Thermodynamic Entropy in Quantum Statistics for Stock Market Networks

Abstract

The stock market is a dynamical system composed of intricate relationships between financial entities, such as banks, corporations, and institutions. Such a complex interactive system can be represented by the network structure. The underlying mechanism of stock exchange establishes a time-evolving network among companies and individuals, which characterise the correlations of stock prices in the time sequential trades. Here, we develop a novel technique in quantum statistics to analyse the financial market evolution. We commence from heat bath analogy where the normalised Laplacian matrix plays the role of the Hamiltonian operator of the network. The eigenvalues of the Hamiltonian specify energy states of the network. These states are occupied by either indistinguishable bosons or fermions with corresponding Bose-Einstein and Fermi-Dirac statistics. Using the relevant partition functions, we develop the thermodynamic entropy to explore dynamic network characterisations. We conduct the experiments to apply this novel method to identify the significant variance in network structure during the financial crisis. The thermodynamic entropy provides an excellent framework to represent the variations taking place in the stock market.

Network geometry and market instability

“…It takes into account the number of communities as well as the size of the communities [ 28 ] to determine the structural entropy, which is then used to continuously monitor the market. The thermodynamical entropy [ 29 ] can also be used to describe the dynamics of stock market networks as it acts like an indicator for the financial system. Very recently, based on the distribution properties of the eigenvector centralities of correlation matrices, Chakraborti & Pharasi [ 30 ] have proposed a computationally cheap yet uniquely defined and non-arbitrary eigen-entropy measure, to show that the financial market undergoes ‘phase separation’ and there exists a new type of scaling behaviour (data collapse) in financial markets.…”

Network edge entropy decomposition with spin statistics

“…Recently, network entropy has attracted increased attention because of its capacity to distinguish the structural properties of different types of networks [1,2,3]. Different varieties of entropy have been extensively used to characterise the salient features of networks, not only in the static domain but also in the domain of time varying or dynamic networks, such as the biological, social and financial networks [4,5,6]. One of the most sophisticated studies involves the von Neumann entropy, which has been successfully used as an effective characterisation to describe the structural properties of random, small-world and scale-free networks [4,7].…”

Weighted Network Analysis Using the Debye Model

“…Recently, sophisticated tools from statistical physics have provided powerful ways to extend this kind of analysis [6,5]. These computationally efficient methods rely on thermodynamic analogies to describe the different structural or topological properties of networks [3]. For example, the Boltzmann distribution provides expressions for the macroscopic thermal characteristics such as temperature, energy and entropy from a microcosmic point of view [7].…”