  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 $\phi_m$. However,
actual adoption of the information only occurs when the cumulative influence
across all communication channels exceeds a second threshold, $\phi_k$. 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 $(\phi_m, \langle m \rangle)$ and $(\phi_k, \langle k
\rangle)$ 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.
