Directed and undirected network evolution from Euler–Lagrange dynamics

In this paper, we investigate both undirected and directed network evolution using the Euler–Lagrange equation. We use the Euler–Lagrange equation to develop a variational principle based on the von Neumann entropy for time-varying network structure. Commencing from recent work to approximate the von Neumann entropy using simple degree statistics, the changes in entropy between different time epochs are determined by correlations in the degree difference in the edge connections. Our Euler–Lagrange equation minimises the change in entropy and allows to develop a dynamic model to simulate the changes of node degree with time. We first explore the effect of network dynamics on the three widely studied complex network models, namely (a) Erdős-Rényi random graphs, (b) Watts-Strogatz small-world networks, and (c) Barabási-Albert scale-free networks. Our model effectively captures both undirected and directed structural transitions in the dynamic network models. We apply our model to a network time sequence representing the evolution of stock prices on the New York Stock Exchange(NYSE) and sequences of Drosophila gene regulatory networks containing different developmental phases of the organism from embryo to adult. Here we use the model to differentiate between periods of stable and unstable stock price trading and to detect periods of anomalous network evolution. Our experiments show that the presented model not only provides an accurate simulation of the degree statistics in time-varying networks but also captures the topological variations taking place when the structure of a network changes violently.