A simple model of global cascades on random hypergraphs

This study introduces a comprehensive framework that situates information cascades within the domain of higher-order interactions, utilizing a double-threshold hypergraph model. We propose that individuals (nodes) gain awareness of information through each communication channel (hyperedge) once the number of information adopters surpasses a threshold ϕ_m. However, actual adoption of the information only occurs when the cumulative influence across all communication channels exceeds a second threshold, ϕ_k. We analytically derive the cascade condition for both the case of a single seed node using percolation methods and the case of any seed size employing mean-field approximation. Our findings underscore that when considering the fractional seed size, r₀∈(0,1], the connectivity pattern of the random hypergraph, characterized by the hyperdegree, k, and cardinality, m, distributions, exerts an asymmetric impact on the global cascade boundary. This asymmetry manifests in the observed differences in the boundaries of the global cascade within the (ϕ_m,〈m〉) and (ϕ_k,〈k〉) planes. However, as r₀→0, this asymmetric effect gradually diminishes. Overall, by elucidating the mechanisms driving information cascades within a broader context of higher-order interactions, our research contributes to theoretical advancements in complex systems theory.