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arXiv · 2310.11661

Extreme local statistics in random graphs: maximum tree extension counts

Pedro Araújo·Simon Griffiths·Matas Šileikis·Lutz Warnke
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Abstract

We consider maximum rooted tree extension counts in random graphs, i.e., we consider M_n = \max_v X_v where X_v counts the number of copies of a given tree in G_{n,p} rooted at vertex v. We determine the asymptotics of M_n when the random graph is not too sparse, specifically when the edge probability p=p(n) satisfies p(1-p)n \gg \log n. The problem is more difficult in the sparser regime 1 \ll pn \ll \log n, where we determine the asymptotics of M_n for specific classes of trees. Interestingly, here our large deviation type optimization arguments reveal that the behavior of M_n changes as we vary p=p(n), due to different mechanisms that can make the maximum large.

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Pedro Araújo, Simon Griffiths, Matas Šileikis, Lutz Warnke. 2023-10-18. Extreme local statistics in random graphs: maximum tree extension counts. https://arxiv.org/abs/2310.11661

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