Cluster Expansion Analysis for Dynamic Networks

Abstract

The structure of networks can be efficiently represented using motifs, which are those subgraphs that recur most frequently. One route to understanding the motif structure of a network is to study the distribution of subgraphs using statistical mechanics. In this paper, we address the use of motifs as network primitives using the cluster expansion from statistical physics. By mapping the network motifs to clusters in the gas model, we derive the partition function for a network and this allows us to calculate global thermodynamic quantities, such as energy and entropy. We present analytical expressions for the numbers of certain types of motifs, and compute their associated entropy. We conduct numerical experiments for synthetic and real-world data-sets and evaluate the qualitative and quantitative characterizations of the motif entropy derived from the partition function. We find that the motif entropy for real-world networks, such as financial stock market networks, is sensitive to the variance in network structure. This is in line with recent evidence that network motifs can be regarded as basic elements with well defined information-processing functions. Our model is capable of detecting abrupt changes or anomalies in network structure and distinguishing different types of time-dependency for different types of anomaly.