Statistical mechanical analysis for unweighted and weighted stock market networks

Financial markets are time-evolving complex systems containing different financial entities, such as banks, corporations and institutions that interact through transactions and respond to external economic and political events. They can be conveniently represented as a network structure. In this paper, we analyse the unweighted and weighted market networks from a statistical mechanical perspective. In particular, we propose a novel thermodynamic analogy to characterise the dynamic structural properties of time-evolving networks. The intricate pattern of edge connections in the network is modelled by using a heat bath analogy in which particles occupy the energy states according to the Boltzmann distribution. According to this analogy the occupation of the energy states is determined by the temperature of the heat bath, and the spectrum of energy states of the network is determined by the number of nodes and edges. For unweighted networks, the binary representation of the elements in the adjacency matrix can be modelled as a statistical ensemble, using the corresponding partition function to compute thermodynamic network characterisations. For weighted networks, on the other hand, the derived thermodynamic quantities together with their distribution of fluctuations identify the salient structure in the network evolution. We conduct experiments on time-evolving stock exchanges using data for the S&P500 Index Stock Exchanges over the past decade. The thermodynamic characterisations provide an excellent framework to identify epochs in which there is significant variance in network structure during financial crises induced by economic and political events.