[论文解读] 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 Biology被引用 3

一句话总结

本文提出了一种基于配对的平均场近似方法，用于在单纯复形上建模SIS传播动力学，能够捕捉个体基于平均场模型所忽略的三元组群体内的动态相关性。与传统方法相比，该方法在预测双稳态区域、转变类型以及感染流行程度的时间演化方面显著提高了准确性。

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.

研究动机与目标

• 为解决个体基于平均场模型在捕捉单纯复形上高阶传播过程中的动态相关性方面的局限性。
• 开发一种能考虑三节点及以上节点群体内部相关性的更精确近似方法。
• 将基于配对的方法与个体基于平均场模型及随机单纯复形上的随机模拟进行比较。
• 评估基于配对模型在预测双稳态范围、转变类型以及感染比例时间演化方面的能力。

提出的方法

• 该方法推导出一组连续时间微分方程，描述单纯复形中配对状态（如感染-感染、易感-感染）的时间演化。
• 通过引入闭合假设，降低矩方程的层级，从而获得与网络规模无关的封闭形式系统。
• 通过单纯形结构显式建模三体相互作用，其中感染传播取决于2-单纯形中所有三个节点的状态。
• 将基于配对的近似方法与个体基于平均场模型及随机单纯复形（RSC）上的随机模拟进行对比。
• 通过将预测的临界阈值（λc 和 λ*）、双稳态区域以及感染比例（ρ(t)）的时间演化与模拟结果对比，对方法进行验证。

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

实验结果

研究问题

• RQ1 在三元组群体内包含动态相关性如何影响单纯传播中流行病临界阈值的预测？
• RQ2 基于配对的近似方法在捕捉单纯复形上SIS动力学的双稳态区域方面，相较于个体基于平均场模型有多大改进？
• RQ3 在不同三体相互作用强度下，基于配对的模型在预测从无病状态到地方性流行状态转变时的准确性如何？
• RQ4 基于配对的模型能否再现随机模拟中观察到的感染流行程度的平均时间演化？

主要发现

• 基于配对的近似更准确地预测了临界阈值λc和λ*随三体相互作用强度λΔ的变化关系，二者均随λΔ增大而减小。
• 该模型正确识别出在个体基于模型预测为双稳态的区域中仅存在一个稳定平衡态，与随机模拟结果一致。
• 基于配对的模型预测的双稳态区域比个体基于模型更窄，后者因低估λc和高估λ*而过度估计了双稳态区域的范围。
• 当(λ, λΔ) = (0.3, 3.5)时，基于配对的模型预测为无病稳态，与随机模拟一致，而个体基于模型错误地预测存在两个稳定平衡态。
• 当(λ, λΔ) = (0.95, 3)时，基于配对的模型正确预测为地方性稳态，与模拟结果一致，而个体基于模型再次未能捕捉到正确的单一平衡态。
• 基于配对的近似方法在匹配随机模拟的ρ(t)平均时间演化方面，优于个体基于方法。

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.