多关系网络上的流行病传播动力学研究*

摘要

多关系网络已经吸引了许多人的注意，目前的研究主要涉及其拓扑结构及其演化的分析、不同类型关系的挖掘、重叠社区的检测、级联失效动力学等。然而，多关系网络上流行病传播的研究还相对较少。由此提出一种双关系网络模型(工作-朋友关系网)，研究多关系对于流行病传播动力学行为的影响。在全接触模式下，多关系的存在会显著降低网络中的爆发阈值，使得疾病更容易流行而难以控制。对比ER (Erd ¨os-R ´enyi)， WS (Watts-Strogatz)， BA (Barab´asi-Albert)三种网络，由于结构异质性的差异， WS网络受到的影响最大， ER网络次之， BA网络最小。有趣的是，其爆发阈值的相对变化大小与网络结构无关。在单点接触模式下，增加强关系的权重将显著提升爆发阈值，降低感染密度；随着强关系的比例变化将出现最优值现象：极大的爆发阈值和极小的感染密度。随着强关系的边权增加，达到最优值的边比例将减少。更为有趣的是，三个网络中优值出现的位置几乎一致，独立于网络结构。这一研究不但有助于理解多关系网络上的病毒传播过程，也为多关系网络研究提供了一个新的视角。%Networks with links representing different relationships have attracted much attention in recent years. Previous studies mostly focused on the analyses of network topology and evolution, multi-relation pattern mining, detection of overlapping communities, and cascading failure. However, epidemic spreading on multi-relation networks remains a largely unexplored area. We propose a binary-relation network model, representing working and friendship relationships, to reveal the effect of multiple relationships on the epidemic spreading. A link representing a closer relationship carries a higher weight. For reactive infection process in a multi-relation network, the threshold of outbreak is suppressed, making the epidemic harder to control. Comparing the networks with different structural heterogeneities such as the Watts-Strogatz (WS), Erd ¨os-R ´enyi and Barab ´asi-Albert networks, the WS network is affected most significantly. Interestingly, the relative changes in the thresholds on the three networks are found to be independent of the structure. For contact infection process, an increase in the weight of the closer relationship can raise the outbreak threshold significantly and reduce the prevalence. As the fraction of closer relationship varies, an optimal fraction corresponding to a maximum outbreak threshold and minimum prevalence emerges. With an increase in the weight of the closer relationship, the proportion of links corresponding to the optimal value decreases. Most interestingly, the optimal proportions of closer-relation links on the three networks are almost the same, and thus they are independent of the network topology. This study not only contributes to the better understanding of epidemic spreading dynamics on multi-relation networks, but also provides a new perspective for research on multi-relation networks.