## Seeding simplicial contagions in hypergraphs with heterogeneous structure

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

A recently proposed model of “simplicial contagion” highlighted the rich behaviour of social spreading phenomena obtained when combining nonlinear interaction rules with an hypergraph representation of social interactions, in this case using simplicial complexes. This new perspective allows us to look beyond pairwise interactions, which are often not the best representation for contagion processes in which social reinforcement mechanisms have a large impact. The original analysis of the simplicial contagion model showed how the hypergraph structure of most real networks leads to interesting dynamical features such as discontinuous transitions and bistability.
Unfortunately, the heterogeneity of the hypergraph structure and features of the time evolution of the contagion have yet to be studied due to constraints in previous mathematical formalisms. This work aims to fill this gap by providing an approximated master equation analysis that is precise enough to capture those features, but yet is simple enough to yield explicit results and important insights into the behavior of complex contagion processes.
One novel contribution opened by our new framework concerns the problem of seeding complex social contagions. Given an initial level of contagion (or adoption of a new norm), does the contagion spread faster across communities if it starts uniformly at random in the hypergraph? Is it more efficient to fully infect a few small communities? Or to partially infect the largest community? Previous mathematical formalisms assumed, for simplicity, that communities of the same size were equivalent and therefore could not follow the heterogeneous initial conditions required to study the seeding problem. Using our new approach, we solve the seeding problem by finding the best set of initial conditions required to optimize the spread of a given simplicial contagion. These results are discussed in the context of viral memes and emerging social norms in online communities.
Our second contribution is to study the impact of an heterogeneous structure on the phase diagram. We parametrize the hypergraph structure using two distributions; the size distribution governing the dimensionality of different simplexes and the membership distribution controlling in how many simplexes a unique vertex can be found. We then ask how these distributions affect the invasion threshold, the point where an infinitesimal initial infection can still reach a macroscopic portion of the population, and the nature of the phase transition found around that point. Surprisingly, we show that the first three moments of the membership distribution are involved in setting the behaviour of the system at its critical point. This is unlike most contagion models which are usually fully characterized by the first two moments of this distribution.
Altogether, our work further highlights how contagion on hypergraphs can differ from the more simple, pairwise, models on regular networks. These results also provide insights into the nature of virality in online media, and the role of a well-distributed critical mass in emerging social movements.← Schedule