Simplicial models of social contagion

Abstract: Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.

• The paper introduces a novel Simplicial Contagion Model (SCM) that integrates pairwise and group interactions to better capture social contagion.
• It demonstrates emergent phenomena such as discontinuous epidemic transitions and bistable regions that traditional models fail to capture.
• Analytical and numerical analyses confirm that incorporating higher-order interactions fundamentally alters contagion dynamics in social systems.

Simplicial Models of Social Contagion: An Essay

The paper entitled "Simplicial models of social contagion" introduces a novel framework for modeling social contagion processes by employing higher-order structures known as simplicial complexes. Traditional approaches to modeling social contagion have largely relied on complex networks that focus on pairwise interactions. However, the authors argue that these models may be insufficient for capturing the intricacies of social influence, reinforcement, and group dynamics prevalent in phenomena such as opinion formation and the adoption of innovations.

Key Contributions

The central contribution of the paper is the introduction of a simplicial contagion model (SCM) that enables the representation and analysis of both pairwise and group interactions within a social system. Simplicial complexes allow for the explicit incorporation of higher-order interactions, i.e., interactions that involve more than two individuals simultaneously. This structural shift from traditional networks to simplicial complexes is critical in capturing the multi-faceted nature of social contagion processes.

Some of the notable findings in the paper include:

1. Model Architecture: The proposed SCM incorporates contagion dynamics that occur through different-dimensional simplices, allowing for both simple and complex contagion processes. In the model, simplices of varying dimensions represent interactions of different group sizes, with each simplex contributing to the contagion process based on its dimensionality.
2. Emergent Phenomena: The model is shown to exhibit novel phenomena not captured by traditional models. These include a discontinuous transition at the epidemic threshold and the existence of a bistable region where both healthy and endemic states can coexist. Such phenomena are attributed to the role of higher-order interactions in propagating contagion.
3. Analytical and Numerical Analysis: The authors support their findings through extensive numerical simulations on both empirical and synthetic simplicial complexes. They provide an analytical mean-field framework to describe the density of infected nodes and demonstrate that higher-order interactions fundamentally alter the nature of contagion dynamics.

Implications and Future Prospects

The implications of this research are multifold:

• Understanding Social Dynamics: By capturing complex contagion dynamics, the model provides insights into how critical mass and collective behavior impact the spread of information or behaviors in social networks. This is particularly relevant for understanding how social norms or innovations gain widespread adoption.
• Applications Beyond Social Systems: While the primary focus of the SCM is on social contagion, the framework can be extended to other dynamical systems and domains, such as epidemiology, neuroscience, and communication networks, where higher-order interactions are prevalent.
• Development of Hypergraph Models: The paper opens avenues for expanding the SCM to hypergraph models, which could incorporate even more intricate interdependencies and multidimensional interactions in various real-world systems.

Conclusion

The introduction of the Simplicial Contagion Model marks a significant advancement in the modeling of social contagion processes by rigorously accounting for higher-order interactions. By leveraging the mathematical elegance of simplicial complexes, the authors provide a robust framework that not only enhances our theoretical understanding of complex contagions but also sets the stage for practical applications in various fields. Future research can build upon these foundations to explore and model the dynamic interplay of higher-order interactions in increasingly complex social and artificial systems.