Analytically solvable processes on networks

TL;DR

This paper introduces a new class of analytically solvable network processes that generalize basic models like random walks and consensus, allowing for diverse node dynamics and complex network structures with explicit equilibrium solutions.

Contribution

It presents a broad, analytically solvable framework for network processes that accommodates heterogeneous node dynamics and arbitrary finite graphs, extending previous models.

Findings

Explicit solutions for equilibrium behavior in complex networks

Model flexibility with heterogeneous node dynamics

Decomposability of the process based on network topology

Abstract

We introduce a broad class of analytically solvable processes on networks. In the special case, they reduce to random walk and consensus process - two most basic processes on networks. Our class differs from previous models of interactions (such as stochastic Ising model, cellular automata, infinite particle system, and voter model) in several ways, two most important being: (i) the model is analytically solvable even when the dynamical equation for each node may be different and the network may have an arbitrary finite graph and influence structure; and (ii) in addition, when local dynamic is described by the same evolution equation, the model is decomposable: the equilibrium behavior of the system can be expressed as an explicit function of network topology and node dynamics