Network entropy analysis using the Maxwell-Boltzmann partition function

Jianjia Wang, Richard C. Wilson, Edwin R. Hancock

Abstract: In this paper, we use the Maxwell-Boltzmann partition function to compute network entropy. The partition function is used to model the energy level population statistics where the network is in thermodynamic equilibrium with a heat-bath. Here the network Hamiltonian operator defines a set of energy levels occupied by particles in thermal equilibrium. These energy levels are given by the eigenvalues of the normalized Laplacian matrix. In other words, we investigate a thermalised version of the system normally studied in spectral graph theory, where the thermalisation accounts for noise in the system. We provide a systematic study of the entropy resulting from this characterization. Compared to previous work based on using von Neumann network entropy, this thermodynamic quantity is effective in characterizing changes of network structure and distinguishing different types of network models (e.g. Erdős-Rényi random graphs, small world networks, and scale free networks). Numerical experiments on real world data-sets are presented to evaluate the qualitative and quantitative differences in performance.