Spin models on random graphs with controlled topologies beyond degree constraints

TL;DR

This paper introduces a method to study Ising spin models on random graphs with customizable topologies, allowing for more precise modeling of real-world systems by controlling both degree distribution and edge attachment preferences.

Contribution

It develops a solvable model for spin systems on graphs with tunable topology beyond degree constraints, using replica theory and a new edge attachment function Q(k,k').

Findings

Phase diagrams depend on degree distribution and attachment function Q(k,k')

Enhanced control over graph topology improves modeling of real-world systems

Model solvable via replica symmetric equilibrium theory

Abstract

We study Ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution $p(k)$ can be chosen arbitrarily, but which allows for further fine-tuning of the topology via preferential attachment of edges on the basis of an arbitrary function Q(k,k') of the degrees of the vertices involved. We solve these models using finite connectivity equilibrium replica theory, within the replica symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution, but also on the choice made for Q(k,k'). The increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real-world interacting particle systems by spin systems on suitably defined random graphs.