Classical random walks over complex networks and network complexity

TL;DR

This paper models classical random walks on complex networks as thermal systems, deriving thermodynamic quantities and proposing a topological entropy measure as a network complexity indicator.

Contribution

It introduces a thermodynamic framework for random walks on complex networks, linking entropy to network topology and complexity, and explores effects of prior resting probabilities.

Findings

Entropy has a topological component increasing with network size.

Average energy exhibits equipartition among nodes.

Topological entropy correlates with network complexity measures.

Abstract

In this paper we view the steady states of classical random walks over complex networks with an arbitrary degree distribution as states in thermal equilibrium. By identifying the distribution of states as a canonical ensemble, we are able to define the temperature and the Hamiltonian for the random walk systems. We then calculate the Helmholtz free energy, the average energy, and the entropy for the thermal equilibrium states. The results shows equipartition of energy for the average energy. The entropy is found to consist of two parts. The first part decreases as the number of walkers increases. The second part of the entropy depends solely on the topology of the network, and will increase when more edges or nodes are added to the network. We compare the topological part of entropy with some of the network descriptors and find that the topological entropy could be used as a measure of network complexity. In addition, we discuss the scenario that a walker has a prior probability of resting on the same node at the next time step, and find that the effect of the prior resting probabilities is equivalent to increasing the degree for every node in the network.