Trajectories entropy in dynamical graphs with memory

TL;DR

This paper explores how non-local graph entropy can characterize evolving dynamical graphs, including scale-free networks and memristive circuits, revealing their structural and self-organizing properties under different conditions.

Contribution

It introduces an entropy measure based on Markov diffusion for dynamical graphs and applies it to reinforcement-decay models and memristive circuits, highlighting their structural features.

Findings

Node entropy characterizes network structure in dynamical graphs.

Entropy based on forward probability suffices for DC memristive circuits.

Both forward and backward probabilities are needed for AC memristive circuits.

Abstract

In this paper we investigate the application of non-local graph entropy to evolving and dynamical graphs. The measure is based upon the notion of Markov diffusion on a graph, and relies on the entropy applied to trajectories originating at a specific node. In particular, we study the model of reinforcement-decay graph dynamics, which leads to scale free graphs. We find that the node entropy characterizes the structure of the network in the two parameter phase-space describing the dynamical evolution of the weighted graph. We then apply an adapted version of the entropy measure to purely memristive circuits. We provide evidence that meanwhile in the case of DC voltage the entropy based on the forward probability is enough to characterize the graph properties, in the case of AC voltage generators one needs to consider both forward and backward based transition probabilities. We provide also evidence that the entropy highlights the self-organizing properties of memristive circuits, which re-organizes itself to satisfy the symmetries of the underlying graph.