Spin statistics, partition functions and network entropy

Abstract

This article explores the thermodynamic characterization of networks using the heat bath analogy when the energy states are occupied under different spin statistics, specified by a partition function. Using the heat bath analogy and a matrix characterization for the Hamiltonian operator, we consider the cases where the energy states are occupied according to Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. We derive expressions for thermodynamic variables, such as entropy, for the system with particles occupying the energy states given by the normalized Laplacian eigenvalues. The chemical potential determines the number of particles at a given temperature. We provide the systematic study of the entropic measurements for network complexity resulting from the different partition functions and specifically those associated with alternative assumptions concerning the spin statistics. Compared with the network von Neumann entropy corresponding to the normalized Laplacian matrix, these entropies are effective in characterizing the significant structural configurations and distinguishing different types of network models (Erdos-Rényi random graphs, Watts-Strogatz small world networks and Barabási-Albert scale-free networks). The effect of the spin statistics is (a) in the case of bosons to allow the particles in the heat bath to congregate in the lower energy levels and (b) in the case of fermions to populate higher energy levels. With normalized Laplacian energy states, this means that bosons are more sensitive to the spectral gap and hence to cluster or community structure, and fermions better sample the distribution of path lengths in a network. Numerical experiments for synthetic and real-world data sets are presented to evaluate the qualitative and quantitative differences of the thermodynamic network characterizations derived from the different occupation statistics, and these confirm these qualitative intuitions.