September 14, 2011
In this paper we extend our previous work on the stochastic block model, a commonly used generative model for social and biological networks, and the problem of inferring functional groups or communities from the topology of the network. We use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram. We describe in detail properties of the detectability/undetectability phase transition and the easy/hard phase transition for the community detection problem. Our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way, i.e., that maximizes the overlap with the underlying group memberships, and learning the underlying parameters of the block model. Finally, we apply the algorithm to two examples of real-world networks and discuss its performance.
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February 6, 2011
We present an asymptotically exact analysis of the problem of detecting communities in sparse random networks. Our results are also applicable to detection of functional modules, partitions, and colorings in noisy planted models. Using a cavity method analysis, we unveil a phase transition from a region where the original group assignment is undetectable to one where detection is possible. In some cases, the detectable region splits into an algorithmically hard region and an ...
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Modularity is a popular measure of community structure. However, maximizing the modularity can lead to many competing partitions, with almost the same modularity, that are poorly correlated with each other. It can also produce illusory "communities" in random graphs where none exist. We address this problem by using the modularity as a Hamiltonian at finite temperature, and using an efficient Belief Propagation algorithm to obtain the consensus of many partitions with high mo...
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Real-world networks usually have community structure, that is, nodes are grouped into densely connected communities. Community detection is one of the most popular and best-studied research topics in network science and has attracted attention in many different fields, including computer science, statistics, social sciences, among others. Numerous approaches for community detection have been proposed in literature, from ad-hoc algorithms to systematic model-based approaches. ...
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We take a whirlwind tour of problems and techniques at the boundary of computer science and statistical physics. We start with a brief description of P, NP, and NP-completeness. We then discuss random graphs, including the emergence of the giant component and the k-core, using techniques from branching processes and differential equations. Using these tools as well as the second moment method, we give upper and lower bounds on the critical clause density for random k-SAT. We ...
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Community structure is one of the most relevant features encountered in numerous real-world applications of networked systems. Despite the tremendous effort of scientists working on this subject over the past few decades to characterize, model, and analyze communities, more investigations are needed to better understand the impact of community structure and its dynamics on networked systems. Here, we first focus on generative models of communities in complex networks and thei...
February 1, 2017
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 undergoing a resurgence of interest due to the need to analyze 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 "ground truth" community structure built into the graph somehow. The task is then ...
March 29, 2017
The stochastic block model (SBM) is a random graph model with different group of vertices connecting differently. It is widely employed as a canonical model to study clustering and community detection, and provides a fertile ground to study the information-theoretic and computational tradeoffs that arise in combinatorial statistics and more generally data science. This monograph surveys the recent developments that establish the fundamental limits for community detection in...
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, p...
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Predicting labels of nodes in a network, such as community memberships or demographic variables, is an important problem with applications in social and biological networks. A recently-discovered phase transition puts fundamental limits on the accuracy of these predictions if we have access only to the network topology. However, if we know the correct labels of some fraction $\alpha$ of the nodes, we can do better. We study the phase diagram of this "semisupervised" learning ...
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We prove the existence of an asymptotic phase transition threshold on community detectability for the spectral modularity method [M. E. J. Newman, Phys. Rev. E 74, 036104 (2006) and Proc. National Academy of Sciences. 103, 8577 (2006)] under a stochastic block model. The phase transition on community detectability occurs as the inter-community edge connection probability $p$ grows. This phase transition separates a sub-critical regime of small $p$, where modularity-based comm...