Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA.
Department of Translational Hematology and Oncology Research and Department of Radiation Oncology, Cleveland Clinic, Cleveland, Ohio 44195, USA.
Phys Rev E. 2017 Jul;96(1-1):012313. doi: 10.1103/PhysRevE.96.012313. Epub 2017 Jul 13.
We study a stochastic model of infection spreading on a network. At each time step a node is chosen at random, along with one of its neighbors. If the node is infected and the neighbor is susceptible, the neighbor becomes infected. How many time steps T does it take to completely infect a network of N nodes, starting from a single infected node? An analogy to the classic "coupon collector" problem of probability theory reveals that the takeover time T is dominated by extremal behavior, either when there are only a few infected nodes near the start of the process or a few susceptible nodes near the end. We show that for N≫1, the takeover time T is distributed as a Gumbel distribution for the star graph, as the convolution of two Gumbel distributions for a complete graph and an Erdős-Rényi random graph, as a normal for a one-dimensional ring and a two-dimensional lattice, and as a family of intermediate skewed distributions for d-dimensional lattices with d≥3 (these distributions approach the convolution of two Gumbel distributions as d approaches infinity). Connections to evolutionary dynamics, cancer, incubation periods of infectious diseases, first-passage percolation, and other spreading phenomena in biology and physics are discussed.
我们研究了网络上感染传播的随机模型。在每个时间步长，随机选择一个节点及其一个邻居。如果节点感染且邻居易感，则邻居会被感染。从单个感染节点开始，需要多少个时间步长 T 才能完全感染 N 个节点的网络？与概率论中的经典“优惠券收集者”问题的类比表明，接管时间 T 主要由极端情况决定，要么在过程开始时只有少数感染节点附近，要么在结束时只有少数易感节点附近。我们表明，对于 N≫1，接管时间 T 的分布为星形图的 Gumbel 分布，为完全图和 Erdős-Rényi 随机图的两个 Gumbel 分布的卷积，为一维环和二维晶格的正态分布，以及对于具有 d≥3 的 d 维晶格的一系列中间偏斜分布（这些分布在 d 接近无穷大时接近两个 Gumbel 分布的卷积）。讨论了与进化动力学、癌症、传染病潜伏期、首次通过渗流和生物学和物理学中的其他传播现象的联系。
