Network Edge Entropy from Maxwell-Boltzmann Statistics

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

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 Maxwell-Boltzmann 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.