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Extreme local statistics in random graphs: maximum tree extension counts

Pedro Araújo, Simon Griffiths, Matas Šileikis, Lutz Warnke

TL;DR

This paper investigates extreme local statistics in G_{n,p} by studying the maximum over vertices of the number of rooted copies of a fixed tree T. It uncovers a dichotomy: in the dense regime p(1-p)n \gg \log n, M_n concentrates tightly around the mean mu_T with fluctuations on the order of sigma_T \sqrt{2\log n}, and the maximum is effectively governed by near-maximum-degree vertices. In the sparser regime 1 \ll pn \ll \log n, the authors develop a fine-grained, tree-class dependent theory, reducing sparse-case analysis to tails of random Galton–Watson trees and solving for specific trees (notably paths P_m and spherically symmetric trees T_{a,b}) via large-deviation optimization; the resulting M_n scales interpolate between distinct mechanisms depending on p, including sums over degree-structure combinations. The work advances the understanding of extreme local statistics in random graphs, connects to known results on maximum degree, and provides precise asymptotics and probabilistic tools that may inform broader extremal-graph and probabilistic-combinatorial analyses.

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.

Extreme local statistics in random graphs: maximum tree extension counts

TL;DR

This paper investigates extreme local statistics in G_{n,p} by studying the maximum over vertices of the number of rooted copies of a fixed tree T. It uncovers a dichotomy: in the dense regime p(1-p)n \gg \log n, M_n concentrates tightly around the mean mu_T with fluctuations on the order of sigma_T \sqrt{2\log n}, and the maximum is effectively governed by near-maximum-degree vertices. In the sparser regime 1 \ll pn \ll \log n, the authors develop a fine-grained, tree-class dependent theory, reducing sparse-case analysis to tails of random Galton–Watson trees and solving for specific trees (notably paths P_m and spherically symmetric trees T_{a,b}) via large-deviation optimization; the resulting M_n scales interpolate between distinct mechanisms depending on p, including sums over degree-structure combinations. The work advances the understanding of extreme local statistics in random graphs, connects to known results on maximum degree, and provides precise asymptotics and probabilistic tools that may inform broader extremal-graph and probabilistic-combinatorial analyses.

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.
Paper Structure (27 sections, 29 theorems, 219 equations)

This paper contains 27 sections, 29 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. How and when does M_n concentrate around the mean?
  3. Can M_n behave differently for smaller edge probabilities?
  4. Background and discussion
  5. Organization of the paper
  6. Preliminaries
  7. Asymptotics of binomial probabilities
  8. Extremal degrees of binomial random graphs
  9. Expectation and variance of tree extension counts
  10. Trees: the dense case
  11. Proof of the main result for trees: Theorem and Corollaries ,
  12. Proof of main technical result for trees: Proposition
  13. Trees: reduction to random trees in the sparse case
  14. Proofs of Lemmas and
  15. Paths: the sparse case
  16. ...and 12 more sections

Key Result

Theorem 1

Fix a rooted tree $T$, with root degree $a$. If $pqn \gg \log n$, then where the variance $\sigma_T^2:=\operatorname{Var} X_v$ satisfies $\sigma_T^2 \sim a^2\mu_T^2 q/(pn)$.

Theorems & Definitions (64)

  • Theorem 1: Maximum for general trees, dense case
  • Proposition 2
  • Corollary 3
  • Theorem 4: Maximum for spherically symmetric trees $T_{a,b}$, simplified
  • Remark 1
  • Theorem 5: Maximum for paths of length two, sparse case
  • Remark 2
  • Corollary 6
  • Corollary 7: Maximum for general trees: Gumbel distribution
  • Proposition 8
  • ...and 54 more