Network Edge Entropy from Maxwell-Boltzmann Statistics

Abstract

Abstract. In prior work, we have shown how to compute global network entropy using a heat bath analogy and Maxwell-Boltzmann statistics. In this work, we show how to project out edge-entropy components so that the detailed distribution of entropy across the edges of a network can be computed. This is particularly useful if the analysis of non-homogeneous networks with a strong community as hub structure is being attempted. To commence, we view the normalized Laplacian matrix as the network Hamiltonian operator which specifies a set of energy states with the Laplacian eigenvalues. The network is assumed to be in thermodynamic equilibrium with a heat bath. According to this heat bath analogy, particles can populate the energy levels according to the classical MaxwellBoltzmann distribution, and this distribution together with the energy states determines thermodynamic variables of the network such as entropy and average energy. We show how the entropy can be decomposed into components arising from individual edges using the eigenvectors of the normalized Laplacian. Compared to previous work based on the von Neumann entropy, this thermodynamic analysis is more effective in characterizing changes of network structure since it better represents the edge entropy variance associated with edges connecting nodes of large degree. Numerical experiments on real-world datasets are presented to evaluate the qualitative and quantitative differences in performance.

Thermodynamic Entropy in Quantum Statistics for Stock Market Networks

“…This forced numerous large corporations to restate earnings and reestablished investors' confidence [7]. This considerably altered the interrelationships among stocks and resulted in a significant fluctuation in the structure of the entire market [16].…”

“…Two kinds of partition function in each quantum case provide different occupation statistics for the energy levels. This gives different thermodynamic entropy for each statistical distribution [16]. The qualitative description is that particles in Fermi-Dirac statistics obey the Pauli exclusion principle with only one particle for each energy state.…”

“…We show that the financial crashes are characterised by the presence of welldefined changes to the thermodynamic entropy [13,14], whereas outside these critical periods this characterisation remains stable for long periods. To do this, we make use of 2 Complexity some recent framework in quantum statistics concerning the normalised Laplacian matrix for the construction of partition functions in Bose-Einstein and Fermi-Dirac statistics [15,16].…”