Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs

Abstract

Several biological and social contagion phenomena, such as superspreading events or social reinforcement, are the results of multi-body interactions, for which hypergraphs offer a natural mathematical description. In this paper, we develop a novel mathematical framework based on approximate master equations to study contagions on random hypergraphs with a heterogeneous structure, both in terms of group size (hyperedge cardinality) and of membership of nodes to groups (hyperdegree). The characterization of the inner dynamics of groups provides an accurate description of the contagion process, without losing the analytical tractability. Using a contagion model where multi-body interactions are mapped onto a nonlinear infection rate, our two main results show how large groups are influential, in the sense that they drive both the early spread of a contagion and its endemic state (i.e., its stationary state). First, we provide a detailed characterization of the phase transition, which can be continuous or discontinuous with a bistable regime, and derive analytical expressions for the critical and tricritical points. We find that large values of the third moment of the membership distribution suppress the emergence of a discontinuous phase transition. Furthermore, the combination of heterogeneous group sizes and nonlinear contagion facilitates the onset of a mesoscopic localization phase, where contagion is sustained only by the largest groups, thereby inhibiting bistability as well. Second, we formulate a simple problem of optimal seeding for hypergraph contagions to compare two strategies: tuning the allocation of seeds according to either node individual properties or according to group properties. We find that, when the contagion is sufficiently nonlinear, groups are more effective seeds of contagion than individual nodes.

Dynamics on higher-order networks: A review

“…So, with the view of exploring the roles of set of groups on the hypergraph contagion dynamics, considering heterogeneity in both hyper-degree and hyperlink cardinality, in the Ref. [88] the authors construct a framework based upon approximate master equations analyzing contagion dynamics on top of random higher-order networks. Assuming the rate of infection as a nonlinear function of the number of infectious individuals in groups, it is shown how influential groups can govern the initial dynamics as well as the final stationary state of the contagion.…”

The physics of higher-order interactions in complex systems

“…This result has been found to be robust and general. Explosive transitions have in fact been observed in heterogeneous [42] and time-varying [43,44] structures, and in the more general setup of hypergraphs [34,45,46], where they can also be related to higher-order discontinuous percolation processes [47].…”