Fixed speed competition on the configuration model with infinite variance degrees: equal speeds

Abstract

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 $\tau\in (2,3)$. 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 $\tau-2$ 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.