Network edge entropy decomposition with spin statistics

In a previous study, we have explored how to decompose the global entropy of a network into edge components using a graph-spectral decomposition technique. Here, we develop this work in more depth to understand the role of edge entropy as an efficient and effective tool in analysing network structure. We use the edge entropy distribution as a network feature or characterisation and combine it with linear discriminant analysis to distinguish different types of network model and structure. Interpreting the normalised Laplacian matrix as the network Hamiltonian (or energy) operator, the network is assumed to be in thermodynamic equilibrium with a heat bath where the energy states correspond to the normalised Laplacian eigenvalues. To model the way in which particles occupy the energy states, we explore the use of three different spin-dependent statistical models to determine the thermodynamic entropy of the network. These are a) the classical spinless Maxwell-Boltzmann distribution, and two models based on quantum mechanical spin-statistics, namely b) the Bose-Einstein model for particles with integer spin, and c) the Fermi-Dirac model for particles with half-integer spin. By using the spectral decomposition of the Laplacian, we illustrate how to project out the edge-entropy components from the global network entropy. In this way, the detailed distribution of entropy across the edges of a network can be constructed. Compared to our previous study of the von Neumann edge entropy, where the edge entropy just depends on the degrees of the nodes forming an edge, in the case of the new statistical mechanical model, there is a more subtle dependence of the edge entropy on the structure of a network. We illustrate how this new edge entropy distribution can be used to more effectively identify variations in network structure, in particular for edges incorporating nodes of large degree. Numerical experiments on synthetic and real-world data-sets are presented to evaluate the qualitative and quantitative differences in performance.