Dynamics of the threshold model on hypergraphs

Abstract

The threshold model has been widely adopted as a prototype for studying contagion processes on social networks. In this paper, we consider individual interactions in groups of three or more vertices and study the threshold model on hypergraphs. To understand how high-order interactions affect the breakdown of the system, we develop a theoretical framework based on generating function technology to derive the cascade condition and the giant component of vulnerable vertices, which depend on both hyperedges and hyperdegrees. First, we find a dual role of the hyperedge in propagation: when the average hyperdegree is small, increasing the size of the hyperedges may make the system fragile, while the average hyperdegree is relatively large, the increase of the hyperedges causes the system to be robust. Then, we identify the effects of threshold, hyperdegree, and hyperedge heterogeneities. The heterogeneity of individual thresholds causes the system to be more fragile, while the heterogeneity of individual hyperdegrees or hyperedges increases the robustness of the system. Finally, we show that the higher hyperdegree a vertex has, the larger possibility and faster speed it will get activated. We verify these results by simulating meme spreading on both random hypergraph models and hypergraphs constructed from empirical data.

Hyper-network models for civil aviation systems

“…The mechanisms involving the Weibull distribution for hyper-networks are never considered before, but might be of value to the analyses of some problems. For an example, the robustness of hyper-graphs, which is always studied by assuming the hyper-degree and hyper-edge cardinality obey the Poisson or power law distributions [36,[41][42][43]. Another example often conducted in the similar way is the dynamical behaviour of the contagion process [44].…”

Novel propagation phenomena: behaviors of local trend imitation on multiple limited contact networks

“…When the initial seed size is very tiny, the percolation theory can be used to estimate the proportion [12,13]. When the adoption threshold is fixed, the average degree scale change results in saddle point bifurcation, and with the increasing of network average degree, the final adoption scope increases continuously at first and then decreases discontinuously [14,15]. It has been found that the social propagation mechanism in the threshold model will be impacted by elements like the seed number [14], clustering coefficient [16], community structure [17], multiple networks [18], network timing [19], and so on.…”