Discrete-Time SIS Contagion Processes on Hypergraphs

Recent research on epidemic processes has revealed the limitations of traditional networks, which capture only pairwise relationships, to characterize complex multiparty relationships and group influences properly. Epidemic processes on higher-order networks (simplicial complexes and general hypergraphs) have therefore emerged as a novel frontier. In this work, we investigate discrete-time Susceptible-Infected-Susceptible (SIS) contagion processes occurring on weighted and directed hypergraphs and their extensions to general higher-order SIS processes with the aid of tensor algebra. Our focus lies in comprehensively characterizing the healthy state and the endemic equilibria within this framework. The emergence of bistability behavior phenomena, where multiple equilibria coexist and are simultaneously locally asymptotically stable, is demonstrated in view of the presence of higher-order interactions. Novel sufficient conditions for different system behaviors, which are determined by both (higher-order) network topology and transition rates, are provided to assess the likelihood of the SIS contagion processes causing an outbreak. More importantly, after establishing the local stability of the equilibrium, an explicit domain of attraction linked to the system parameters is constructed. Moreover, a learning method to estimate the transition rates is presented. In the end, the attained theoretical results are supplemented via numerical examples. Specifically, we validate the approximation efficacy of the mean-field SIS contagion processes compared with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2^{n}$</tex-math></inline-formula>-state Markov chain models. These numerical examples are given to highlight the performance of parameter learning algorithms and the system behaviors of the discrete-time SIS contagion processes.