Loops, not groups: Long cycles are responsible for discontinuous phase transitions in higher-order network contagions

TL;DR

This paper introduces a self-consistent model that incorporates local group structures and global cycles to explain how long cycles, rather than just group effects, cause discontinuous phase transitions in complex contagion processes on networks.

Contribution

It presents a novel self-consistent approach that accounts for global cycles and local group effects, clarifying their distinct roles in phase transitions of contagion models.

Findings

Long cycles are responsible for discontinuous phase transitions.

Group effects alone lead to continuous transitions.

The model distinguishes multiple exposures from different transmission chains.

Abstract

We study a self-consistent approach to introduce higher-order effects in a branching process model of complex contagion on clustered networks. Branching processes operate over an infinite population such that they never circle back and interact with previously exposed parts of the system. This infinite, treelike, structure makes it tricky to account for complex contagion mechanisms such as group effects, peer pressure, or social reinforcement where multiple exposures interact in synergistic ways. Here we present a self-consistent solution that accounts for local group structure and global cycles where the process can feedback on itself. This allows us to distinguish multiple exposures that stem from a single transmission chain, from those occurring at the intersection of different transmission chains. We find that only the latter mechanism can give rise to a discontinuous phase transition in the size of global cascades, which is a defining feature of complex contagions. Group effects alone, without long cycles, produce standard continuous phase transitions.