[Paper Review] Local Limit Theorems and Number of Connected Hypergraphs

Michael Behrisch, Amin Coja‐Oghlan+1 more|arXiv (Cornell University)|Jun 4, 2007

Stochastic processes and statistical mechanics17 references9 citations

TL;DR

This paper presents a novel probabilistic approach to derive local limit theorems for the joint distribution of the size and order of the largest component in sparse random $d$-uniform hypergraphs $H_d(n,p)$ and $H_d(n,m)$, establishing asymptotic formulas for the probability of connectivity and the conditional edge distribution given connectivity. The key contribution is a non-combinatoric, probabilistic framework yielding precise asymptotic expressions for the number of connected $d$-uniform hypergraphs and the distribution of edges in connected components near the phase transition threshold.

ABSTRACT

Let $H_d(n,p)$ signify a random $d$-uniform hypergraph with $n$ vertices in which each of the ${n}\choose{d}$ possible edges is present with probability $p=p(n)$ independently, and let $H_d(n,m)$ denote a uniformly distributed with $n$ vertices and $m$ edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of $H_d(n,p)$ and $H_d(n,m)$ for the regime ${{n-1}\choose{d-1}} p,dm/n >(d-1)^{-1}+ε$. As an application, we obtain an asymptotic formula for the probability that $H_d(n,p)$ or $H_d(n,m)$ is connected. In addition, we infer a local limit theorem for the conditional distribution of the number of edges in $H_d(n,p)$ given connectivity. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.

Motivation & Objective

• To develop a new, purely probabilistic framework for analyzing the component structure of random $d$-uniform hypergraphs, avoiding reliance on enumerative combinatorics.
• To derive local limit theorems for the joint distribution of the number of vertices and edges in the largest component of $H_d(n,p)$ and $H_d(n,m)$ in the supercritical regime.
• To obtain an asymptotic formula for the probability that $H_d(n,p)$ or $H_d(n,m)$ is connected, extending classical results from random graphs to hypergraphs.
• To establish a local limit theorem for the number of edges in $H_d(n,p)$ conditioned on the hypergraph being connected, providing a precise edge distribution in the connected regime.

Proposed method

• Derive local limit theorems for the joint distribution of component order $\mathcal{N}(H)$ and size $\mathcal{M}(H)$ in $H_d(n,p)$ and $H_d(n,m)$ using a probabilistic approach based on generating functions and Fourier analysis.
• Use the solution $\rho$ to the transcendental equation $\rho = \exp(c(\rho^{d-1} - 1))$ to characterize the giant component size, where $c = \binom{n-1}{d-1}p$ or $c = d(d-1)m/n$.
• Apply asymptotic expansions and saddle-point methods to approximate the probability that the largest component has a given size and order, leveraging the fact that $\binom{n}{d}p^2 = o(1)$ in the supercritical regime.
• Condition on the largest component having size $\nu$ and derive the conditional distribution of the number of edges $\mu$ in $H_d(\nu,p)$ given connectivity, using multivariate normal approximation via joint cumulants.
• Express the final asymptotic formulas in terms of $\varrho$, the solution to $\varrho = \exp(\zeta(\varrho^{d-1} - 1))$, where $\zeta = \binom{\nu-1}{d-1}p$, to parameterize the results in terms of the component size.
• Use the relation $|n - (1 - \varrho)^{-1}\nu| = O(n^{-1})$ and $|c - \zeta(1 - \varrho)^{1-d}| = O(n^{-1})$ to reparameterize results from $n$ and $c$ to $\nu$ and $\zeta$ for the final formulas.

Experimental results

Research questions

• RQ1 What is the asymptotic probability that a random $d$-uniform hypergraph $H_d(n,p)$ or $H_d(n,m)$ is connected in the supercritical regime?
• RQ2 How does the joint distribution of the order and size of the largest component behave in $H_d(n,p)$ and $H_d(n,m)$ near the phase transition?
• RQ3 What is the conditional distribution of the number of edges in a $d$-uniform hypergraph given that it is connected and has a fixed number of vertices?
• RQ4 Can a purely probabilistic method replace classical enumerative combinatorics in deriving asymptotic formulas for connected hypergraphs?

Key findings

• The probability that $H_d(n,p)$ or $H_d(n,m)$ is connected is asymptotically $\Psi_d(\varrho, \zeta)^n \cdot \left(1 + o(1)\right)$, where $\varrho$ solves $\varrho = \exp(\zeta(\varrho^{d-1} - 1))$ and $\Psi_d$ is a computable function depending on $d$ and $\zeta$.
• For $d=2$, the asymptotic probability of connectivity reduces to $\exp(-\zeta) \cdot \Psi_2(\varrho, \zeta)$, matching known results for random graphs.
• The conditional distribution of the number of edges $\mu$ in a connected $H_d(\nu,p)$ is asymptotically normal with mean and variance derived from the joint cumulants of $\mathcal{N}(H)$ and $\mathcal{M}(H)$, with explicit formulas in terms of $\varrho$ and $\zeta$.
• The asymptotic formula for the number of connected $d$-uniform hypergraphs of order $n$ and size $m$ is derived as a consequence of the local limit theorems, providing a non-combinatoric derivation of the asymptotic count.
• The derived local limit theorems are valid in the supercritical regime where $c > (d-1)^{-1} + \varepsilon$, with $\varepsilon > 0$ fixed as $n \to \infty$, and hold uniformly over compact sets of $\zeta$ and $\nu$.
• The method avoids generating functions and complex asymptotic enumeration, instead using probabilistic tools such as moment generating functions, characteristic functions, and multivariate normal approximation to derive the results.