Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs

Guillaume St-Onge,Laurent Hébert-Dufresne,Alain Barrat,Antoine Allard,Giovanni Petri,Iacopo Iacopini,Vito Latora

doi:10.1038/s42005-021-00788-w

Abstract

Contagion phenomena are often the results of multibody interactions—such as superspreading events or social reinforcement—describable as hypergraphs. We develop an approximate master equation framework to study contagions on hypergraphs with a heterogeneous structure in terms of group size (hyperedge cardinality) and of node membership (hyperdegree). By mapping multibody interactions to nonlinear infection rates, we demonstrate the influence of large groups in two ways. First, we characterize the phase transition, which can be continuous or discontinuous with a bistable regime. Our analytical expressions for the critical and tricritical points highlight the influence of the first three moments of the membership distribution. We also show that heterogeneous group sizes and nonlinear contagion promote a mesoscopic localization regime where contagion is sustained by the largest groups, thereby inhibiting bistability. Second, we formulate an optimal seeding problem for hypergraph contagion and compare two strategies: allocating seeds according to node or group properties. We find that, when the contagion is sufficiently nonlinear, groups are more effective seeds than individual hubs.

Highlights

Contagion phenomena are often the results of multibody interactions—such as superspreading events or social reinforcement—describable as hypergraphs
We look at the phase transition for a regular hypergraph with fixed group size, pn = δn,[4], and a perturbed version of it, where we introduce a small proportion of larger groups, pn = (1 − ε)δn,4 + εδn,[15] with ε = 10−3
We have introduced group-based approximate master equations (AMEs) to describe hypergraph contagions

Summary

Introduction

Contagion phenomena are often the results of multibody interactions—such as superspreading events or social reinforcement—describable as hypergraphs. More generally hypergraph contagions, a susceptible individual can become infected because of a multibody process, i.e., through exposure to an infectious group[31] In this way, the node is simultaneously exposed to the state of the entire group, whose effect can be interpreted as a mechanism of social reinforcement[33]. Most approaches follow a heterogeneous mean-field (HMF) framework in which nodes are divided into hyperdegree classes These descriptions are analytically tractable, but do not consider the details of the structure and ignore the dynamical correlations within groups, which are especially important for hypergraph contagions since multibody interactions naturally reinforce these correlations

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Conclusion