Discrete Network Dynamics. Part 1: Operator Theory

TL;DR

This paper introduces an operator algebra framework for modeling discrete network dynamics, enabling the simulation of Markov random fields with properties akin to quantum field theory, and unifies different network theories.

Contribution

It presents a novel operator algebra approach to discrete network dynamics, connecting Markov chain Monte Carlo algorithms with quantum field theory concepts.

Findings

Equivalence between equilibrium behaviors of generalized MRFs and ACEnet.

Framework allows reuse of quantum field theory structures in network dynamics.

Provides a unified view of different network models.

Abstract

An operator algebra implementation of Markov chain Monte Carlo algorithms for simulating Markov random fields is proposed. It allows the dynamics of networks whose nodes have discrete state spaces to be specified by the action of an update operator that is composed of creation and annihilation operators. This formulation of discrete network dynamics has properties that are similar to those of a quantum field theory of bosons, which allows reuse of many conceptual and theoretical structures from QFT. The equilibrium behaviour of one of these generalised MRFs and of the adaptive cluster expansion network (ACEnet) are shown to be equivalent, which provides a way of unifying these two theories.