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Upper tails for homomorphism counts in sparse random hypergraphs

Nicholas A. Cook, Nguyen Nguyen

TL;DR

This work studies the upper tails of H-counts in sparse random r-graphs by a two-step Naive Mean-Field (NMF) program: first reducing the tail problem to a weighted-graph entropy optimization, and then solving a finite-dimensional variational problem to identify the leading rate function ρ_H(δ). The authors prove the conjecture of Liu–Zhao for broad classes of H, including complete r-partite graphs, tight cycles, and the Fano plane, and derive a general large-deviation upper bound for H with certain edge-covering properties. By exploiting fractional hypergraph theory and strict stable labelings, they isolate dominant planted structures that realize the upper-tail event and match lower bounds to obtain sharp asymptotics: R_H(n,p,δ) ∼ (1/r!) ρ_H(δ) n^r p^{Δ(H)} log(1/p) in regimes where p → 0 slowly enough. The results provide a unified framework extending known 2-graph theory to hypergraphs, clarifying phase-transition phenomena and enabling explicit rate functions for several natural H classes, with potential applications to sparse random structures and large deviations in nonlinear settings.

Abstract

The "infamous upper tail problem" for $r$-uniform hypergraphs is to estimate the probability that the number of copies of a fixed hypergraph $H$ in a large binomial $r$-uniform hypergraph $\boldsymbol{G}$ exceeds its expectation by a constant factor. The problem was popularized by Janson and Ruciński and, particularly in the case of graphs ($r=2$), has been a driving example in the development of nonlinear large deviations theory. Recent work of the first author with Dembo and Pham has accomplished the \emph{naive mean-field reduction step}, reducing the upper tail problem to an entropic variational problem on a space of weighted graphs. The latter was resolved for counts of $r$-uniform cliques and a certain linear 3-uniform hypergraph by Liu and Zhao, who also conjectured a general formula. We confirm their conjecture for other classes of hypergraphs, including complete $r$-partite $r$-graphs, tight cycles, and the Fano plane. We also prove a general large deviation upper bound for counts of $r$-graphs $H$ satisfying certain edge covering properties.

Upper tails for homomorphism counts in sparse random hypergraphs

TL;DR

This work studies the upper tails of H-counts in sparse random r-graphs by a two-step Naive Mean-Field (NMF) program: first reducing the tail problem to a weighted-graph entropy optimization, and then solving a finite-dimensional variational problem to identify the leading rate function ρ_H(δ). The authors prove the conjecture of Liu–Zhao for broad classes of H, including complete r-partite graphs, tight cycles, and the Fano plane, and derive a general large-deviation upper bound for H with certain edge-covering properties. By exploiting fractional hypergraph theory and strict stable labelings, they isolate dominant planted structures that realize the upper-tail event and match lower bounds to obtain sharp asymptotics: R_H(n,p,δ) ∼ (1/r!) ρ_H(δ) n^r p^{Δ(H)} log(1/p) in regimes where p → 0 slowly enough. The results provide a unified framework extending known 2-graph theory to hypergraphs, clarifying phase-transition phenomena and enabling explicit rate functions for several natural H classes, with potential applications to sparse random structures and large deviations in nonlinear settings.

Abstract

The "infamous upper tail problem" for -uniform hypergraphs is to estimate the probability that the number of copies of a fixed hypergraph in a large binomial -uniform hypergraph exceeds its expectation by a constant factor. The problem was popularized by Janson and Ruciński and, particularly in the case of graphs (), has been a driving example in the development of nonlinear large deviations theory. Recent work of the first author with Dembo and Pham has accomplished the \emph{naive mean-field reduction step}, reducing the upper tail problem to an entropic variational problem on a space of weighted graphs. The latter was resolved for counts of -uniform cliques and a certain linear 3-uniform hypergraph by Liu and Zhao, who also conjectured a general formula. We confirm their conjecture for other classes of hypergraphs, including complete -partite -graphs, tight cycles, and the Fano plane. We also prove a general large deviation upper bound for counts of -graphs satisfying certain edge covering properties.

Paper Structure

This paper contains 37 sections, 31 theorems, 230 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Background
  4. Step 1: NMF approximation
  5. Step 2: NMF variational problem
  6. Organization of the paper
  7. Notational conventions
  8. Asymptotic notation
  9. Graphs and graph parameters
  10. Homomorphism densities
  11. Preliminary estimates
  12. Relative entropy
  13. Homomorphism densities
  14. Optimization
  15. Bounds from fractional hypergraph theory
  16. ...and 22 more sections

Key Result

Theorem 1.1

Fix an $r$-graph $H$ and $\delta>0$. Assume $p=o(1)$.

Figures (6)

  • Figure 1: 3-graphs for which the large deviations rate function ${\overline\rho}_H$ is not given by the function ${\sigma}_H$ from \ref{['def:rbi']}. Left: The 3-graph from \ref{['thm:LiZh']}(b). The 6 vertices are represented by circles and the 4 edges by straight lines. Center: The Fano plane from \ref{['thm:main']}(c) is a 3-regular 3-graph with 7 vertices and 7 edges. Vertices are by represented small circles and edges by the 6 straight lines and one large circle. Right: (Also from \ref{['thm:main']}(\ref{['main.Fano']})) the subgraph of the Fano plane obtained by removing a single edge.
  • Figure 2: Top: The tight 3-uniform 7-cycle $C_7^{(3)}$. Vertices are vertical lines, cyclically labeled on the number line, and edges are diagonal lines. Bottom: A subgraph of $C_7^{(3)}$in which vertex 1 has degree 3 and all other vertices have degree 2.
  • Figure 3: Illustration of a subgraph $F'\subset F$ as in \ref{['def:good']}(VG\ref{['VG2']}). Vertices are circles and edges are straight lines. Vertices in $S$ are solid black. Vertex $v\in S$ has degree 2 in $F'$, while all other degree-2 vertices in $F'$ (open circles) lie outside $S$. (Other vertices of $F'$ are not depicted.) In this example $F'$ consists of 3 loose paths of lengths 3, 1 and 5.
  • Figure 4: To verify (VG\ref{['VG2']}) of \ref{['def:good']}, we select the edges of $F'$ as depicted. In this example $H$ has parts of size 6, and $F$ is the subgraph consisting of all edges incident to the set $S$ of 5 red vertices in $V_1$. The lines depict the choice of $|S|+1=6$ edges for $F'$.
  • Figure 5: Stable labelings of the Fano plane. Empty vertices are labeled 0. There are $7$ stable labelings of the first type, $7$ stable labelings of the second type, and $1$ stable labeling of the third type. Note that the middle labeling is supported on a proper subgraph, obtained by removing the circular edge, and takes fractional values. This contrasts with the $r$-graphs considered in Sections \ref{['sec:gen']}--\ref{['sec:cycles']}, where all labelings supported on proper subgraphs were indicators of independent sets, like the labeling on the left above.
  • ...and 1 more figures

Theorems & Definitions (74)

  • Theorem 1.1: LiZh, CDP
  • Theorem 1.2
  • Remark 1.3: Cycles case
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7: NMF approximation CDP
  • Conjecture 1.8: Liu--Zhao LiZh
  • Proposition 1.9
  • Lemma 2.1: LiZh
  • ...and 64 more