A simple model of global cascades on random hypergraphs

Abstract

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 $\ddot{I}{•}_m$. However, actual adoption of the information only occurs when the cumulative influence across all communication channels exceeds a second threshold, $\ddot{I}{•}_k$. 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, $r_0\in (0,1]$, 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 $(\ddot{I}{•}_m,⟨m⟩)$ and $(\ddot{I}{•}_k,⟨k⟩)$ planes. However, as $r_0\to 0$, 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.

11 pages, 10 figures