Thermodynamic Network Analysis with Quantum Spin Statistics

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

Abstract: In this paper, we explore the thermodynamic analysis of networks using a heat-bath analogy and different choices of quantum spin statistics for the occupation of energy levels defined by the network. We commence from the set of energy states given by the eigenvalues of the normalized Laplacian matrix, which plays the role of the Hamiltonian operator of the network. We explore a heat bath analogy in which the network is in thermodynamic equilibrium with a heat-bath and its energy levels are occupied by either indistinguishable bosons or fermions obeying the Pauli exclusion principle. To compute thermodynamic characterization of this system, i.e. the entropy and energy, we analyse the partition functions relevant to Bose-Einstein and Fermi-Dirac statistics. At high temperatures, the effects of quantum spin statistics are disrupted by thermalisation and correspond to the classical Maxwell-Boltzmann case. However, at low temperatures the Bose-Einstein system condenses into a state where the particles occupy the lowest energy state, while in the Fermi-Dirac system there is only one particle per energy state. These two models produce quite different entropic characterizations of network structure, which are appropriate to different types of structure. We experiment with the two different models on both synthetic and real world imagery, and compare and contrast their performance.