Information Dimension of Stochastic Processes on Networks: Relating Entropy Production to Spectral Properties

TL;DR

This paper establishes a relationship between the entropy growth of stochastic processes on networks and the spectral properties of their governing matrices, introducing an information dimension concept that characterizes network dynamics.

Contribution

It derives a novel relation linking entropy growth to spectral density and introduces the stochastic information dimension for network processes.

Findings

Entropy growth is proportional to log(time) in scaling spectral density cases.

The spectral density influences the rate of information loss in network processes.

Examples confirm the theoretical relation on regular and fractal networks.

Abstract

We consider discrete stochastic processes, modeled by classical master equations, on networks. The temporal growth of the lack of information about the system is captured by its non-equilibrium entropy, defined via the transition probabilities between different nodes of the network. We derive a relation between the entropy and the spectrum of the master equation's transfer matrix. Our findings indicate that the temporal growth of the entropy is proportional to the logarithm of time if the spectral density shows scaling. In analogy to chaos theory, the proportionality factor is called (stochastic) information dimension and gives a global characterization of the dynamics on the network. These general results are corroborated by examples of regular and of fractal networks.