Thermodynamic motif analysis for directed stock market networks

In this paper, we present a novel thermodynamically based analysis method for directed networks, and in particular for time-evolving networks in the finance domain. Based on an analogy with a dilute gas in statistical mechanics, we develop a partition function for a network composed of directed motifs. The method relies on the decomposition of directed networks into a series of frequently occurring graphlets, or motifs. According to the connection between a directed network and the dilute gas, the network motifs have the same topological structure as the low-order interactions between particles in the gas. This means that we can use the so-called cluster expansion from statistical mechanics to develop a partition function for the motif decomposition. In prior work, we have reported a detailed analysis of the cluster expansion for the case of undirected graphs, and showed how the resulting motif entropy can be used to analyse time evolving networks [1]. In this paper we extend this work to the case of directed graphs to compute thermodynamic quantities including energy, entropy and temperature for the directed network. The three thermodynamic quantities constitute the thermodynamic framework for the analysis of directed network evolution. We apply our thermodynamic framework to the financial and biological domains to represent real world complex systems as time-varying directed networks. Experimental results successfully demonstrate the effectiveness of the thermodynamic framework in representing the evolution of directed network structure and anomalous event detection.