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Voters, Games, and Epidemics on Random Graphs

随机图上的选民、游戏和流行病

基本信息

批准号：
1809967
负责人：
Richard Durrett
金额：
$ 29.97万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1809967&HistoricalAwards=false
关键词：
Voters Games Epidemics Random Graphs

项目摘要

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.

这项研究的主要重点将是空间随机过程。受生态学应用的启发，我们将考虑在二维和三维空间中发生的过程。我们还将研究随机图上发生的过程，这些过程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，我们还将研究进化游戏的行为，这些游戏在多年前被引入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。尽管事实上，它已经清楚地建立了空间结构改变进化游戏的结果，大多数应用程序假设均匀混合，并使用复制因子方程来确定动态。有一个良好的发展理论，预测空间游戏的行为将是有用的应用。这一理论的大部分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。工作将在进化博弈、选民模型的变体、合并随机游走和流行病方面进行。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，以推广已知的d维格上的结果。此外，新的现象发生在这些结构上，如严格可证明的不连续相变。具体的研究目标包括（i）研究随机图上合并随机游动密度的衰减率。(ii)反驳物理学家？证明了有限图上的临界传染概率在邻接矩阵的最大特征值上为1。(iii)证明了如果度分布具有指数尾，则SIS的临界阈值为正。(iv)研究SIR和SIS模型的进化图，其中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。在SIR版本中，我们希望确定临界值。在SIS的情况下，我们想表明，当重新布线率是固定的，感染率是变化的，有一个不连续的相变。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。