Abstract
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.
Highlights
Many complex systems can be represented as networks, that is, as a set of elements characterized by pairwise interactions
We measure the accuracy of the inferred labels by the overlap between the true assignment gà and the inferred one g. This is the fraction of nodes labeled correctly, averaged over all nodes and all times, normalized so that it is 1 if g 1⁄4 gà and 0 if gis uniformly random. (To break the permutation symmetry, we maximize over all k! permutations of the groups.) In Fig. 2, we show the overlap obtained by belief propagation (BP) for dynamic networks as a function of ε for several choices of η, with n 1⁄4 512, T 1⁄4 40, k 1⁄4 2, and c 1⁄4 16
We derive a mathematically precise understanding of the limits of detectability for communities in dynamic networks, under a model in which group memberships are correlated over time but where the edges at each time are generated independently
Summary
Many complex systems can be represented as networks, that is, as a set of elements characterized by pairwise interactions. Approaches for detecting communities in multiplex networks are relevant [36,37,38,39,40,41,42,43,44,45], as dynamic networks are a special case of multiplex networks, in which the layers are organized in a linear sequence Despite these varied efforts, up to now we have lacked a theoretical understanding of the optimality of these techniques, when or how they tend to fail, or whether there are fundamental limits to detecting community structure in dynamic networks. Consider a network of phone calls or emails where the autocorrelation time governing conversations (or sequences of successive calls) is on the order of days, but where each network snapshot aggregates these calls over a month In this case, belief propagation (BP) algorithms, like the ones we develop here, are often asymptotically optimal, and we may use the cavity method to compute the detectability threshold exactly. We confirm our theoretical calculations by showing that these algorithms accurately recover the true community structure in dynamic networks all the way down to the generalized detectability threshold
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