Community detection in temporal multilayer networks, with an application to correlation networks

TL;DR

This paper develops a multilayer modularity maximization framework for detecting communities in temporal networks, with a focus on financial correlation networks, and introduces a measure for community persistence over time.

Contribution

It clarifies the role of null models in modularity, formulates a multilayer community detection problem, and proposes a persistence diagnostic for temporal networks.

Findings

Multilayer modularity maximization captures trade-offs between static and persistent community structures.

The proposed diagnostic measures the persistence of communities across layers.

Discussion of computational challenges and mitigation strategies for the Louvain heuristic.

Abstract

Networks are a convenient way to represent complex systems of interacting entities. Many networks contain "communities" of nodes that are more densely connected to each other than to nodes in the rest of the network. In this paper, we investigate the detection of communities in temporal networks represented as multilayer networks. As a focal example, we study time-dependent financial-asset correlation networks. We first argue that the use of the "modularity" quality function---which is defined by comparing edge weights in an observed network to expected edge weights in a "null network"---is application-dependent. We differentiate between "null networks" and "null models" in our discussion of modularity maximization, and we highlight that the same null network can correspond to different null models. We then investigate a multilayer modularity-maximization problem to identify communities in temporal networks. Our multilayer analysis only depends on the form of the maximization problem and not on the specific quality function that one chooses. We introduce a diagnostic to measure \emph{persistence} of community structure in a multilayer network partition. We prove several results that describe how the multilayer maximization problem measures a trade-off between static community structure within layers and larger values of persistence across layers. We also discuss some computational issues that the popular "Louvain" heuristic faces with temporal multilayer networks and suggest ways to mitigate them.