具有无穷大变异度的构形模型上的定速竞争：等速
Fixed speed competition on the configuration model with infinite variance degrees: equal speeds
摘要：
在具有独立同分布的组态模型上，研究了两种扩展色从单个光源开始的竞争。度遵循指数为$τ\in (2,3)$的幂律分布。在这个模型中，两种颜色在未加权的随机图上以固定且相等的速度传播。我们分析这两种颜色最终绘制了多少个顶点。我们发现，共存敏感地依赖于源顶点的初始局部邻域：如果这些邻域是“足够不同的”，那么就没有共存，和“失败者”的颜色画一个多项式分数的顶点与随机指数。如果起始顶点的局部邻域“足够相似”，则存在共存，即，两种颜色都绘制严格正比例的顶点。我们给出了一个定量表征的“相似”的本地社区：两个随机变量描述的源顶点的本地社区的双指数增长性必须在一个因素$τ-2$的其他。这两个结果都以正概率发生，并且具有明确可计算的渐近值。这张图片强化了人们的普遍看法，即位置是广告的一个重要特征。这篇论文是一个后续的类似命名的论文，处理的情况下，两种颜色的速度不相等。在那里，我们已经表明，更快的颜色几乎所有的顶点，而较慢的颜色只画一个随机的子多项式分数的顶点。
英文摘要：
We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent $τ\in (2,3)$. In this model two colors spread with a fixed and equal speed on the unweighted random graph. We analyse how many vertices the two colors paint eventually. We show that coexistence sensitively depends on the initial local neighborhoods of the source vertices: if these neighborhoods are `dissimilar enough', then there is no coexistence, and the `loser' color paints a polynomial fraction of the vertices with a random exponent. If the local neighborhoods of the starting vertices are `similar enough', then there is coexistence, i.e., both colors paint a strictly positive proportion of vertices. We give a quantitative characterization of `similar' local neighborhoods: two random variables describing the double exponential growth of local neighborhoods of the source vertices must be within a factor $τ-2$ of each other. Both of the two outcomes happen with positive probability with asymptotic value that is explicitly computable. This picture reinforces the common belief that location is an important feature in advertising. This paper is a follow-up of the similarly named paper that handles the case when the speeds of the two colors are not equal. There, we have shown that the faster color paints almost all vertices, while the slower color paints only a random sub-polynomial fraction of the vertices.
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