Minimising entropy changes in dynamic network evolution

The modelling of time-varying network evolution is critical to understanding the function of complex systems. The key to such models is a variational principle. In this paper we explore how to use the Euler-Lagrange equation to investigate the variation of entropy in time evolving networks. We commence from recent work where the von Neumman entropy can be approximated using simple degree statistics, and show that the changes in entropy in a network between different time epochs are determined by correlations in the changes in degree statistics of nodes connected by edges. Our variational principle is that the evolution of the structure of the network minimises the change in entropy with time. Using the Euler-Lagrange equation we develop a dynamic model for the evolution of node degrees. We apply our model to a time sequence of networks representing the evolution of stock prices on the New York Stock Exchange (NYSE). Our model allows us to understand periods of stability and instability in stock prices, and to predict how the degree distribution evolves with time. We show that the framework presented here provides allows accurate simulation of the time variation of degree statistics, and also captures the topological variations that take place when the structure of a network changes violently.