Higher-order non-Markovian social contagions in simplicial complexes

Social contagion, the spread of behaviors or information through a population, is a ubiquitous phenomenon. Traditional models often represent social networks as graphs, focusing on pairwise interactions between individuals. However, real-world interactions are often more complex, involving higher-order structures where groups of individuals interact simultaneously. These higher-order structures can be effectively represented using simplicial complexes, mathematical structures that generalize graphs to include higher-dimensional simplices (nodes, edges, triangles, tetrahedra, etc.). Empirical observations show that social contagion is not solely determined by pairwise interactions but also significantly influenced by group interactions within these higher-order structures. Furthermore, the temporal dynamics of contagion are often non-Markovian, meaning that the probability of an event (infection or recovery) depends not only on the current state but also on the history of the process. Classic models, like the threshold model, fail to capture the nuances of these higher-order interactions and the non-Markovian aspects of contagion. Therefore, understanding the dynamics of higher-order non-Markovian social contagions within simplicial complexes is crucial for accurately modeling real-world phenomena such as information diffusion, rumor spreading, and the adoption of innovations. This research addresses this gap by proposing a comprehensive model that explicitly incorporates both higher-order interactions and non-Markovian dynamics. The study provides a theoretical framework for analyzing the spread of information and behaviors in scenarios where these complex interactions are significant.

Numerous studies have explored spreading dynamics in complex networks, predominantly focusing on pairwise interactions represented by graphs. However, the limitations of these approaches in capturing group interactions have become increasingly apparent. The importance of higher-order structures, such as those found in simplicial complexes, is increasingly recognized in various fields. For example, feed-forward loops are important structures in gene regulatory networks. Triangles frequently appear in networks of scientific collaboration and social communication, reflecting teamwork and intragroup dynamics. Simplicial complexes have also been identified in brain networks, email networks, and human contact networks. Research into higher-order interactions has shown their impact on phase transitions in social contagion, the dynamics of competing epidemics, and the influence of temporal properties. Studies in multilayer networks highlight how higher-order structures, especially 2-simplices, can suppress epidemic spreading. Classical spreading models typically assume Markovian processes (constant infection/recovery rates), which is a simplification. Real-world processes often exhibit non-Markovian characteristics, where inter-event times don't follow an exponential distribution but rather more complex distributions. This non-Markovian nature has been observed in epidemic spreading and innovation diffusion, showing the need for more sophisticated models. While there is growing interest in non-Markovian processes in networks, our understanding of these dynamics within higher-order structures remains limited. This paper bridges this gap by proposing a model that encompasses both aspects.

The authors propose a higher-order non-Markovian social contagion model within simplicial complexes. The model uses a susceptible-infected-susceptible (SIS) framework, where each node can be in either a susceptible (S) or infected (I) state. The key innovation is the introduction of "z-dimensional virtual nodes," which represent groups of z real nodes within a z-dimensional simplex. A virtual node is considered infected if all its constituent real nodes are infected. The age of a virtual or real node tracks the time elapsed since it entered its current state. The infection process is non-Markovian, with infection rates η(τ) varying with the infection age τ. The recovery process is also non-Markovian, with a recovery rate ωrec(τ) dependent on the duration in the infected state. Information propagation occurs through simplices (edges, triangles, etc.). A z-dimensional infected virtual node attempts to infect its remaining neighbor at a rate η(τ). The authors develop a mean-field theory to describe the temporal evolution of the model. This theory considers the probability density functions for nodes in the I and S states with given ages. These equations are simplified under the assumption of a homogeneous network, leading to a set of equations governing the temporal evolution of the infected fraction. For more accurate predictions, especially when higher-order simplices are involved, a pairwise approximation method is employed, which considers correlations between node pairs. This involves modeling the dynamics of different types of node pairs (e.g., susceptible-infected pairs). The Weibull distribution is used to model the infection and recovery times due to its flexibility in capturing various shapes. Monte Carlo simulations are conducted on random simplicial complexes with Poisson degree distributions to validate the theoretical framework. The simulations start with a small fraction of randomly infected nodes, and the time evolution of the infected fraction is tracked. The average infection rate is defined considering the expected infection and recovery times from the Weibull distribution, and its relation to the Markovian effective infection rate is explored. The authors then investigate the equivalence between higher-order non-Markovian and Markovian models by comparing steady-state results obtained using both simulation and theory.

The study's key findings include the successful development and validation of a higher-order non-Markovian social contagion model. The mean-field theory developed provides a good prediction of the steady-state behavior of the system, although deviations are observed in transient dynamics, especially when higher-order interactions are strong due to nodal-pair dynamic correlations. A significant finding is the demonstration of an equivalence between higher-order non-Markovian and higher-order Markovian social contagions. This equivalence is established theoretically and verified through simulations. The results show a clear overlap between the steady-state infection density curves for non-Markovian and Markovian models when the appropriate effective infection rate is used. Furthermore, the research highlights the role of higher-order interactions in promoting spreading dynamics. When considering higher-order structures (e.g., triangles), information propagates more readily compared to simple pairwise interactions. The impact of non-Markovian recovery processes on system resilience is explored. The simulations reveal that non-Markovian recovery can enhance network resilience to large-scale or small-scale infections under different conditions. Specifically, longer, more homogeneous recovery times (αr > 1) improve resilience to large-scale infections, while faster, more heterogeneous recovery times (αr < 1) improve resilience to small-scale infections. A phase transition from a continuous to a discontinuous phase transition is observed as the average 2-simplicial degree increases. This transition is similar to that observed in Markovian models. The simulations show that the theoretical results and simulation results align well in the steady state.

The findings of this paper significantly advance our understanding of social contagion dynamics. The equivalence between non-Markovian and Markovian models under specific conditions simplifies theoretical analysis and provides a powerful tool for predicting steady-state behavior. The model's ability to accurately predict steady-state behavior despite the complexities of higher-order interactions and non-Markovian dynamics demonstrates the efficacy of the mean-field theory and the pairwise approximation. The observed promotion of spreading by higher-order interactions underscores the importance of considering these structures in realistic models. The findings on resilience are particularly significant. The demonstration that non-Markovian recovery can enhance resilience under various infection scenarios opens new avenues for controlling the spread of information and behaviors in real-world networks. This has implications for interventions aimed at enhancing the resilience of social systems against undesired contagion such as rumor spreading or misinformation.

This study provides a comprehensive framework for understanding higher-order non-Markovian social contagion in simplicial complexes. The proposed model, supported by theoretical analysis and simulations, reveals the equivalence between non-Markovian and Markovian dynamics under certain conditions and highlights the critical role of higher-order interactions and non-Markovian recovery in shaping contagion dynamics and system resilience. Future research could explore more complex network structures, investigate the effects of different non-Markovian distributions, and delve into the impact of network heterogeneity on the observed phenomena.

One limitation of this study is the deviation between theoretical predictions and simulation results during the transient phase, particularly in networks with strong higher-order interactions. This deviation arises from the limitations of the mean-field and pairwise approximations in capturing nodal-pair dynamic correlations. Further refinement of the theoretical framework might be needed for highly complex scenarios. Additionally, the study focuses primarily on random simplicial complexes with Poisson degree distributions. The applicability of the findings to other types of networks with different topological properties warrants further investigation.