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.
A parallel line of work studied the problem. In this case, the estimated community structure is only required to be asymptotically positively correlated with the ground truth. For this requirement, two independent groups @cite_44 @cite_24 proved that detection is solvable (and so efficiently) if and only if @math , when @math , @math . This settles a conjecture made in @cite_13 and improves on earlier work @cite_21 . Results for detection with more than two communities were recently obtained in @cite_37 @cite_28 @cite_49 @cite_30 . A variant of community detection with a single hidden community in a sparse graph was studied in @cite_23 .
@cite_30 @cite_37 @cite_28 @cite_21 @cite_44 @cite_24 @cite_49 @cite_23 @cite_13
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We consider the problem of community detection in the Stochastic Block Model with a finite number @math of communities of sizes linearly growing with the network size @math . This model consists in a random graph such that each pair of vertices is connected independently with probability @math within communities and @math across communities. One observes a realization of this random graph, and the objective is to reconstruct the communities from this observation. We show that under spectral algorithms, the number of misclassified vertices does not exceed @math with high probability as @math grows large, whenever @math , @math and where @math and @math denote the (fixed) proportions of vertices in the two smallest communities. In view of recent work by and , this establishes that the proposed spectral algorithms are able to exactly recover communities whenever this is at all possible in the case of networks with two communities with equal sizes. We conjecture that condition (1) is actually necessary to obtain less than @math misclassified vertices asymptotically, which would establish the optimality of spectral method in more general scenarios. Community detection in graphs is the problem of finding groups of vertices which are more densely connected than they are to the rest of the graph. This problem has a long history, but it is currently motivated by social and biological networks. While there are many ways to formalize it, one of the most popular is as an inference problem, where there is a planted “ground truth” community structure around which the graph is generated probabilistically. Our task is then to recover the ground truth knowing only the graph. Recently it was discovered, first h euristically i n p hysics a nd then rigorously in probability and computer science, that this problem has a phase transition at which it suddenly becomes impossible. Namely, if the graph is too sparse, or the probabilistic process that generates it is too noisy, then no algorithm can find a partition t hat i s correlated with the planted one—or even tell if there are communities, i.e., distinguish the graph from a purely random one with high probability. Above this information-theoretic threshold, there is a second threshold beyond which polynomial-time algorithms are known to succeed; in between, there is a regime in which community detection is possible, but conjectured to be exponentially hard. For computer scientists, this field offers a wealth of new ideas and open questions, with connections to probability and combinatorics, message-passing algorithms, and random matrix theory. Perhaps more importantly, it provides a window into the cultures of statistical physics and statistical inference, and how those cultures think about distributions of instances, landscapes of solutions, and hardness. We consider the problem of community detection from observed interactions between individuals, in the context where multiple types of interaction are possible. We use labelled stochastic block models to represent the observed data, where labels correspond to interaction types. Focusing on a two-community scenario, we conjecture a threshold for the problem of reconstructing the hidden communities in a way that is correlated with the true partition. To substantiate the conjecture, we prove that the given threshold correctly identifies a transition on the behaviour of belief propagation from insensitive to sensitive. We further prove that the same threshold corresponds to the transition in a related inference problem on a tree model from infeasible to feasible. Finally, numerical results using belief propagation for community detection give further support to the conjecture. Detecting community structures in graphs is a well studied problem in graph data analytics. Unprecedented growth in graph structured data due to the development of the world wide web and social networks in the past decade emphasizes the need for fast graph data analytics techniques. In this paper we present a simple yet efficient approach to detect communities in large scale graphs by modifying the sequential Louvain algorithm for community detection. The proposed distributed memory parallel algorithm targets the costly first iteration of the initial method by parallelizing it. Experimental results on a MPI setup with 128 parallel processes shows that up to ≈5× performance improvement is achieved as compared to the sequential version while not compromising the correctness of the final result.