June 20, 2015
We study the fundamental limits on learning latent community structure in dynamic networks. Specifically, we study dynamic stochastic block models where nodes change their community membership over time, but where edges are generated independently at each time step. In this setting (which is a special case of several existing models), we are able to derive the detectability threshold exactly, as a function of the rate of change and the strength of the communities. Below this threshold, we claim that no algorithm can identify the communities better than chance. We then give two algorithms that are optimal in the sense that they succeed all the way down to this limit. The first uses belief propagation (BP), which gives asymptotically optimal accuracy, and the second is a fast spectral clustering algorithm, based on linearizing the BP equations. We verify our analytic and algorithmic results via numerical simulation, and close with a brief discussion of extensions and open questions.
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November 4, 2019
Networks are useful representations of many systems with interacting entities, such as social, biological and physical systems. Characterizing the meso-scale organization, i.e. the community structure, is an important problem in network science. Community detection aims to partition the network into sets of nodes that are densely connected internally but sparsely connected to other dense sets of nodes. Current work on community detection mostly focuses on static networks. How...
June 9, 2021
The community detection problem requires to cluster the nodes of a network into a small number of well-connected "communities". There has been substantial recent progress in characterizing the fundamental statistical limits of community detection under simple stochastic block models. However, in real-world applications, the network structure is typically dynamic, with nodes that join over time. In this setting, we would like a detection algorithm to perform only a limited num...
We study the inference of a model of dynamic networks in which both communities and links keep memory of previous network states. By considering maximum likelihood inference from single snapshot observations of the network, we show that link persistence makes the inference of communities harder, decreasing the detectability threshold, while community persistence tends to make it easier. We analytically show that communities inferred from single network snapshot can share a ma...
June 3, 2020
This article considers the problem of community detection in sparse dynamical graphs in which the community structure evolves over time. A fast spectral algorithm based on an extension of the Bethe-Hessian matrix is proposed, which benefits from the positive correlation in the class labels and in their temporal evolution and is designed to be applicable to any dynamical graph with a community structure. Under the dynamical degree-corrected stochastic block model, in the case ...
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...
July 25, 2017
Dynamic networks, especially those representing social networks, undergo constant evolution of their community structure over time. Nodes can migrate between different communities, communities can split into multiple new communities, communities can merge together, etc. In order to represent dynamic networks with evolving communities it is essential to use a dynamic model rather than a static one. Here we use a dynamic stochastic block model where the underlying block model i...
February 7, 2020
In this paper, we analyse classical variants of the Spectral Clustering (SC) algorithm in the Dynamic Stochastic Block Model (DSBM). Existing results show that, in the relatively sparse case where the expected degree grows logarithmically with the number of nodes, guarantees in the static case can be extended to the dynamic case and yield improved error bounds when the DSBM is sufficiently smooth in time, that is, the communities do not change too much between two time steps....
April 26, 2019
We discuss a variant of `blind' community detection, in which we aim to partition an unobserved network from the observation of a (dynamical) graph signal defined on the network. We consider a scenario where our observed graph signals are obtained by filtering white noise input, and the underlying network is different for every observation. In this fashion, the filtered graph signals can be interpreted as defined on a time-varying network. We model each of the underlying netw...
December 7, 2018
We consider the problem of estimating the location of a single change point in a dynamic stochastic block model. We propose two methods of estimating the change point, together with the model parameters. The first employs a least squares criterion function and takes into consideration the full structure of the stochastic block model and is evaluated at each point in time. Hence, as an intermediate step, it requires estimating the community structure based on a clustering algo...
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...