Directed Graph Evolution from Euler-Lagrange Dynamics

In this paper, we develop a variational principle from the von Neumann entropy for directed graph evolution. We minimise the change of entropy over time to investigate how directed networks evolve under the Euler-Lagrange equation. We commence from our recent work in which we show how to compute the approximate von Neumann entropy for a directed graph based on simple in and out degree statistics. To formulate our variational principle we commence by computing the directed graph entropy difference between different time epochs. This is controlled by the ratios of the in-degree and out-degrees at the two nodes forming a directed edge. It also reveals how the entropy change is related to correlations between the changes in-degree ratio and in-degree, and their initial values. We conduct synthetic experiments with three widely studied complex network models, namely Erdos-Renyi random graphs, Watts-Strogatz small-world networks, and Barabasi-Albert scale-free networks, to simulate the in-degree and out-degree distribution. Our model effectively captures the directed structural transitions in the dynamic network models. We also apply the method to the real-world financial networks. These networks reflect stock price correlations on the New York Stock Exchange(NYSE) and can be used to characterise stable and unstable trading periods. Our model not only effectively captures how the directed network structure evolves with time, but also allows us to detect periods of anomalous network behaviour.