# Critical behaviors in contagion dynamics

## TL;DR

This paper analyzes the critical behavior of a general contagion model with nodes switching states due to spontaneous, neighbor-induced, and reverse transitions, revealing three universal regimes and deepening the mathematical understanding of contagion dynamics.

## Contribution

It provides a unifying mean-field theory that classifies the critical regimes of complex contagion dynamics, resolving a long-standing debate.

## Key findings

- Identifies three universal regimes: uncorrelated, contact process, and cusp catastrophes.
- Derives a mean-field theory that unifies different contagion behaviors.
- Deepens the mathematical understanding of contagion phase transitions.

## Abstract

We study the critical behavior of a general contagion model where nodes are either active (e.g. with opinion A, or functioning) or inactive (e.g. with opinion B, or damaged). The transitions between these two states are determined by (i) spontaneous transitions independent of the neighborhood, (ii) transitions induced by neighboring nodes and (iii) spontaneous reverse transitions. The resulting dynamics is extremely rich including limit cycles and random phase switching. We derive a unifying mean-field theory. Specifically, we analytically show that the critical behavior of systems whose dynamics is governed by processes (i-iii) can only exhibit three distinct regimes: (a) uncorrelated spontaneous transition dynamics (b) contact process dynamics and (c) cusp catastrophes. This ends a long-standing debate on the universality classes of complex contagion dynamics in mean-field and substantially deepens its mathematical understanding.