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Discrete-time SIS Social Contagion Processes on Hypergraphs

Lidan Liang, Shaoxuan Cui, Fangzhou Liu

TL;DR

This work investigates discrete-time Susceptible-Infected-Susceptible dynamics on weighted directed hypergraphs to capture higher-order social contagion effects. It develops a mean-field, tensor-based framework that relates micro-level Markov-chain transitions to macro-level infection probabilities, and derives conditions for healthy-state stability, bistability, and endemic equilibria, including explicit domains of attraction. The analysis highlights the pivotal role of higher-order interactions in generating multiple stable equilibria and rich dynamical behavior, while providing a practical parameter-learning method and numerical validation against $2^n$-state Markov-chain models. The results offer theoretical and computational tools for understanding and controlling contagion processes in group-influenced social networks with higher-order structure.

Abstract

Recent research on social contagion processes has revealed the limitations of traditional networks, which capture only pairwise relationships, to characterize complex multiparty relationships and group influences properly. Social contagion 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) social contagion processes occurring on weighted and directed hypergraphs and their extensions to bivirus cases and general higher-order SIS processes with the aid of tensor algebra. Our focus lies in comprehensively characterizing the healthy state and endemic equilibria within this framework. The emergence of bistability or multistability behavior phenomena, where multiple equilibria coexist and are simultaneously locally asymptotically stable, is demonstrated in view of the presence of the higher-order interaction. The novel sufficient conditions of the appearance for system behaviors, which are determined by both (higher-order) network topology and transition rates, are provided to assess the likelihood of the SIS social contagion processes causing an outbreak. More importantly, given the equilibrium is locally stable, an explicit domain of attraction associated with 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 evaluate the effectiveness of the networked SIS social contagion process by comparing it with the $2^n$-state Markov chain model. These numerical examples are given to highlight the performance of parameter learning algorithms and the system behaviors of the discrete-time SIS social contagion process.

Discrete-time SIS Social Contagion Processes on Hypergraphs

TL;DR

This work investigates discrete-time Susceptible-Infected-Susceptible dynamics on weighted directed hypergraphs to capture higher-order social contagion effects. It develops a mean-field, tensor-based framework that relates micro-level Markov-chain transitions to macro-level infection probabilities, and derives conditions for healthy-state stability, bistability, and endemic equilibria, including explicit domains of attraction. The analysis highlights the pivotal role of higher-order interactions in generating multiple stable equilibria and rich dynamical behavior, while providing a practical parameter-learning method and numerical validation against -state Markov-chain models. The results offer theoretical and computational tools for understanding and controlling contagion processes in group-influenced social networks with higher-order structure.

Abstract

Recent research on social contagion processes has revealed the limitations of traditional networks, which capture only pairwise relationships, to characterize complex multiparty relationships and group influences properly. Social contagion 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) social contagion processes occurring on weighted and directed hypergraphs and their extensions to bivirus cases and general higher-order SIS processes with the aid of tensor algebra. Our focus lies in comprehensively characterizing the healthy state and endemic equilibria within this framework. The emergence of bistability or multistability behavior phenomena, where multiple equilibria coexist and are simultaneously locally asymptotically stable, is demonstrated in view of the presence of the higher-order interaction. The novel sufficient conditions of the appearance for system behaviors, which are determined by both (higher-order) network topology and transition rates, are provided to assess the likelihood of the SIS social contagion processes causing an outbreak. More importantly, given the equilibrium is locally stable, an explicit domain of attraction associated with 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 evaluate the effectiveness of the networked SIS social contagion process by comparing it with the -state Markov chain model. These numerical examples are given to highlight the performance of parameter learning algorithms and the system behaviors of the discrete-time SIS social contagion process.
Paper Structure (25 sections, 21 theorems, 46 equations, 6 figures, 1 table)

This paper contains 25 sections, 21 theorems, 46 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Model Setup
  3. Weighted directed hypergraph and tensor
  4. Network Model
  5. The network SIS social contagion processes involving both 2nd-order and 3rd-order hyperedges
  6. Analytical Results
  7. Parameter Learning
  8. Extension to competitive bi-virus SIS social contagion processes with high-order interactions
  9. Extension to general higher-order SIS social contagion processes
  10. Numerical examples
  11. Approximate performance of the mean-field SIS social contagion processes corresponding to the $2^n$-state Markov chain models
  12. Convergence Performance of the Discrete-Time SIS Social Contagion Processes
  13. Conventional higher-order single SIS Social Contagion Process
  14. Bi-virus SIS Social Contagion Processes over hypergraphs
  15. Conclusions
  16. ...and 10 more sections

Key Result

Lemma 1

chen2022explicit Given a one dimensional homogenous polynomial function $f(x(t))$: $\mathbb{R}^n\rightarrow\mathbb{R}.$ It can be uniquely determined by $Ax^m$, $m$ is the order of $A$, $A$ is super-symmetric. Given a $n$-dimensional homogenous polynomial function $g(x(t))$: $\mathbb{R}^n\rightarrow

Figures (6)

  • Figure 1: (a) and (b) show the state trajectories of the infection level for the mean-field SIS social contagion processes and Markov chain model when the reproduction number $\rho(I-h\mathcal{D}+h\mathcal{B})$ equals 0.9995 and 1.0056, respectively.
  • Figure 2: The weighted and directed hypergraph with 2nd-order and 3rd-order hyperedges
  • Figure 3: (a) State trajectory of the infection level $x(t)$ when the reproduction number equals 0.9986. From 3 different random initial conditions, the system (\ref{['matrix_form']}) always converges to the healthy state. (b) State trajectory of the infection level $x(t)$ when the reproduction number equals 0.9995. From 3 different random initial conditions, the system (\ref{['matrix_form']}) either converges to the corresponding endemic equilibrium (0.6306,0.5926,0.6559,0.6371,0.6679) or to the healthy state. (c) State trajectory of the infection level $x(t)$ when the reproduction number equals 1.0021. From 3 different random initial conditions, the system (\ref{['matrix_form']}) always converges to the corresponding endemic equilibrium (0.4913,0.5663,0.5547,0.4750,0.6455).
  • Figure 4: State trajectory of the given simulated infection level $x(t)$
  • Figure 5: State trajectory $x_{[\ell]}(t)(\ell=1,2)$ of the bi-virus system (\ref{['bi_virus_2']}) for different random initial conditions when the reproduction numbers equal 0.9970 and 0.9960.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • Remark 3
  • Proposition 1
  • Proposition 2
  • Remark 4
  • Proposition 3
  • Remark 5
  • ...and 23 more