Troubles with community detection in networks: No ground truth, no free lunch, and the complex coupling of metadata with structure.

The detection of communities within a dynamic network is a common means for obtaining a coarsegrained view of a complex system and for investigating its underlying processes. While a number of methods have been proposed in the machine learning and physics literature, we lack a theoretical analysis of their strengths and weaknesses, or of the ultimate limits on when communities can be detected. Here, we study the fundamental limits of detecting community structure in dynamic networks. Specifically, we analyze the limits of detectability for a dynamic stochastic block model where nodes change their community memberships over time, but where edges are generated independently at each time step. Using the cavity method, we derive a precise detectability threshold as a function of the rate of change and the strength of the communities. Below this sharp threshold, we claim that no efficient 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 threshold. The first uses belief propagation, which gives asymptotically optimal accuracy, and the second is a fast spectral clustering algorithm, based on linearizing the belief propagation equations. These results extend our understanding of the limits of community detection in an important direction, and introduce new mathematical tools for similar extensions to networks with other types of auxiliary information.

SignificanceA central theme of network science is the heterogeneity present in real-life systems, for instance through the absence of a characteristic degree for the nodes. Despite their small-worldness, networks may present other types of heterogeneous patterns, with different parts of the network exhibiting different behaviors. Here we focus on assortativity, a network analogue of correlation used to describe how the presence and absence of edges covaries with the properties of nodes. We design a method to characterize the heterogeneity and local variations of assortativity within a network and exhibit, in a variety of empirical data, rich mixing patterns that would be obscured by summarizing assortativity with a single statistic.