Thermodynamics of time evolving networks

Abstract

In this paper, we present a novel and effective method for better understanding the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure. We commence from the spectrum of the normalized Laplacian of a network. We show that by defining the normalized Laplacian eigenvalues as the microstate occupation probabilities of a complex system, the recently developed von Neumann entropy can be interpreted as the thermodynamic entropy of the network. Then, we give an expression for the internal energy of a network and derive a formula for the network temperature as the ratio of change of entropy and change in energy. We show how these thermodynamic variables can be computed in terms of node degree statistics for nodes connected by edges. We apply the thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in evolving network evolution.