In this paper, we present a novel way to summarize the structure of large graphs, based on non-parametric estimation of edge density in directed multigraphs. Following coclustering approach, we use a clustering of the vertices, with a piecewise constant estimation of the density of the edges across the clusters, and address the problem of automatically and reliably inferring the number of clusters, which is the granularity of the coclustering. We use a model selection technique with data-dependent prior and obtain an exact evaluation criterion for the posterior probability of edge density estimation models. We demonstrate, both theoretically and empirically, that our data-dependent modeling technique is consistent, resilient to noise, valid non asymptotically and asymptotically behaves as an universal approximator of the true edge density in directed multigraphs. We evaluate our method using artificial graphs and present its practical interest on real world graphs. The method is both robust and scalable. It is able to extract insightful patterns in the unsupervised learning setting and to provide state of the art accuracy when used as a preparation step for supervised learning.
More expressive graph models aim at searching a partition of both the source and target vertices into clusters, with different types of interaction between clusters. The cross-product of the two partitions of vertices form a partition of the edges into or . This modeling approach is called blockmodeling and has been thoroughly studied for decades. In early approaches @cite_40 @cite_14 , non-stochastic blocks are considered, with a focus on predefined types of block patterns. The blockmodel is searched either indirectly using a (dis)similarity measure between pairs of vertices and then applying a standard clustering algorithm, or directly by optimizing an ad hoc function measuring the fit of real blocks to the corresponding predefined types of blocks. The limit of these approaches is that they do not cope with the stochastic nature of many real world datasets.
@cite_40 @cite_14
2508909698 2116061856 2206841333 1825700702
We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph @math from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and --in particular-- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. Given a background graph with n vertices, the binary censored block model assumes that vertices are partitioned into two clusters, and every edge is labeled independently at random with labels drawn from Bern(1 − e) if two endpoints are in the same cluster, or from Bern(e) otherwise, where e ∈ [0; 1 2] is a fixed constant. For Erdős-Renyi graphs with edge probability p = a log n n and fixed a, we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold equation for exactly recovering the partition from the labeled graph with probability tending to one as n → ∞. For random regular graphs with degree scaling as a log n, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold aD(Bern(1 2)∥Bern(e)) > 1, where D denotes the Kullback-Leibler divergence. The binary symmetric stochastic block model deals with a random graph of @math vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability @math within clusters and @math across clusters. In the asymptotic regime of @math and @math for fixed @math and @math , we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe14 . Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to @math .