高阶网络上传染动力学介观局域化的主方程分析
Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks
摘要：
传染病的简单模型倾向于假设个体的随机混合，但真正的相互作用不是随机的成对相遇：它们发生在各种类型的聚集体中，如工作场所、家庭、学校和音乐会，最好用高阶网络结构来描述。我们使用基于组的近似主方程对高阶网络上的传染进行建模，在该方程中，我们跟踪一组节点内的所有状态和相互作用，并假设它们之间存在平均场耦合。使用易感-感染-易感动力学，我们的方法揭示了一个介观局部化制度的存在，其中疾病可以集中和自我维持，只有在网络整体组织中的大群体周围。在这种情况下，相变是模糊的，其特征在于基团的不均匀活化。在中观层次上，我们观察到相同大小的组内感染节点的分布可以非常分散，甚至是双峰。当考虑异构网络，无论是在节点和组的水平，我们分析表征与介观局部化的结构参数空间中的区域。我们把这种现象的角度与特征向量本地化，并讨论如何重点高阶结构需要辨别更微妙的本地化在介观层次。最后，我们讨论了介观本地化如何影响结构干预的反应，以及这个框架如何为广泛的动态提供重要的见解。
英文摘要：
Simple models of infectious diseases tend to assume random mixing of individuals, but real interactions are not random pairwise encounters: they occur within various types of gatherings such as workplaces, households, schools, and concerts, best described by a higher-order network structure. We model contagions on higher-order networks using group-based approximate master equations, in which we track all states and interactions within a group of nodes and assume a mean-field coupling between them. Using the Susceptible-Infected-Susceptible dynamics, our approach reveals the existence of a mesoscopic localization regime, where a disease can concentrate and self-sustain only around large groups in the network overall organization. In this regime, the phase transition is smeared, characterized by an inhomogeneous activation of the groups. At the mesoscopic level, we observe that the distribution of infected nodes within groups of a same size can be very dispersed, even bimodal. When considering heterogeneous networks, both at the level of nodes and groups, we characterize analytically the region associated with mesoscopic localization in the structural parameter space. We put in perspective this phenomenon with eigenvector localization and discuss how a focus on higher-order structures is needed to discern the more subtle localization at the mesoscopic level. Finally, we discuss how mesoscopic localization affects the response to structural interventions and how this framework could provide important insights for a broad range of dynamics.
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