论文标题
获胜者除了一个
The winner takes it all but one
论文作者
论文摘要
我们研究了具有无限均值学位的配置模型产生的图表上的竞争第一通道渗透。最初，两个均匀选择的顶点分别被1型和2型感染感染，然后感染通过图中最近的邻居传播。 1型1（分别为2）感染所需的时间是通过随机变量$ x_1（e）$（分别$ x_2（e）$）给出的边缘$ e $，并且，如果边缘另一端的顶点仍未感染，则它会被感染，然后将其感染并免疫到另一种类型。假设该学位遵循带有指数$τ\（1,2）$的幂律分布，我们表明，由于顶点趋于无穷大的概率，但其中一种感染类型占据了所有顶点，除其他类型的起点外。此外，无论通行时间分布如何，这两种感染都有赢得胜利的阳性概率。结果还显示出对擦除的配置模型的保留，在该模型中，自宽的删除并合并了多个边缘，并且在某些$α> 0 $的条件下，该学位的条件小于$ n^α$。
We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with type 1 and type 2 infection, respectively, and the infection then spreads via nearest neighbors in the graph. The time it takes for the type 1 (resp. 2) infection to traverse an edge $e$ is given by a random variable $X_1(e)$ (resp. $X_2(e)$) and, if the vertex at the other end of the edge is still uninfected, it then becomes type 1 (resp. 2) infected and immune to the other type. Assuming that the degrees follow a power-law distribution with exponent $τ\in (1,2)$, we show that, with high probability as the number of vertices tends to infinity, one of the infection types occupies all vertices except for the starting point of the other type. Moreover, both infections have a positive probability of winning regardless of the passage times distribution. The result is also shown to hold for the erased configuration model, where self-loops are erased and multiple edges are merged, and when the degrees are conditioned to be smaller than $n^α$ for some $α> 0$.
