[Paper Review] A pair-based approximation for simplicial contagion

Federico Malizia, Luca Gallo+4 more|arXiv (Cornell University)|Jul 19, 2023

Evolution and Genetic DynamicsBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper introduces a pair-based mean-field approximation for modeling SIS contagion dynamics on simplicial complexes, capturing dynamical correlations within triadic groups that individual-based mean-field models miss. The method improves accuracy in predicting the bistable region, transition type, and time evolution of infection prevalence compared to traditional approaches.

ABSTRACT

Higher-order interactions play an important role in complex contagion processes. Mean-field approximations have been used to characterize the onset of spreading in the presence of group interactions. However, individual-based mean-field models are unable to capture correlations between different subsets of nodes, which can significantly influence the dynamics of a contagion process. In this paper, we introduce a pair-based mean-field approximation that allows to study the dynamics of a SIS model on simplicial complexes by taking into account correlations at the level of pairs of nodes. %by taking into account dynamical correlations emerging in groups of nodes. Compared to individual-based mean-field approaches, the proposed approximation yields more accurate predictions of the dynamics of contagion processes on simplicial complexes. Specifically, the pair-based mean-field approximation provides higher accuracy in predicting the extent of the region of bistability, the type of transition from disease-free to endemic state, and the average time evolution of the fraction of infected individuals. Crucially, the pair-based approximation correctly predicts that the onset of the epidemic outbreak in simplicial complexes depends on the strength of higher-order interactions. Overall, our findings highlight the importance of accounting for pair correlations when investigating contagion processes in the presence of higher-order interactions.

Motivation & Objective

• To address the limitations of individual-based mean-field models in capturing dynamical correlations in higher-order contagion processes on simplicial complexes.
• To develop a more accurate approximation method that accounts for correlations within groups of three or more nodes.
• To compare the pair-based approach with individual-based mean-field models and stochastic simulations on random simplicial complexes.
• To evaluate the performance of the pair-based model in predicting the extent of bistability, transition types, and time evolution of infection fractions.

Proposed method

• The method derives a system of continuous-time differential equations describing the time evolution of pair states (e.g., infected-infected, susceptible-infected) in a simplicial complex.
• It introduces closure assumptions to reduce the hierarchy of moment equations, enabling a closed-form system independent of network size.
• The model explicitly accounts for three-body interactions via simplicial structures, where infection transmission depends on the state of all three nodes in a 2-simplex.
• The pair-based approximation is contrasted with the individual-based mean-field model and stochastic simulations on random simplicial complexes (RSC).
• The approach is validated by comparing predicted thresholds (λc and λ*), bistable regions, and time evolution of infection fractions (ρ(t)) with simulation results.

Figure 1: Pictorial representation of infection processes among susceptible (in blue) and infected (in red) individuals in SIS models. (a) Infection of a node connected to an infected node through a link. This is the only infection process occurring in the individual-based SIS model. (b) Infection i

Experimental results

Research questions

• RQ1 How does the inclusion of dynamical correlations within triadic groups affect the prediction of epidemic thresholds in simplicial contagion?
• RQ2 To what extent does the pair-based approximation improve upon individual-based mean-field models in capturing the bistable regime of SIS dynamics on simplicial complexes?
• RQ3 How accurately does the pair-based model predict the transition from disease-free to endemic states under varying three-body interaction strengths?
• RQ4 Can the pair-based model reproduce the average time evolution of infection prevalence observed in stochastic simulations?

Key findings

• The pair-based approximation more accurately predicts the dependence of the critical thresholds λc and λ* on the three-body interaction strength λΔ, showing both decrease with increasing λΔ.
• The model correctly identifies the existence of a single stable equilibrium in regions where the individual-based model predicts bistability, matching stochastic simulation outcomes.
• The pair-based model predicts a narrower region of bistability than the individual-based model, which overestimates its extent by underestimating λc and overestimating λ*.
• For (λ, λΔ) = (0.3, 3.5), the pair-based model predicts a disease-free steady state, consistent with stochastic simulations, while the individual-based model incorrectly predicts two stable equilibria.
• For (λ, λΔ) = (0.95, 3), the pair-based model correctly predicts an endemic steady state, aligning with simulations, whereas the individual-based model again fails to capture the correct single equilibrium.
• The pair-based approximation provides a better match to the average time evolution of ρ(t) from stochastic simulations than the individual-based approach.

Figure 2: Graphical representation of the three possible microscopical configurations of four-node motif states $(I,I,S,S)$ (on top) and $(I,I,S,I)$ (bottom). Square brackets refer to the expected number of the singular configurations.