聚类系数对单纯复形网络上SIS模型多稳态的影响

Effect of Clustering Coefficient on Multistability of SIS Model on Simplicial Complex Networks

摘要: 本文基于单纯复形网络上SIS对逼近模型，系统研究了聚类系数对传染病传播的影响。 证明了不存 在聚类肘，平凡平衡点为鞍点，系统存在唯一全局渐近稳定的非平凡平衡点，且严格证明了不变 直线的存在性，澄清了原模型的近似性。 研究表明，当存在聚类系数肘，系统发生Isola分支，多 稳态行为显著。 随着聚类系数增大，Isola分支范围扩张，多稳态参数区域增大，高聚类系数还会 放大系统对初始感染条件的敏感性。进一步研究表明，高阶结构参数（2-单纯形的平均数量）对多 稳态具有抑制作用，相同聚类系数下，2-单纯形的平均数量越小，Isola分支范围越大；聚类系数 越高，2-单纯形的平均数量的抑制效应越显著。综上，我们的研究结果揭示了聚类系数是诱发系统 多稳态与Isola分支的关键因素，并明确了2-单纯形的平均数量在聚类调控中的作用，为针对聚集 性活动的非极端防控策略提供了理论依据。

Abstract: We systematically investigate the influence of the clustering coefficient on epidemic spreading based on a pairwise SIS approximation model on simplicial complexes. We prove that in the absence of clustering, the trivial equilibrium is a saddle point, the system admits a unique globally asymptotically stable nontrivial equilibrium, and the existence of the invariant line is rigorously established, thereby clarifying the approximation used in the original model. The results show that in the presence of a clustering, the system exhibits an Isola bifurcation and significant multistability. As the clustering coefficient increases, the Isola bifurcation range expands, the region of multistability enlarges, and a higher clustering coefficient also amplifies the system’s sensitivity to initial infection conditions. Further analysis indicates that the higher- order structural parameter (the average number of 2-simplices per node) suppresses multistability: for a fixed clustering coefficient, a smaller average number of 2-simplices leads to a larger Isola bifurcation range; the higher the clustering coefficient, the more pronounced the suppressive effect of the average number of 2-simplices. In summary, our findings reveal that the clustering coefficient is a key factor inducing multistability and Isola bifurcation in the system, and clarify the role of the average number of 2-simplices in clustering modulation, providing a theoretical basis for non-extreme intervention strategies targeting gathering activities.