Voters, Games, and Epidemics on Random Graphs

这项研究的主要焦点将是空间随机过程。在生态学应用的激励下，我们将考虑发生在二维和三维空间中的过程。我们还将研究发生在随机图上的过程，这为社交网络提供了模型。我们将考虑通过这些结构传播信息和疾病。此外，我们还将研究进化博弈的行为，进化博弈在许多年前被引入以帮助理解动物的行为，最近被用来理解癌症建模中不同类型细胞的相互作用。尽管已经清楚地确定空间结构会改变进化博弈的结果，但大多数应用都假设均匀混合，并使用复制者方程来确定动力学。拥有一个预测空间游戏行为的成熟理论将对应用程序有用。这一理论在很大程度上是在弱选择的假设下发展起来的，因此理解在选择不弱的情况下结果是否有效是很重要的。工作将在进化博弈、选民模型的变体、联合随机游走和流行病上进行。这些过程在大多数情况下将发生在由配置模型生成的静态随机图上，在该配置模型中，顶点被分配I.I.D.。度，但在一种情况下，我们将让SIS流行病的状态和图形共同进化。随机图的度异质性迫使新技术的发展，以推广d维格上的已知结果。此外，在这些结构上出现了新的现象，例如严格可证明的不连续相变。具体的研究目标包括：(I)研究随机图上聚合随机游动密度的衰减率。(Ii)反驳物理学家？声称有限图上的临界感染概率在邻接矩阵的最大特征值上为1。(3)证明了当度分布具有指数尾部时，SIS的临界阈值为正。(Iv)研究进化图上的SIR和SIS模型，在该模型中，易感人群切断了与受感染邻居的联系，并重新连接到随机选择的个人。在SIR版本中，我们要确定临界值。在SIS的案例中，我们想要证明，当重新布线率固定而感染率不同时，存在不连续的相变。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势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.