From Signed Networks to Group Graphs

TL;DR

This paper introduces 'group graphs' that generalize signed networks by allowing links labeled with any group, and shows that for balanced group graphs, the network topology determines the process evolution, unifying various network theories.

Contribution

It extends signed networks to general group labels, unifies existing theories, and explores the implications for network dynamics and symmetry-driven modeling.

Findings

Time evolution depends only on topology for balanced group graphs.

Unifies and extends theories of signed, voltage, and gain graphs.

Discusses applications in network dynamics and symmetry modeling.

Abstract

I define a "group graph" which encodes the symmetry in a dynamical process on a network. Group graphs extend signed networks, where links are labelled with plus or minus one, by allowing link labels from any group and generalising the standard notion of balance. I show that for processes on a balanced group graph the time evolution is completely determined by the network topology, not by the group structure. This unifies and extends recent findings on signed networks (Tian \& Lambiotte, 2024a) and complex networks (Tian \& Lambiotte, 2024b). I will also relate the results discussed here to related work such as the "group graph" of Harary (1982), a "voltage graph" (Gross, 1974) and a "gain graph" (Zaslavsky 1989). Finally, I will review some promising applications for network dynamics and symmetry-driven modelling including status, edges with a zero label, weak balance, unbalanced group graphs and using monoids.