairs of nodes that are in different communities in the ground-truth. The more we add such values, the harder it is for the algorithm to find the real communities, even though these values are very close to 0.

Finally, we can observe that, using the graph distance, the NMI is maximised when all entries corresponding to pairs of nodes at distance 1 are present in the consensus matrix, plus some of the entries at distance 2 (the peak is located a little after the vertical blue bar). Even if Tandon [START_REF] Tandon | Fast consensus clustering in complex networks[END_REF] and EGC [START_REF] Poulin | Ensemble clustering for graphs[END_REF] do not use exactly the same method as we do, it might be possible to do a little better than these algorithms by adding a few more entries in the consensus matrix.

To improve these algorithms we would therefore need to select some at distance 2. However, since the graph distance can only take integer values, we are limited by its precision and we therefore need another criterion that would help discriminate such entries.

Edge Clustering Coefficient

Intuitively we want to use a distance that will first add entries corresponding to intra-community pairs of nodes. Several criteria have been used to identify such pairs and they are often used to find communities. We can cite, among others, the Jaccard coefficient [START_REF] Jaccard | Distribution de la flore alpine dans le bassin des dranses et dans quelques régions voisines[END_REF], the Cosine similarity, the Hamming distance [START_REF] Hamming | Error detecting and error correcting codes[END_REF] or the Edge Clustering Coefficient (ECC ) [START_REF] Radicchi | Defining and identifying communities in networks[END_REF] that measures the similarity between nodes i and j, and is defined as

ECC(i, j) = nb common neigh min(deg(i), deg(j)) ,
where nb common neigh is the number of neighbours that nodes i and j have in common, while deg() is the degree of a node.

Most of these similarities give good results, even though we ruled out Cosine because of its higher computing time. We present here the results for the ECC.

We proceed as before: we generate an LFR graph G, but time we use the ECC as dist. We compute the ECC of all pairs of nodes that are at distance one or two. Nodes at distance three or higher cannot have any neighbour in common, thus their ECC will be zero. They should be avoided to maximise the NMI anyway, as shown in figures 1. Finally, we use algorithm 1 but insert values in decreasing order of the ECC.

Figure 2 shows the NMI as a function of the number of entries in P , for 3 different values of µ of LFR graphs with 10 000 nodes. The graphs were generated with the same implementation and the same parameters as figure 1. The ECC value being a real number, we can now precisely choose which entry to add in the consensus matrix, based on their ECC. We also notice that the maximum NMI is much greater for the ECC than graph distance. They are respectively .9994, .9910 and .8864 for µ from .5 to .7 (compared to .93, .904 and .874).

In real scenarios, we do not know the ground truth communities, so we cannot compute the NMI. We therefore need to decide beforehand how many entries need to be inserted in P to get as close as possible to the maximum NMI. The next section focuses on finding such an optimum number of entries. Fig. 2: NMI vs number of entries of P using the ECC in decreasing order. LFR graphs with 10.000 nodes and different values of µ.

Finding an optimum number of entries -Filtering out the noise

This subsection is devoted to showing how, based on some correlations, we can estimate the number of entries to insert in the consensus matrix. Figure 3a shows that the average modularity obtained across multiple executions of A is linearly correlated with the mixing parameters µ on LFR graphs. We use the correlation:

Q = 1 -µ
Kaminski et al studied such phenomenon on the ABCD graph model, which is very close to LFR [START_REF] Kamiński | Modularity of the abcd random graph model with community structure[END_REF]. They shown that the modularity of the ground truth partition of an ABCD graph with a mixing parameter µ asymptotically reaches 1 -µ as the number of nodes n grows. They also observe that the modularity is smaller for smaller graphs, and converges as n grows.

Figure 3b shows that the mixing parameter µ is correlated to the ECC value τ above which pairs of nodes should be added to the consensus matrix to maximise the NMI. That is, all the pairs of nodes {i, j} whose ECC is greater than τ should have their consensus coefficient put in the consensus matrix.

τ = 1.414µ + 0.991
Combining these two correlations it is therefore possible to deduce the optimum number of entries in the consensus matrix from the first n p executions of A . We compute several partitions (which is anyway the first step to compute the consensus matrix), we compute the average modularity and deduce the value of µ from which we deduce the lower limit on ECC. These correlations are made on LFR synthetic graphs and may not be valid for real graphs. However, we will see in Section 6 that they give good results even in these situations.

Another potential limitation to these correlations is that the average modularity over several executions increases with the size of the graph and stabilizes once the number of nodes of a graph reaches 20 000 (see Figure 3c). A solution would be to artificially increase the average modularity for graphs with less than 20 000 nodes. Figure 3c shows the modularity Q as a function of n, for LFR graphs with µ = 0.4. We repeat the operation for different values of µ. We get a fit in the general case Qmax×n (K+n) , where Q max is the modularity for a sufficiently large graph, and K = 71.21. Therefore:

Q max = KQ n + Q
The threshold τ should therefore be deduced from Q max if the graph has less than 20 000 nodes.

Improving existing algorithms

In this subsection, we show how our observations can be included into current algorithms. To do so, we pick an existing algorithm that is representative of existing consensual community detection algorithms and modify it. We also build a generic algorithm that uses our filtering method.

Consider algorithm 2, a generic algorithm implementing our filter. This algorithm takes an input graph G, filters it based on the ECC metric, and our correlation, based on the modularity of n p executions of A (G). It outputs a filtered graph G L .

Algorithm 2 Filter (G, A , n p ) 1: Execute np times algorithm A on G 2: Compute the average modularity Q of the np partitions 3: Deduce a threshold τ from Q 4: Compute the ECC of the pairs of nodes (i, j) whose distance is ≤ 2 5: Build a list L of pairs (i, j) whose ECC(i, j) > τ 6: Build a graph GL whose edges set is L. The weight of each edge is its corresponding consensus coefficient 7: return GL This filtered graph G L can be used as input for other consensual community detection algorithms. We call the Generic Filter approach, which consists in running A (G L ), where the edges of G L are weighted by their consensus coefficient. Consider also the ECG Filter algorithm which improves on ECG [START_REF] Poulin | Ensemble clustering for graphs[END_REF]. After applying the filter algorithm on G, we feed G L into ECG, then return the consensual communities found by ECG. We also study the Tandon Filter algorithm, which builds G L in the same way, then feeds it to Tandon. Note that since Tandon end ECG work on unweighted graphs, we don't bother weighting them.

The next section focuses on studying the performance of both algorithms.

Experiments

First we run our algorithm on several LFR graphs and compare the NMI along with the running time compared to other state of the art algorithms, then we repeat the experiment on real-world graphs. In this section, we chose A = Louvain and n p = 64 as we chose to parallelize the executions of A and our machine has got 64 cores.

Implementation details with correlations and correction for small graphs can be found on Software Heritage. The ordering criteria (based on the ECC) along with such correlations allows to improve on some current consensual community detection algorithms. The implementation behind figures 1 and 2 is available on Software Heritage

Synthetic Graphs

First, we generate LFR graphs with 10 000 nodes, and the same parameters as in Section 4.1 µ varies from 0.4 to 0.7. Figure 4a shows the NMI as a function of µ, while figure 4b shows the running time as a function of the size of the graph (number of nodes). We compare the filtered version of Tandon and ECG against the regular version of Tandon and ECG. We also the performance of our generic filter, along with the Louvain and Infomap algorithms as baselines. Fow low values of µ, the filtered version of ECG provides a higher NMI, at the cost of a higher running time compared the regular ECG algorithm. For high values of µ, the NMI rapidly decreases, providing an NMI of 0.42 and 0.30 for µ = 0.7 and 0.75 respectively, which is a lot lower than the regular ECG. Our generic filter algorithm gives a higher NMI compared to the ECG approach, but is outperformed by Tandon's algorithm. However, it uses a lot less time and memory, allowing it to work on bigger graphs. We also observe a sharp decrease in NMI for high values of µ, with an NMI of 0.75 and 0.72 for µ = 0.7 and 0.75 respectively. As for Tandon's algorithm, the filtered version provides a slim imporvement in the NMI, at the cost of a slightly higher running time.

Note that since a fair amount of the running time is spent filling the consensus matrix, the memory consumption grows with the running time. Even though classical community detection algorithms suffer from the limits described before, we also applied the same protocol to Louvain and Infomap. As one could have predicted the NMI is much lower with these two algorithms: Louvain (resp. Infomap) goes from 0.84 to 0.76 (resp. from 0.81 to 0.78) for µ from 0.4 to 0.7.

Limitations

Overall, figure 4a shows that the filtered algorithms tend to perform better than their non-filtered counterparts for low values of µ, and perform worse on high values of µ. We believe that this is due to a combination of several effects.

As µ grows, the number of edges inside communities decreases, while the number of inter-community edges increases. This means that the ECC of pairs of nodes located in different communities will also tend to increase. Therefore, our filter is more likely to add noise in the consensus matrix with noise, which will amplify the problem, if µ is sufficiently large.

Second, as µ grows, the modularity of the ground-truth partition tends to decrease. Therefore, as observed by Aynaud et al. [START_REF] Aynaud | Multilevel local optimization of modularity[END_REF] a community detection algorithm that focuses on modularity maximisation may find a partition with a higher modularity than the ground-truth. In this case it might be interesting to use non-modularity based algorithms.

Last, figure 5a shows the NMI as a function of the number of entries added in the consensus matrix, along with the value of the ECC pair that is being added to the matrix. For low values of µ, we observe a sharp decrease in the ECC values, which corresponds to the maximum NMI value. This allows some imprecision in our threshold on the ECC value. As long as the threshold falls within the sharp decrease, we should be close to the maximum NMI possible. However, as µ grows, the sharp decrease tends to happen after the maximum NMI. The decrease tends to be less sharp, thus requiring the threshold to be more precise. This makes the ECC threshold harder to set as a small imprecision might make a big change in the number of entries added to the consensus matrix. All in all, communities are harder to find and the threshold needs to be more precise. According to Orman et al [START_REF] Orman | A comparison of community detection algorithms on artificial networks[END_REF], in LFR graphs, when µ is greater than 0.5, the communities are less well defined, and as µ increases we are left with a scale-free network with little to no community structure. At one extreme, if µ is very low, the partition is very easy to find. At the other extreme, if µ is too high, there is no more community structure in the graph. Thus, in a real scenario, what is important is to be able to find the communities when they exist and the consensus methods are particularly useful in intermediate ranges of µ.

Real Graphs

Football Dataset First we consider the football dataset [START_REF] Girvan | Community structure in social and biological networks[END_REF], where the nodes represent US football teams, and the edges represent games played between the teams. The communities are the conference in which the team play. There are 115 nodes, 613 games and 12 communities in this dataset. The results are summarized in table 1. We see that our generic filter algorithm provides the highest NMI along with the lowest running time, while our filtered version of ECG provides a higher NMI, but a higher running time than the classical ECG algorithm. We compare the performance against two classical community detection algorithms: the Louvain method [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF], which gives a lower NMI than the other algorithms we tested, and the Infomap algorithm [START_REF] Pons | Computing communities in large networks using random walks[END_REF], which gives the second highest NMI on this dataset.

Conclusion

Most community detection algorithms are non-deterministic and it is often necessary to aggregate the results of multiple runs to find a consensus, summarized in a consensus matrix. We have shown that the information in this matrix can be noisy but that it is possible to filter out some of this noise. We then used these observations to improve existing algorithms, and verified the effectiveness of our approach on synthetic and real graphs. ECG Filter tends to yield a higher NMI than ECG at the cost of a higher running time. Our generic filter, Generic filter, also provides a high NMI in most of our tests, while being more scalable than Tandon's algorithm.

A closer look at existing consensus community detection algorithms would also be relevant. Indeed, we believe that our general observations would be useful for most algorithms that use a consensus matrix and that could therefore benefit from noise filtering. Moreover, some algorithms like the LF procedure perform community detection in several steps, it could be worthwhile to filter the graph at each step.

7 Fig. 1 :

 71 Fig.1: NMI vs number of entries of P filled using the graph distance ordering. LFR graphs with 10.000 nodes, values of the mixing parameter µ from .5 to .7. The vertical bars represent change of distance: entries that are added before the blue bar correspond to pairs of nodes at distance 1 (using a random order as a tie-breaker). Between the blue and orange bars are pairs at distance 2. . .

Fig. 3 :

 3 Fig. 3: Average modularity as a function of µ (top left). Threshold τ as a function of µ (top right). Q vs the number of nodes n (bottom left). τ vs µ at max NMI, computed with our filter and measured (bottom right)

Fig. 4 :

 4 Fig. 4: NMI as a function of µ on LFR graphs with 10 000 nodes, with different algorithms (left). Running time as a function of n on LFR graphs with µ = 0.6 (right)

Fig. 5 :

 5 Fig. 5: NMI and ECC threshold as a function of the number of entries in the consensus matrix, for µ = 0.4 (left). NMI and ECC threshold as a function of the number of entries in the consensus matrix, for µ = 0.7 (right)

Table 1 :

 1 NMI and running time of several algorithms for the football and DBLP datasets DBLP Dataset The Digital Bibliography & Library Project (DBLP) dataset[START_REF] Yang | Defining and evaluating network communities based on ground-truth[END_REF] is a co-authorship network where nodes are authors and two authors are linked by an edge if they co-published a paper. It is a larger graph than football, with 317 thousands nodes and about 1 million edges. The ground truth communities are the publication venues. Results are summarized in table 1. First, we note a big difference in running time between the different algorithms, ranging from 12 seconds to 35 minutes. The filtered algorithms are the longest because of the computation of the ECC and the generation of the consensus graph. The highest NMI is provided by Infomap, at the cost of a high running time. The filtered version of the ECG algorithm provides the second highest NMI, but also has a high running time. Our generic filter algorithm performs poorly on this dataset, while Louvain gives the lowest NMI but is the fastest by far. We could not run Tandon's algorithm because the running time was too high.

			Football	DBLP
	Algorithm	NMI Running Time NMI Running Time
	ECG	0.9079	1.10 s	0.6030	171 s
	Tandon	0.8976	1.29 s	-	-
	Tandon Filter 0.8981	2.54 s	-	-
	Generic Filter 0.9354	1.21 s	0.4613	2236 s
	ECG Filter 0.9349	1.99 s	0.6288	1278 s
	Louvain	0.8879	1.02 s	0,4769	12 s
	Infomap	0.9242	1.12 s	0.7738	1305 s

⋆ The authors would like to sincerely thank Bivas Mitra (from IIT Kharagpur, India)for the insightful discussions in the early stages of the article. This work has been partially funded by the ANR MITIK project, French National Research Agency (ANR), PRC AAPG2019.


M Bossert

A Grosman

I Trimaille

F Souris

V Doebele

A Benoit-Gonin

L Cagnon

P Spathis

P E Wolf

E Rolley

Evaporation process in porous silicon: cavitation vs pore-blocking

We have measured sorption isotherms for helium and nitrogen in wide temperature ranges and for a series of porous silicon samples, both native samples and samples with reduced pore mouth so that the pores have an ink-bottle shape. Combining volumetric measurements and sensitive optical techniques, we show that, at high temperature, homogeneous cavitation is the relevant evaporation mechanism for all samples. At low temperature, the evaporation is controlled by meniscus recession, the detailed mechanism being dependent on the pore length and on the mouth reduction. Native samples and samples with ink-bottle pores shorter than one micrometer behave as an array of independent pores. In contrast, samples with long ink-bottle pores exhibit long-range correlations between pores. In this latter case, evaporation takes place by a collective percolation process and not by heterogeneous cavitation as previously proposed. The variety of evaporation mechanisms points to porous silicon being an anisotropic three dimensional pore network rather than an array of straight independent pores.

INTRODUCTION

For a porous material where the pore transverse dimension is larger than a few tens of nanometers, it is usually assumed that cavitation in the liquid condensed in the pores is similar to cavitation in bulk liquid, provided that the liquid wets the pore walls. Evaporation by cavitation is expected to occur when the pores have a so-called ink-bottle shape, that is, if they are connected to the outer gas reservoir through small enough apertures [START_REF] Ravikovitch | Experimental confirmation of different mechanisms of evaporation from ink-bottle type pores: Equilibrium, pore blocking, and cavitation[END_REF][START_REF] Vishnyakov | Monte Carlo Simulation Test of Pore Blocking Effects[END_REF][START_REF] Morishige | Adsorption hysteresis in inkbottle pore[END_REF]. Experiments when decreasing the gas pressure indeed show that evaporation takes place at a well defined pressure threshold [START_REF] Morishige | Change in Desorption Mechanism from Pore Blocking to Cavitation with Temperature for Nitrogen in Ordered Silica with Cagelike Pores[END_REF][START_REF] Rasmussen | Cavitation in Metastable Liquid Nitrogen Confined to Nanoscale Pores[END_REF]. For large enough pores [START_REF] Rasmussen | Cavitation in Metastable Liquid Nitrogen Confined to Nanoscale Pores[END_REF], this evaporation pressure is found to be consistent with the classical nucleation theory (CNT) for bulk liquid [START_REF] Blander | Bubble nucleation in liquids[END_REF]. Using porous alumina as a model system, we have recently shown that the evaporation threshold is independent of the pore aperture diameter and consistent with CNT, providing a direct evidence for homogeneous cavitation [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF].

Similar studies of the evaporation mechanism have been reported for porous silicon which, like porous alumina, is often described as an array of parallel independent pores [START_REF] Grosman | Capillary condensation in porous materials. Hysteresis and interaction mechanism without pore blocking/percolation process[END_REF]. As in the case of porous alumina, for small pore apertures, evaporation of hexane at room temperature also takes place at a well defined threshold, independent of the pore aperture [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF]. However, this threshold is much larger than predicted by CNT. A similar discrepancy has been previously found using nitrogen at 77 K [START_REF] Grosman | Cavitation in Metastable Fluids Confined to Linear Mesopores[END_REF]. In both cases, it was argued that this discrepancy could result from heterogeneous cavitation. However, this hypothesis is contradictory with the fact that both fluids wet silicon. Moreover, it does not account for the surprising strong dependence of the evaporation threshold on the pore length and diameter [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF][START_REF] Grosman | Cavitation in Metastable Fluids Confined to Linear Mesopores[END_REF]. Another classical explanation for a well defined evaporation threshold is that it follows from a percolation phenomenon in a disordered pore network [START_REF] Mason | A model of adsorptiondesorption hysteresis in which hysteresis is primarily developed by the interconnections in a network of pores[END_REF]. This mechanism has been directly checked by optical measurements in Vycor [START_REF] Page | Adsorption and desorption of a wetting fluid in Vycor studied by acoustic and optical techniques[END_REF][START_REF] Soprunyuk | Freezing and melting of Ar in mesopores studied by optical transmission[END_REF][START_REF] Bonnet | Light scattering study of collective effects during evaporation and condensation in a disordered porous material[END_REF][START_REF] Bonnet | On Condensation and Evaporation Mechanisms in Disordered Porous Materials[END_REF]. However, this scenario requires interconnected pores. While this hypothesis seemed ruled out by measurements performed twenty years ago [START_REF] Coasne | Adsorption in Noninterconnected Pores Open at One or at Both Ends: A Reconsideration of the Origin of the Hysteresis Phenomenon[END_REF], a growing number of recent experiments (NMR diffusion measurements and Xray tomography [START_REF] Puibasset | Structure and Permeability of Porous Silicon Investigated by Self-Diffusion NMR Measurements of Ethanol and Heptane[END_REF][START_REF] Kondrashova | Scaledependent diffusion anisotropy in nanoporous silicon[END_REF]) suggest that there is indeed a large density of connections between neighboring pores. Similarly, while early investigation of adsorption isotherms [START_REF] Grosman | Capillary condensation in porous materials. Hysteresis and interaction mechanism without pore blocking/percolation process[END_REF] favored the picture of straight pores almost invariant along their axis, recent measurements of NMR diffusion [START_REF] Puibasset | Structure and Permeability of Porous Silicon Investigated by Self-Diffusion NMR Measurements of Ethanol and Heptane[END_REF][START_REF] Kondrashova | Scaledependent diffusion anisotropy in nanoporous silicon[END_REF], permeability [START_REF] Puibasset | Structure and Permeability of Porous Silicon Investigated by Self-Diffusion NMR Measurements of Ethanol and Heptane[END_REF], and elastic constant anisotropy [START_REF] Rolley | Using capillary forces to determine the elastic properties of mesoporous materials[END_REF] suggest that constrictions are present along the pores.

In order to elucidate the true nature of the evaporation mechanism in porous silicon, we study in this paper its dependence on the temperature T and the pore length l, for both native silicon and silicon ink-bottles. We find that homogeneous cavitation is always dominant at high temperature, with a threshold consistent with the CNT prediction. As the temperature is decreased, an increasing fraction of the pores empties through meniscus recession from the sample's surface. Pore blocking is then sensitive to the pore network geometry. For native samples and short ink-bottles (l 1 µm), the recession occurs independently in the different pores, as expected for non-interconnected pores. In contrast, for long inkbottle pores (l 5 µm), we observe that evaporation presents long-range correlations between pores, directly evidencing the existence of interconnections. The steep evaporation event is then controlled by vapor percolation in the disordered pore network. This complex picture was built after systematic experiments. First, we used a sensitive optical method to measure sorption isotherms of nitrogen at a fixed temperature (77 K) for ink-bottle samples of increasing pore length l and evidence the role of l. Then, we measured evaporation isotherms over a broad range of temperatures. While the results in the short pore case pointed to the conventional picture of independent pores, this was not true for long pores. In order to elucidate the mechanisms of evaporation at play in this case, we combined the results of several experiments. In a first step, by comparing the evaporation of nitrogen and helium 4 (a perfectly wetting fluid), we ruled out the earlier hypothesis of heterogeneous cavitation in long pores. Second, by studying the temperature dependence of the evaporation pressure, both for native and ink-bottle samples, we evidenced a crossover between homogeneous cavitation at high temperature and meniscus recession at low temperature. Third, in the latter case, an optical interferometric technique [START_REF] Bossert | Stress or Strain Does Not Impact Sorption in Stiff Mesoporous Materials[END_REF] allowed us to reveal long-range spatial correlations between ink-bottle pores, and show the collective nature of evaporation by meniscus recession. Finally, putting together the results of these different experiments, we could propose a network geometry for thick porous silicon.

MATERIALS AND METHODS

Porous Silicon Samples. Porous silicon samples are fabricated using the same procedure as in Ref. [START_REF] Grosman | Capillary condensation in porous materials. Hysteresis and interaction mechanism without pore blocking/percolation process[END_REF], where the preparation process, the morphology and the physical properties of the samples are reported in detail. Samples are prepared by electro-etching of highly boron-doped (100) Si single crystal in hydrofluoric acid solution. One obtains parallel pores, perpendicular to the wafer surface, with a cross-section of polygonal shape. The pore size distribution is broad and depends on the concentration of the solution. Native samples used for synthesis of ink-bottles have two different porosities: for a nominal porosity of 70%, the pore transverse dimension d is 12 < d < 40 nm, with < d >= 26 nm. For a nominal porosity of 85%, d is 23 < d < 80 nm, with < d >= 50 nm. In addition, we also studied a 50% porosity native sample (7 < d < 19 nm, < d >= 13 nm).

We use monolithic samples still attached to the underlying wafer. The lateral size is of the order of 1 cm. The thickness of the porous layer, hence the pore length l, is proportional to the etching time. In this work, l is varied from 1 to 40 µm. Note that, in almost all earlier studies of adsorption in porous silicon, l was larger than 10 µm [START_REF] Grosman | Capillary condensation in porous materials. Hysteresis and interaction mechanism without pore blocking/percolation process[END_REF][START_REF] Coasne | Adsorption in Noninterconnected Pores Open at One or at Both Ends: A Reconsideration of the Origin of the Hysteresis Phenomenon[END_REF][START_REF] Naumov | Understanding capillary condensation and hysteresis in porous silicon: Network effects within independent pores[END_REF][START_REF] Gor | Adsorption-induced deformation of nanoporous materials -A review[END_REF][START_REF] Grosman | Adsorptioninduced strain of a nanoscale silicon honeycomb[END_REF]. The list of samples with their main characteristics is summarized in table I.

Reduction of the pore mouth. In order to observe cavitation, the pores have to present an ink-bottle geometry. For most samples, the reduction of the pore mouth at the outer surface of the sample is obtained by multiple deposition of atomic layers of alumina (Atomic Layer Deposition). We used a ANRIC -AT400 reactor, mostly in continuous mode. In this case, trimethyl-aluminum and water exposition time is short so that the alumina layer is formed only at the pore mouth. Based on isotherms measurements, we empirically fixed the number of ALD cycles in such a way that the accessible porous volume is reduced, but only slightly. Most pores, except the ones with the smallest apertures, are then still accessible to the fluid. For the present purpose, we do not need to know the resulting size distribution of (reduced) pore mouth, but we stress that it is presumably as broad as the native distribution. Indeed, ALD is ideally a conformal process, so that the diameter distribution of apertures is simply the native one downshifted by twice the (uniform) alumina thickness. The "short pore" behavior is observed for the two samples 70% -1 µm and 85% -2 µm. The reduction of pore mouth is obtained through ALD deposition as described in the text except for the 85% -2 µm. For this particular sample, the exposition time is of the order of 10 s, much larger than the one for the continuous mode (smaller than 1 s).

Sorption Isotherms Nitrogen volumetric isotherms -For samples with long pores, nitrogen sorption isotherms were performed over a large range of temperatures, 64 K < T < 116 K. The isotherms are measured volumetrically using a commercial apparatus (HPVA II Micromeritics) equipped with 1000 Torr and 100 bar pressure transducers. The manifold at room temperature is connected to the cell containing the sample. The cell is cooled at low temperature by a homemade cryostat. We measure the admitted quantity as a function of pressure, up to the vapor saturation pressure, both with and without the sample for each temperature of interest. The difference between the two measurements yields the adsorbed amount. This procedure is required due to the large dead volume between the manifold and the cell. The associated uncertainty prevents us to study temperatures higher than 116 K, where the amount condensed in the sample is only 3% of the total fluid amount (see typical isotherms in the Supporting Information). In the whole temperature range, the adsorbed quantity is computed taking into account the non-ideality of the gas phase through the compressibility factor [START_REF] Span | A Reference Quality Equation of State for Nitrogen[END_REF][START_REF] Span | A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000 K and Pressures to 2200 MPa[END_REF].

Nitrogen optical isotherms -The porous silicon layer being transparent above a wavelength 500 nm, the condensed mass could be determined using white light interferometry (WLI) [START_REF] Bossert | Stress or Strain Does Not Impact Sorption in Stiff Mesoporous Materials[END_REF][START_REF] Casanova | Gas adsorption and capillary condensation in nanoporous alumina films[END_REF] rather than volumetric measurements. The key advantage of the method is its better sensitivity, which enables measurement of thin samples with very small porous volumes. The porous layer is illuminated by a white light beam (500 to 900 nm). The interference between the beams reflected at both sides of the porous layer creates fringes in the spectrum of the reflected light. As the mean distance between pores is less than 100 nm, the porous layer appears as an homogeneous film (provided that the pore liquid content is statistically uniform), and the Fourier transform of the spectrum exhibits a single peak corresponding to the optical thickness L of the layer. Along the sorption isotherm, the shift ∆L of this thickness is proportional to the adsorbed amount within a few percent. Optical sorption isotherms ∆L = f (P/P sat ) thus have a shape very similar to that of ordinary sorption isotherms obtained through volumetry [START_REF] Bossert | Stress or Strain Does Not Impact Sorption in Stiff Mesoporous Materials[END_REF]. This method is extremely sensitive almost up to the critical bulk point as, in contrast to volumetry, the interference does not depend on the contents of dead volumes outside the membrane. In practice, using a specially designed optical cryostat, it allows us to measure sorption isotherms for the thinnest porous layers (1 µm) with a very small porous volume (less than ∼ 10 -9 cm 3 ) up to 124 K and 30 bar.

Helium volumetric isotherms -Helium sorption isotherms were performed over a large range of temperatures, 2.38 K < T < 4.65 K. The isotherms are obtained with a homemade setup using a continuous volumetric technique. The sample is placed in an annular copper cell inside a modified commercial 4 He circulation cryostat. The cell is filled or emptied through a capillary line connected to a gas-handling manifold at room temperature. The pressure inside the cell is measured at room temperature using a separate capillary line, so as to suppress, for the used flow rates, the viscous pressure drop along the capillary. The cell temperature is regulated with a typical r.m.s. stability of 1 mK and measured using a Cernox™thermometer calibrated against the saturated vapor pressure of helium. Condensation and evaporation isotherms are respectively obtained by connecting, through a voltage-controlled precision microvalve (Pfeiffer EVR-116), the cell to a reservoir of pressurized gas at constant pressure or to a pump, and measuring the time evolution of the cell pressure P (t). The helium mass inside the membrane can be retrieved once the total flow rate through the microvalve and the effective dead vol-ume are known. In practice, as evaporation inside the membrane occurs over a narrow pressure range, the evaporation pressure is simply determined as the pressure for which the time derivative | Ṗ (t)| is minimal (see Supporting Information).

Determination of the diameters of constrictions and interconnections. For evaporation by meniscus recession, the evaporation pressure for nitrogen and helium was converted into an effective pore diameter by assuming that recession takes place at equilibrium. To relate the equilibrium pressure to the pore diameter, we use the model developed by Saam and Cole for a cylindrical pore [START_REF] Saam | Excitations and thermodynamics for liquid-helium films[END_REF], which takes into account the van der Waals interaction between the fluid and the substrate. We assume that the interaction between those fluids and porous silicon is close to that with quartz, computed to be around 4400 and 1200 K • Å3 respectively for N 2 and 4 He [START_REF] Cheng | Retardation and many-body effects in multilayer-film adsorption[END_REF]. The calculated pore diameters are quite insensitive to the assumed interaction strength. Changing this interaction strength by a factor of 10 only modifies the calculated pore diameter by about 1 nm. Note that, since the pores in porous silicon have a polygonal shape, the calculated diameters should rather be considered as a typical transverse dimension (this also holds for the estimated diameter of connections between pores).

CNT prediction for the cavitation threshold. From the measured isotherms, we deduce the vapor pressure at which evaporation takes place inside the sample, and compare this pressure with the Classical Nucleation Theory (CNT) prediction for bulk cavitation. Here, we sketch how we evaluate this CNT prediction.

According to CNT [START_REF] Blander | Bubble nucleation in liquids[END_REF], the nucleation rate per unit volume is given by:

J = J 0 exp (-E b /k B T ) (1)
where J 0 is a prefactor. The energy barrier E b for nucleating the critical bubble is:

E b = 16π 3 γ 3 (P V -P L ) 2 (2)
where γ is the surface tension, P L the pressure of the metastable liquid phase, and P V the pressure of the vapor inside the critical bubble. As the critical bubble is in (unstable) equilibrium with the surrounding liquid phase, the chemical potentials µ L and µ V are equal:

µ L (P L ) = µ V (P V ) (3)
The vapor and liquid chemical potentials are respectively computed from the fugacity of the vapor [START_REF] Span | A Reference Quality Equation of State for Nitrogen[END_REF][START_REF] Span | A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000 K and Pressures to 2200 MPa[END_REF], and by assuming a constant compressibility of the liquid (between saturation and P L ). This allows us to compute E b as a function of P V , which coincides with the reservoir pressure, since this reservoir is at phase equilibrium with the liquid.

The cavitation pressure P cav V is then deduced from the value of E b at cavitation, itself determined by the condition J(P cav V )V τ 1, where V is the volume of liquid (here the pore volume) and τ the observation time, typically 100 s for the experiments using nitrogen. P cav V is thus set by the implicit equation:

E b (P cav V ) = ln (J 0 V τ ) k B T (4)
The prefactor J 0 depends on the dynamics of the critical bubble [START_REF] Blander | Bubble nucleation in liquids[END_REF].

For nitrogen experiments, the dynamics is controlled by viscous dissipation [START_REF] Blander | Bubble nucleation in liquids[END_REF][START_REF] Tanaka | Simple improvements to classical bubble nucleation models[END_REF][START_REF] Baidakov | Homogeneous nucleation in liquid nitrogen at negative pressures[END_REF] and we use the corresponding expression for J 0 . In the middle of the experimental temperature range (T = 100 K), J 0 10 38 s -1 m -3 and E b 46k B T . For helium experiments, E b lies in the range 40-45 k B T , depending on the estimation of J 0 [START_REF] Blander | Bubble nucleation in liquids[END_REF][START_REF] Maris | Nucleation of bubbles in liquid helium at negative pressure[END_REF]. For both fluids, because of the logarithmic dependance, the precise values of J 0 , τ , and V have a small impact on the predicted value for P cav V .

RESULTS

We first report on the experiments in porous silicon samples with alumina deposition : in the first subsections, it is implied that all samples have a reduced pore mouth. Only the last subsection is devoted to native samples.

Impact of pore length on the shape of the isotherm

We have measured optical isotherms for several pore lengths and two porosities with N 2 at 77 K. Figure 1 shows the measured isotherms for the samples of 70% porosity with pore lengths of 1, 5, 20 and 40 µm.

For the 1 µm-long pores (blue curve in Fig. 1), evaporation takes place in two steps. Starting from the filled membrane, a progressive emptying is followed by a steep evaporation step at a pressure P c ≈ 0.49P sat . This "short pore" behaviour is also observed for the 2 µm/85% porosity sample. As shown by the 77.8 K isotherm in Fig. 2, an evaporation step takes place at the same value of P c . This pressure is very close to the value measured by Baidakov et al. in cavitation experiments on superheated bulk nitrogen [START_REF] Baidakov | Homogeneous nucleation in liquid nitrogen at negative pressures[END_REF]. This shows that, for short pores terminated by narrow enough apertures, evaporation takes place by cavitation. Moreover, for these pores whose mean diameter is a few tens of nanometers, no significant confinement I for the respective mouth reduction). The inset shows the secondary evaporation event around 0.48Psat which we ascribe to homogeneous cavitation.

effect is observable, in agreement with earlier results on evaporation in cage-like porous silicas with large pores [START_REF] Rasmussen | Cavitation in Metastable Liquid Nitrogen Confined to Nanoscale Pores[END_REF].

This "short pore" behaviour contrasts with that observed for longer pores in Fig. 1 (5 to 40 µm). In this case, a sharp evaporation event directly occurs at a threshold P p > P c which depends on the pore length. Such a "long pore" behaviour is observed in all samples at 77 K, for pore lengths ranging from 4 to 60 µm and porosities of 70 % and 85%. The general trend is that P p increases with the length and mean diameter of the pores. P p depends very weakly on the thickness of the alumina deposithence on the reduction of the pore mouth. Below P p , the membrane contents slowly decreases down to around 0.48P sat , very close to P c for short pores. As shown by the inset of Fig. 1, optical isotherms reveal a second evaporation event around this pressure, albeit with a much smaller amplitude than the first event.

Short pores: changing the temperature Further evidence that evaporation at P c is due to cavitation can be obtained by studying the temperature evolution of "short pore" sorption isotherms. In Fig. 2, we show optical isotherms for a "short pore" sample (2 µm/85%). The amplitude of the cavitation event, small at 77 K, increases with the temperature T . The corresponding cavitation pressure P c (T ) is compared to the CNT prediction in Fig. 3. The agreement is good at high temperature, confirming the cavitation nature of evaporation. At low temperature, the cavitation threshold is slightly higher than predicted. This result is in qualitative agreement with earlier measurements of cavitation in bulk superheated nitrogen [START_REF] Baidakov | Homogeneous nucleation in liquid nitrogen at negative pressures[END_REF] as well as in other fluids far below their critical point (hexane [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF], heptane [START_REF] Arvengas | Cavitation in heavy water and other liquids[END_REF], ethanol [START_REF] Arvengas | Cavitation in heavy water and other liquids[END_REF][START_REF] Bruot | Curvature dependence of the liquid-vapor surface tension beyond the Tolman approximation[END_REF], water [START_REF] Azouzi | A coherent picture of water at extreme negative pressure[END_REF]). The temperature evolution of the isotherm shape shown by Fig. 2 is consistent with the large distribution of pore apertures. At 77 K, most pores have large enough apertures to empty through meniscus recession before reaching the metastability limit in the bulk of the pore, and cavitation only takes place in the narrowest pores with the narrowest apertures. As the temperature increases, the cavitation pressure increases faster than the equilibrium pressure in the apertures, and an increasing number of pores empty by cavitation. This accounts for the increase in relative height of the cavitation step. Above 120 K, all pores empty through cavitation.

Long pores: changing the temperature

We now turn to the "long pore" samples. Here, we have used volumetric setups in order to measure the temperature dependence of the sorption isotherms.

Nitrogen isotherms are shown in figure 4 for a sample of porosity 70%, thickness 20 µm, coated with 55 ALD cycles. At all temperatures, we observe that nitrogen mostly evaporates at a well defined threshold. Note that the sensitivity of the commercial apparatus is too low to detect a possible small secondary cavitation event similar to that detected at 77 K by WLI (inset of Fig 1).

FIG. 4.

Temperature dependence of nitrogen sorption isotherms for a porous silicon sample of porosity 70%, thickness 20 µm, and 55 ALD cycles. The non-physical shape of the isotherm at 116 K at low pressure is an artefact due to the poor sensitivity at high temperature, but it remains easy to determine the pressure of the evaporation step. FIG. 5. Helium sorption hysteresis loops for a porous silicon sample of porosity 70%, thickness 20 µm, with pore mouth reduced as described in the text. The vertical scale has been adjusted to be respectively 0 and 1 at the lower and upper closure points of the hysteresis loop. The two temperatures lie on both sides of the crossover temperature T * . At the lower temperature, the secondary evaporation event at Pc is similar to that detected with nitrogen using WLI.

Similar experiments were performed with helium-4 using a different sample with the same 70% porosity and thickness 20 µm. This sample had been previously used for the hexane experiments reported in Ref. [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF]. In this case, instead of using ALD, the pore mouth was reduced by evaporating a 2 nm Al layer which is subsequently oxidized, this procedure being repeated 8 times [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF]. As illustrated by the hysteresis loops in Fig. 5, evaporation mostly takes place at a well defined threshold whatever the temperature (see Supporting Information for details). However, at low temperature, the continuous volumetric technique used in this case also reveals a small evaporation event at P c . As discussed below, this event is similar to that detected with nitrogen using WLI, and corresponds to cavitation. FIG. 6. Temperature dependence of the evaporation pressure (P/Psat) for N2 (A) and 4 He (B) in 70% porosity and 20 µm thick porous silicon samples with various alumina deposits. Diamonds correspond to the steep evaporation, while crosses correspond to the evaporation of a small fraction of the pores via homogeneous cavitation (the cross in A corresponds to the event shown in the inset of Fig 1). For both fluids, the solid lines represent the CNT prediction for a constant energy barrier of 46kBT , and the dashed lines are the theoretical curves for evaporation through recession of a liquid-vapor meniscus in a straight cylindrical pore of 9 nm diameter, calculated from Ref. [START_REF] Saam | Excitations and thermodynamics for liquid-helium films[END_REF][START_REF] Cheng | Retardation and many-body effects in multilayer-film adsorption[END_REF]. T * is the crossover temperature between the two evaporation processes.

Results for N 2 and He are respectively summarized in Figs. 6 (a) and (b), which show the pressures of the main evaporation step (black diamonds) and of the secondary event (crosses). The full lines in Figs. 6(a) and (b) correspond to the CNT predictions for the cavitation pressure, using a constant energy barrier of 46k B T , and the dashed lines to the equilibrium pressure in a straight cylindrical pore 9-nm in diameter. In a similar way to an earlier study of evaporation in ordered large-cage mesoporous silicas [START_REF] Morishige | Change in Desorption Mechanism from Pore Blocking to Cavitation with Temperature for Nitrogen in Ordered Silica with Cagelike Pores[END_REF], comparison of the experimental points to the theoretical lines demonstrates the existence of a temperature crossover between two evaporation mechanisms. Above a temperature crossover T * , evaporation mostly takes place at the same pressure P c as in "short pore" samples, consistent with homogeneous cavitation. Below T * , the pressure threshold P p strongly deviates from the CNT curve. For both helium and nitrogen, P p is found equal to the equilibrium pressure in 9-nm diameter cylindrical pores, suggesting that it is controlled by meniscus recession in such apertures. Nevertheless, the pressure of the secondary event is consistent with CNT, showing that a small fraction of the fluid still evaporates through cavitation.

A closer look at evaporation in "long pore" samples

In order to shed light on the evaporation mechanism at play in "long pores" samples, we use an original approach, which consists in monitoring the pressure dependence of the FFT spectrum of the reflected light. An example is shown in Fig. 7 for a sample with porosity 70% and thickness 20 µm. On the saturation plateau (point A in the inset of Fig. 7), and at the end of the evaporation step (point C in the inset of Fig. 7), the Fourier transform of the reflected intensity versus optical wavelength shows a single peak at optical thicknesses corresponding respectively to a saturated or a nearly empty porous layer. If pores would evaporate independently, one would expect the peak to continuously shift between A and C on the evaporation branch. Indeed, the distance between pores being much smaller than the optical wavelength, the reflected light spectrum measures the average of the fluid density over many pores, which, since neighboring pores have uncorrelated diameters, should vary continuously during evaporation. Such a behaviour is indeed observed for "long pore" native samples at porosities of 70% and 85% (see Supporting Information). In contrast, the behaviour for "long pore" ink-bottles is strikingly different. As shown by Fig. 7, the spectrum at point B in the inset of Fig. 7 is a linear combination of the two peaks above: during evaporation, the amplitude of the "saturated" peak (L f ) decreases while that of the "empty" peak (L e ) grows. This peak splitting, rather than the emergence of a single peak corresponding to an intermediate fluid content in the porous layer, means that evaporation is correlated over distances much larger than the pore spacing, directly evidencing the collective nature of evaporation between neighboring pores.

We have confirmed this long range correlation of evaporation in "long pore" samples by direct, high spatial resolution, optical imaging. Imaging the evaporation process is indeed possible using hexane at room temperature instead of nitrogen. In this case, a spatial resolution of 1 µm can be reached because of the smaller distance between the sample and the imaging lens, compared to the low temperature setup, combined with the enhanced contrast between saturated and empty regions due to the larger optical index of hexane compared to nitrogen. Figure 8 shows a series of successive pictures taken during the main evaporation event for a "long pore sample". Evaporation clearly proceeds from a few spots through the emptying of large patches of neighbouring pores. The correlation length of the resulting pattern is of the order of 10 µm, about 2 orders of magnitude larger than the mean distance between pores.

Evaporation in native "long pore" samples

We now turn to native "long pore" samples, i.e. without pore mouth reduction. For such samples, isotherms (presented in Supporting Information) show a steep evaporation event at high temperature. At low temperature, desorption is not as steep and we define the evaporation pressure at mid-height of the evaporation step. The temperature dependence of the normalized evaporation pressure is shown in figure 9 for a 70% porosity sample 20 µm thick. We find that the behavior of the native sample is similar to that of the same sample with alumina deposition (Fig. 6): at high temperature, evaporation occurs through cavitation while below a crossover temperature T * ≈ 114 K, evaporation occurs through meniscus recession in an effective aperture of diameter D eff = 10.9 nm. This value is significantly larger than d eff = 9 nm found for the sample with alumina deposition. This difference in effective diameters suggests that, at low temperature, evaporation by meniscus recession follows different paths in native samples and ink-bottle samples. This is directly confirmed by the study of the reflected light spectrum for the native sample. In contrast to the ink-bottle samples, the Fourier transform of this spectrum does not reveal any peak splitting during evaporation (see Supporting Information). This means that native samples do not show any long-range correlation during evaporation.

For native 50% porosity samples, we have also measured the evaporation pressure as a function of the temperature and we find the same crossover from meniscus recession at low temperature to cavitation at high temperature (see Supporting Information). For 50% porosity, we find D eff = 6.2 nm and T * = 97 K. For native 85% porosity samples, we have only measured the evaporation pressure at 77 K, which gives D eff = 18 nm.

DISCUSSION

For "short pore" samples, all our observations are consistent with the porous layer behaving as independent pores. Depending on the diameter of the alumina constrictions at the pore mouth, and depending on the temperature, evaporation in a pore may occur either through the recession of the meniscus or by homogeneous cavita-FIG. 9. Temperature dependence of the normalized evaporation pressure for the same PoSi sample as in Fig. 6, but prior to alumina deposition. Black symbols correspond to the evaporation pression for the native state and grey ones to the evaporation pressure for reduced pore mouth. As in Fig. 6, the solid line is the CNT prediction for an energy barrier of 46 kBT and the dashed line is the equilibrium pressure in an aperture of diameter 10.9 nm.

tion.

For "long pore" samples and above a crossover temperature T * which is sample dependent, we also observe cavitation. This was expected for samples with alumina deposit: above T * , the alumina constrictions at the pore mouth are narrow enough to block the meniscii and preclude evaporation from the sample surface. Thus, the liquid can reach the metastability limit for bulk cavitation in the whole volume, independently of the topology of the pore network. The fact that cavitation is also observed in native samples is more surprising. The simplest interpretation is that pores are not straight and that native constrictions in the pores play the same role as alumina constrictions at the aperture, i.e. control the recession of the meniscus below T * . The existence of constrictions distributed along the pores is also suggested by NMR measurements of the diffusion constant along the pore axis [START_REF] Puibasset | Structure and Permeability of Porous Silicon Investigated by Self-Diffusion NMR Measurements of Ethanol and Heptane[END_REF][START_REF] Kondrashova | Scaledependent diffusion anisotropy in nanoporous silicon[END_REF], and by permeability measurements [START_REF] Puibasset | Structure and Permeability of Porous Silicon Investigated by Self-Diffusion NMR Measurements of Ethanol and Heptane[END_REF], the values of both quantities being much smaller than expected for straight pores.

In this interpretation, D eff is a typical diameter of these constrictions. This is supported by the value of the alumina thickness t ob which is required to obstruct the pores. Assuming that constrictions exist close to the pore aperture, one expects D eff ∼ 2t ob . This relationship is consistent with our measurements of D eff and estimation of t ob (see Supporting Information). Values of effective diameters for different porosities are gathered in Table II.

Turning now to evaporation below T * , we find that native and ink-bottles samples exhibit different evaporation mechanisms. For native samples, we do not observe longrange correlation between pores. It is thus likely that meniscus recession occurs independently in each pore and is controlled by the disorder induced by the constrictions. Then, each pore behaves as a one-dimensional disordered system, as proposed in Refs. [START_REF] Naumov | Understanding capillary condensation and hysteresis in porous silicon: Network effects within independent pores[END_REF][START_REF] Puibasset | Adsorption-desorption hysteresis of simple fluids confined in realistic heterogeneous silica mesopores of micrometric length: A new analysis exploiting a multiscale Monte Carlo approach[END_REF] to account for the H2 shape of isotherms. Discussing the shape of the isotherms and relating precisely the value of D eff to the pore geometry would require to characterize the disorder along the pores and to take into account the distribution of pore diameters. This is well beyond the scope of the present work. TABLE II. Average pore diameter d (after [START_REF] Grosman | Capillary condensation in porous materials. Hysteresis and interaction mechanism without pore blocking/percolation process[END_REF]), effective diameter D eff of constrictions along pores and effective diameter d eff for connections between pores for several porosities, obtained from the evaporation pressure. We also give the crossover temperature T * between meniscus recession and cavitation for native samples. For 50% porosity, only native samples were studied so that d eff could not be determined. For 85% porosity, measurements were only performed at 77 K, and T * is the estimated intercept of the cavitation curve and of the equilibrium curve for D eff = 18 nm.

Finally, coming to the evaporation mechanism for "long pore" ink-bottles below T * , the very steep evaporation event observed in this case has specific features. First, the evaporation pressure is almost independent of the reduction of the pore mouth. This was demonstrated for hexane at room temperature [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF]. We have checked that this holds for nitrogen at 77 K. As shown in Fig. 10 for a 85% porosity sample, we find that evaporation occurs at P p 0.8P sat whatever the alumina thickness, up to a value where the whole porous volume is almost completely disconnected from the vapor reservoir. Similar results were previouly interpreted as a proof for cavitation [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF][START_REF] Grosman | Cavitation in Metastable Fluids Confined to Linear Mesopores[END_REF], which could only be heterogeneous since the homogeneous cavitation would occur close to 0.5P sat , well below P p . This hypothesis is now ruled out: despite being a perfectly wetting fluid, helium indeed behaves as nitrogen, excluding the heterogeneous cavitation scenario.

The second feature is the long-range correlations between pores. While the simplest explanation is the existence of connections between neighbouring pores, another possibility, put forward by Grosman and Ortega [START_REF] Grosman | Influence of elastic strains on the adsorption process in porous materials: an experimental approach[END_REF], would be an indirect mechanical coupling. However, in view of our recent experiments on strain-and stress-induced adsorption [START_REF] Bossert | Stress or Strain Does Not Impact Sorption in Stiff Mesoporous Materials[END_REF], this mechanical coupling is negligible. Moreover, the existence of connections between pores is also suggested by the moderate anisotropy of the diffusion constant measured by NMR [START_REF] Puibasset | Structure and Permeability of Porous Silicon Investigated by Self-Diffusion NMR Measurements of Ethanol and Heptane[END_REF][START_REF] Kondrashova | Scaledependent diffusion anisotropy in nanoporous silicon[END_REF]. All together, this makes the first explanation the most likely.

The third and last feature is the temperature depen- dence of the evaporation pressure, which is found to be the same as that for a meniscus receding through an aperture of effective diameter d eff , smaller than the typical constriction size D eff . As d eff is independent on the fluid and on the thickness of the alumina coating, it has to be a characteristic dimension of the porous material, most probably related to connections between pores. Bringing all these results together leads to the following scenario for long pores samples of porosities 70% and 85%. Without alumina deposit, the fluid evaporates mainly along the pores as the constrictions are, on average, wider than the connections between pores. The existence of connections has no impact on the evaporation mechanism. In contrast, when the pore mouths (or the constrictions close to the mouth) are reduced by alumina deposition, only the largest pores or defects in the pore network still present a mouth wider than the interconnections. It then becomes more favorable for the meniscii to recede first in these largest pores, then through interconnections to finally reach the narrower pores. The observed long-range correlation during evaporation shows that this process is a collective one. By analogy with 3D disordered porous materials, we suggest that it involves a percolation phenomenon. Assuming that all constrictions within a pore are wider than the interconnections, a given pore will totally empty as soon as one of the neighboring pores to which it is connected will itself empty. The random nature of interconnections then gives rise to an invasion percolation phenomenon, the largest pores acting as germs. Because the underlying sites are the pores themselves, this percolation is likely 2D rather than 3D. Predicting the corresponding evaporation threshold (that is the value of d eff ) would require a detailed knowledge of the network topology, which is well beyond our current capability. However, we stress that this scenario provides a natural explanation to the dependence of d eff on the details of the pore geometry, a feature which was difficult to explain in the hypothesis of heterogeneous cavitation. For instance, as shown in Fig. 1, the evaporation pressure increases with the pore length l. This can be explained by an increase in the diameter of the interconnections with the etching time, a reasonable hypothesis in view of an earlier work were we observed that the isotherms change with the etching time [START_REF] Bossert | Stress or Strain Does Not Impact Sorption in Stiff Mesoporous Materials[END_REF].

In this picture, the observation in the same samples of a small cavitation event (about 1 % of the total volume for the 20 µm/70% sample at 77 K) implies that some pores or some sections of the pores are connected to their neighbours only through some very narrow constrictions.

To summarize, the features of evaporation from "long pores" samples are consistent with porous silicon being a disordered and interconnected 3D pore network -with some anisotropy-rather than an array of straight independent pores. Actually, very few experiments point to the opposite conclusion. The crucial one [START_REF] Coasne | Adsorption in Noninterconnected Pores Open at One or at Both Ends: A Reconsideration of the Origin of the Hysteresis Phenomenon[END_REF], which initially promoted the model of independent pores, is an experiment performed using a 1-cm-size sample prepared in the same way and with a similar geometry as the present "long pore" samples (pore length about 20 µm and porous layer still attached to the underlying wafer). When the sample was half covered with a thick aluminum layer -totally blocking the pores below the deposit -and subsequently exposed to an oxygen atmosphere, oxygen did not enter the pores below the Al layer , which was interpreted as a proof that pores are non interconnected. However, a major difference between this experiment and ours is the typical length scale at which communication between pores is tested. In the earlier experiment, no oxygen was detected in the pores a few mm away from the Al boundary. In contrast, present optical experiments reveal connections up to the µm scale. A precise modeling of the oxygen flow at large scale in a realistic pore network is then necessary to check whether the two observations may be reconciled or not.

CONCLUSION

We have shown that evaporation in porous silicon membranes occurs either through homogeneous cavitation or by meniscus recession. In the latter case, poreblocking can trigger a local or collective evaporation process. This variety of evaporation mechanisms is not consistent with a description of porous silicon as an array of independent straight pores aligned along the etching direction. This is in agreeement with recent measurements of diffusion constant, permability, elastic moduli as well as images obtained by X-ray tomography. One of the major outcomes of the present work is a consistent picture, for all samples and all temperatures, of how the evaporation process is impacted by the defects of the pore array, these defects being schematically described as constrictions along the pore axis and interconnections between pores.

In this picture, the existence of interconnections is responsible for the spectacular long-range nature of evaporation, consistent with a percolation process, observed at low temperature for the long pores samples with reduced pore mouth. Such a behavior, which was previously misinterpreted as heterogeneous cavitation, is however restricted to those particular samples. In all other situations, and particularly for native samples, interconnections have no impact on the evaporation process, which occurs either through menicus recession along the pore direction, or by cavitation. This reflects the fact that, on average, constrictions are larger than interconnections. Hence, the resulting structure of porous silicon and the evaporation process are highly anisotropic (as are elastic moduli, diffusion constants...).

The second important result is that, for long pore samples, we always observe a crossover between meniscus recession at low temperature and homogeneous cavitation at high temperature. The crossover temperature is controlled by the typical diameter of the constrictions for native samples or by that of the interconnections for samples with alumina deposition. Such a crossover is similar to the one observed in ordered cage-like porous silicas [START_REF] Morishige | Change in Desorption Mechanism from Pore Blocking to Cavitation with Temperature for Nitrogen in Ordered Silica with Cagelike Pores[END_REF] whose behavior is now well understood [START_REF] Morishige | Revisiting the Nature of Adsorption and Desorption Branches: Temperature Dependence of Adsorption Hysteresis in Ordered Mesoporous Silica[END_REF] . For these systems, constrictions and interconnections are identical and monodisperse, so that d eff =D eff inferred from the evaporation pressure can be compared to their determination from structural studies. This contrasts with the case of Vycor, a prototype of tridimensional disordered porous material. Although a similar crossover between percolation and cavitation has been previously demonstrated for this material [START_REF] Bonnet | Light scattering study of collective effects during evaporation and condensation in a disordered porous material[END_REF][START_REF] Bonnet | On Condensation and Evaporation Mechanisms in Disordered Porous Materials[END_REF], a direct comparison between d eff and its complex structure seems out of reach. In other words, unlike in ordered cage-like porous silica, T * cannot be precisely predicted from the material structure. The situation for porous silicon is similar. Nevertheless, we can safely expect that, as long as the pore network does not consist in an array of non-interconnected straight pores, cavitation should be the dominant evaporation process at high enough temperature.

Ultimately, the simplest situation is the one of short pore samples with alumina deposit, which can be considered as a array of independent pores. The fraction of such pores emptying through homogeneous cavitation depends on the pore aperture distribution and on the temperature. Since a strong reduction of the pore mouth allows to observe cavitation in a wide temperature range, such samples are powerful tools for studying homogeneous cavitation. Moreover, as a single sample consists in 10 11 independant pores, it is straightforward to determine the cavitation rate, as demonstrated in porous alumina [START_REF] Doebele | Direct Observation of Homogeneous Cavitation in Nanopores[END_REF]. Compared to porous alumina, porous silicon layers are simpler to synthetize and have better optical properties as the surfaces have a better flatness. The wide pore size distribution is not an obstacle for studying homogeneous cavitation as long as the diameter of the critical nucleus is much smaller than the mean transverse pore size, a condition which is met for 85% porosity sample except in the very close vicinity of the bulk critical point. Experiments are currently in progress with the aim to obtain a more precise understanding why, as seen in Fig. 2, the cavitation pressure is shifted with respect to the CNT prediction at low temperature.

FIG. 1 .

 1 FIG.1. Normalized nitrogen optical isotherms at 77.8 K for three porous silicon samples of 70% porosity. Blue: l = 1µm -red: l = 5µm -black: l = 20µm -green: l = 40µm (see TableIfor the respective mouth reduction). The inset shows the secondary evaporation event around 0.48Psat which we ascribe to homogeneous cavitation.

FIG. 2 .

 2 FIG.2. Normalized nitrogen optical isotherms for a "short pore" sample 2 µm thick (85 % porosity, 20 ALD cycles).

FIG. 7 .

 7 FIG. 7. Fourier transform of the reflected light spectrum during N2 evaporation at 77.8 K for a "long pore" sample (porosity 70%, thickness 20 µm and 55 ALD layers). A saturated sample just before evaporation (dashed line), B during evaporation (red) and C at the end of evaporation (black solid line). Inset: corresponding optical isotherm with points A, B, C.

FIG. 8 .

 8 FIG.8. Successive images of a "long pore" sample during the main evaporation event with hexane (same sample as the one used for4 He experiments, T = 18 • C). Using the small difference (about 1%) of effective optical indices, hence reflectivities, between the filled and empty porous layer, contrast enhancing reveals the filled parts of the sample (bright) and the empty ones (dark). The pressure relative to Psat is 0.58 in the middle of the evaporation step and changes by 7 • 10 -4 from one image to the next.

  porosity d (nm) D eff (nm) T * (K) d eff (nm)

FIG. 10 .

 10 FIG.10. Effect of successive alumina deposition on nitrogen isotherms at 77.4 K for a 85% porosity sample ( d = 50 nm, l = 20 µm). Whatever the alumina thickness, hence the pore aperture reduction, the steep evaporation step occurs at PV /Psat 0.8, corresponding to an effective diameter d eff = 12 nm. For 160 ALD cycles, the extent in L of the hysteresis loop is much reduced, implying that most of the pores are obstructed.

ACKNOWLEDGMENTS

We thank J. Puibasset for stimulating discussions, I. Vickridge for careful reading of the manuscript, the INSP workshop for designing the high pressure cell for the volumetric nitrogen experiment, the LPENS workshop for designing and building the entire high-pressure lowtemperature optical setup, and the NEEL workshops for designing and machining parts of the helium set-up. This work, including the funding of M. B and F. S., has been financially supported by Agence Nationale de la Recherche through the project CavConf, ANR-17-CE30-0002.

SUPPORTING INFORMATION

Calibration of ALD deposition. Additional details on the determination of optical and volumetric isotherms for nitrogen and helium. Additionnal experimental data for native porous silicon samples. Determination of the helium cavitation pressure from volumetric isotherms.