Upper tails for irregular graphs beyond the mean-field regime

Abstract: Let $G_{n,p}$ be the binomial random graph of density $p$ and let $X_H$ be the number of copies of a fixed graph $H$ in $G_{n,p}$. We prove asymptotically tight bounds on the logarithmic upper-tail probability of $X_H$ whenever $H$ is a connected, irregular graph with maximum degree $Δ\ge 2$ and $$p \ge n^{-1/Δ- \varepsilon_H} (\log n)<sup>{ω(1)}$$ for an explicit $\varepsilon_H &gt;0$. These bounds are expressed in terms of a new variational problem that generalises the combinatorial optimisation problem arising from the naïve mean-field approximation. This new variational problem includes an entropy term that corresponds to the large number of embeddings of certain highly structured graphs in $K_n$. For a certain class of irregular graphs $H$ that we call stable, we show that this description of the upper-tail probability is valid in a range of densities that is optimal up to a poly($\log\log n$) factor. For a further subclass of stable graphs, which includes all irregular complete bipartite graphs, we show that this range of densities is optimal up to a multiplicative constant.

• The paper introduces an entropic variational principle that accurately quantifies the upper-tail rate for irregular subgraph counts beyond traditional mean-field predictions.
• It establishes asymptotically tight bounds on the probability of rare subgraph occurrences by leveraging innovative hub-core and tight-core constructions.
• The study distinguishes between stable and clean graphs, demonstrating how entropy corrections critically impact upper-tail deviations in sparse random graph models.

Upper Tails for Irregular Graphs Beyond the Mean-Field Regime

Introduction and Motivation

This paper addresses the upper-tail large deviation problem for subgraph counts in the binomial random graph model $G_{n,p}$. For a fixed graph $H$, let $X_H$ denote the number of copies of $H$ in $G_{n,p}$. While the typical behavior of $X_H$ and lower-tail deviations are well-understood via moment and concentration methods, the upper-tail problem—in particular, the asymptotics of $$\log \mathbb{P}(X_H \geq (1+\delta) \mathbb{E}[X_H])$$ for fixed $\delta>0$—remains challenging, especially for irregular $H$ and in the sparse regime $p \ll 1$.

Earlier works, primarily on regular $H$0 or specific motifs (like triangles), established sharp asymptotics for the logarithmic upper-tail probability and introduced the mean-field variational principle as a heuristic and technical tool. However, the irregular case—where $H$1 is not regular and has maximum degree $H$2—presents substantially different combinatorial and probabilistic phenomena, particularly as $H$3 drops below $H$4.

Key Contributions and Results

The principal achievements of this paper are:

• Refinement of Variational Principles: The authors identify the failure of naive mean-field bounds (based on product measure large deviation heuristics) for irregular graphs in the truly sparse regime, and instead develop a new entropic variational principle $H$5, which includes an explicit entropy correction capturing the combinatorial richness of seeds (subgraphs whose conditioning triggers the upper tail) and their embeddings in $H$6.
• Asymptotically Tight Bounds: They establish that this new variational problem yields the correct exponential rate for the upper-tail probability:

$H$7

for connected, irregular $H$8 with $H$9 and densities $X_H$0, where $X_H$1 is explicitly computed.

• Introduction of Hub-Cores and Tight Cores: The authors introduce combinatorial objects—hub-cores and tight cores—that generalize the notion of the minimal seeds, capturing the structure of subgraphs whose appearance triggers the upper-tail event. The asymptotic minimizers in $X_H$2 are hub-cores, and the paper rigorously analyzes their entropy contribution to the large deviations rate.
• Structural Dichotomy: Stable and Clean Graphs: Within irregular graphs, a detailed classification is provided:
  • Stable graphs: those for which the leading order of the upper tail is determined by densest subgraphs.
  • Clean graphs (typically irregular complete bipartite graphs and stars): for which the entropy correction leads to optimal bounds up to a constant factor for all $X_H$3 above the appearance threshold.
  • For one-dimensional stable graphs, the variational rate is shown to be sharp up to polylogarithmic factors in $X_H$4.
• Optimality and Counterexamples: The paper constructs examples (non-stable irregular graphs) showing that the naive mean-field upper bound can be off by orders-of-magnitude in the sparse regime, giving a definitive negative answer to conjectures in the literature regarding possible general asymptotics for upper tails, such as the DeMarco-Kahn conjecture (see also [vsileikis2019counterexample]).

Technical Approach and Methods

The analysis utilizes a mixture of probabilistic large deviation techniques, entropy methods, and detailed combinatorial construction:

• Combinatorial Optimization and Entropy Cost: The entropy correction is rigorously formulated in the variational problem by analyzing the number of embeddings of minimal seeds (hub-cores) into $X_H$5. This produces a negative entropy term $X_H$6 in the variational formula, where $X_H$7 is a hub-core, $X_H$8 is its number of vertices, and $X_H$9 is a parameter controlling the entropic regime.
• High-Moment and Planting Techniques: Conditional second-moment and high-moment arguments are used to show that conditioning on the presence of a minimal seed or hub-core essentially brings about the upper-tail event. The analysis is refined to handle the plethora of embeddings and to show those seeds dominate the union bound.
• Sharp Control for Stable and Clean Graphs: The proof exploits structural properties of $H$0 (via fractional independence numbers and extremal subgraphs) to identify for which irregular graphs the upper-tail rate transitions from being dictated by product-type configuration to the combinatorial entropy-dominated regime.
• Optimal Range of $H$1: For stable and clean graphs, the authors show that the entropic variational principle is sharp for $H$2 down to the appearance threshold $H$3 (where $H$4 is the maximal edge-to-vertex ratio for proper subgraphs of $H$5), up to at most polylogarithmic factors in $H$6 or $H$7.

Notable Results and Quantitative Claims

• For irregular $H$8 and $H$9, the following holds:

$G_{n,p}$0

where

$G_{n,p}$1

• For stable and clean graphs, this asymptotic remains sharp for $G_{n,p}$2, matching known lower bounds up to constants.
• In ranges where $G_{n,p}$3, $G_{n,p}$4 can be strictly smaller than the naive mean-field rate function $G_{n,p}$5, sometimes by polynomial factors, demonstrating the qualitative breakdown of mean-field predictions.

Implications and Broader Context

This work decisively clarifies the correct rate function governing rare upper deviations for subgraph counts in sparse random graphs above the Poisson regime, particularly for irregular graphs. The findings reveal that combinatorial entropy—i.e., the number of possible locations of rare "seed" structures—can dramatically alter large deviation rates, sometimes lowering the cost by polynomially many orders compared to naive heuristics.

From a theoretical standpoint, this paper answers several long-standing open questions and closes gaps left by earlier "mean-field regime" analyses. On the practical side, the refined understanding of upper-tail probabilities for irregular structures is critical for rare-event simulation, threshold phenomena in random graph processes, and probabilistic combinatorics relying on precise control of higher moments and deviations.

The techniques and variational reasoning introduced here could have further applications across areas studying rare structure-induced deviations in high-dimensional or networked random environments.

Future Directions

Future research directions suggested by this work include:

• Extending the upper-tail analysis to more general (possibly inhomogeneous or hypergraph) models using similar entropic variational principles.
• Investigating tight asymptotics and possible entropy corrections below the appearance threshold and in the Poisson regime.
• Further combinatorial classification of irregular graphs where more exotic entropy phenomena may occur, or where the extremal seeds are not uniquely determined.
• Exploring algorithmic implications for sampling or counting rare subgraph appearances in random graphs.

Conclusion

The paper establishes a comprehensive and precise characterization of upper-tail large deviations for counts of irregular subgraphs in sparse random graphs, beyond the mean-field regime. By incorporating a combinatorial entropy correction into the variational principle and introducing the concept of hub-cores and tight cores, the analysis not only yields sharp logarithmic asymptotics but highlights the nuanced interplay between combinatorial structure and probabilistic rare events. The work sets a new standard for the theory of upper tails in random graphs and provides robust techniques likely to impact a range of future investigations.