Voters, Games, and Epidemics on Random Graphs

本研究的重点将放在空间随机过程上。在生态学应用的激励下，我们将考虑在二维和三维空间中发生的过程。我们还将研究发生在随机图上的过程，这为社会网络提供了模型。我们将考虑通过这些结构传播信息和疾病。此外，我们将研究进化博弈的行为，这是多年前引入的，以帮助理解动物行为，最近被用于理解癌症建模中不同细胞类型的相互作用。尽管空间结构改变了进化博弈的结果这一事实已经被明确确立，但大多数应用都假设是均匀混合，并使用复制因子方程来确定动力学。拥有一个完善的理论来预测空间游戏的行为将有助于应用。这个理论的大部分都是在弱选择的假设下发展起来的，所以理解当选择不弱时结果是否有效是很重要的。研究工作将涉及进化博弈、选民模型的变体、合并随机漫步和流行病。在大多数情况下，这些过程将发生在由配置模型生成的静态随机图上，其中顶点被分配为i.i.d度，但在一种情况下，我们将让SIS流行病的状态和图共同进化。随机图的程度异质性迫使新技术的发展，以扩展在d维晶格上已知的结果。此外，在这些结构上还出现了新的现象，如严格可证明的不连续相变。具体的研究目标包括：(i)研究随机图上聚并随机游走密度的衰减速率。（ii）反驳物理学家？声明有限图上的临界感染概率是1除以邻接矩阵的最大特征值。（iii）表明如果程度分布有指数尾，则SIS的临界阈值为正。（iv）在易感个体切断与受感染邻居的联系并重新连接到随机选择的个体的进化图上研究SIR和SIS模型。在SIR版本中，我们想要确定临界值。在SIS病例中，我们想要证明当重布线率固定而感染率变化时存在不连续的相变。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main focus of this research will be on spatial random processes. Motivated by applications in ecology we will consider processes that take place in two and three dimensional space. We will also study processes that take place on random graphs, which provide models for social networks. We will consider the spread of information and of diseases through these structures. In addition we will study of the behavior of evolutionary games, which were introduced many years ago to help understand animal behavior and have recently been used to understand the interactions of different cell types in cancer modeling. Despite the fact that it has been clearly established that spatial structure changes the outcome of evolutionary games, most applications assume homogeneous mixing and use the replicator equation to determine the dynamics. Having a well developed theory that predicts the behavior of spatial games will be useful for applications. Much of this theory has been developed under the assumption of weak selection, so it is important to understand whether the results are valid when selection is not weak.Work will be carried out on evolutionary games, variants of the voter model, coalescing random walk, and epidemics. These processes will in most cases take place on static random graphs generated by the configuration model in which vertices are assigned i.i.d. degrees, but in one situation we will let the states of an SIS epidemic and the graph co-evolve. The degree heterogeneity of random graphs forces the development of new techniques in order to extend results known on the d-dimensional lattice. In addition, new phenomena occur on these structures, such as rigorously provable discontinuous phase transitions. Specific research goals include (i) Study the rate of decay of the density of coalescing random walks on a random graph. (ii) Disprove physicists? claim that the critical infection probability on a finite graph is 1 over the largest eigenvalue of the adjacency matrix. (iii) Show that the critical threshold of the SIS is positive if the degree distribution has an exponential tail. (iv) Study SIR and SIS models on evolving graphs where susceptible individuals cut their ties to infected neighbors and rewire to a randomly chosen individual. In the SIR version we want to determine the critical value. In the SIS case we want to show that when the rewiring rate is fixed and the infection rate is varied there is a discontinuous phase transition.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.