[Paper Review] Spaces of completions of elementary theories and convergence laws for random hypergraphs

Nicolau C. Saldanha, Márcio Telles|arXiv (Cornell University)|Feb 21, 2016

Limits and Structures in Graph Theory10 references3 citations

TL;DR

This paper establishes convergence laws for random (d+1)-uniform hypergraphs in the binomial model $G^{d+1}(n,p)$, showing that for logarithmo-exponential edge probabilities $p \ll n^{-d+\epsilon}$, the probabilities of all elementary properties converge to limits in $[0,1]$. The key contribution is a topological framework: the space of completions of the almost sure theory is compact, metrizable, and totally disconnected, with structure ranging from a point to a Cantor space depending on $p$, and convergence is governed by a Borel probability measure on this space.

ABSTRACT

Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N} o[0,1]$. We prove that, for all logarithmo-exponential $p\ll n^{-d+ε}$, the probabilities of all elementary properties of hypergraphs converge, with particular emphasis in the ranges $p(n)\sim C/n^d$ and $p(n) \sim C\log(n)/n^d$. The exposition is unified by constructing, for each such function $p$, the topological space of all completions of its almost sure theory. This space turns out to be compact, metrizable and totally disconnected, but further properties depend on the range of $p$. The convergence of the probabilities of elementary properties is associated with a borelian probability measure on the space.

Motivation & Objective

• To complete the classification of convergence laws in the Double Jump window $p \sim C \log n / n^d$ for $C > 0$, which had gaps in prior work.
• To provide a detailed and systematic analysis of convergence laws in the Double Jump regime, extending Spencer's sketch in *The Strange Logic of Random Graphs*.
• To introduce and formalize a topological framework—spaces of completions of almost sure theories—for understanding convergence behavior in random hypergraphs.
• To generalize zero-one and convergence laws to all logarithmo-exponential functions $p(n)$, showing no further 'gaps' exist in this class.

Proposed method

• Construct the topological space $\mathcal{K}(\Theta_p)$ of all completions of the almost sure theory $\Theta_p$ for a given edge probability function $p(n)$, which is compact, metrizable, and totally disconnected.
• Associate a Borel probability measure on $\mathcal{K}(\Theta_p)$ that governs the limiting probabilities of elementary properties.
• Use factorial moment methods to show that the distributions of $(r,s)$-patterns of edges and cycles in balls of radius $r$ converge to independent Poisson distributions with explicit means.
• Prove that for $p \sim \lambda / n^d$, the space $\mathcal{K}(\Theta_p)$ is a Cantor space, implying uncountably many completions and non-trivial convergence behavior.
• Apply the method of asymptotic independence of patterns via symmetrization and cycle counting, showing $\mathbb{P}[A \triangle B] = o(1)$ for symmetric differences of edge sets.
• Use the structure of spanning trees of elementary properties to show that the space of completions has no isolated points when $p$ is in the Double Jump regime.

Experimental results

Research questions

• RQ1 For which logarithmo-exponential functions $p(n)$ do the probabilities of all elementary properties in $G^{d+1}(n,p)$ converge to limits in $[0,1]$?
• RQ2 What topological structure does the space of completions $\mathcal{K}(\Theta_p)$ of the almost sure theory $\Theta_p$ possess for different ranges of $p(n)$?
• RQ3 How does the space $\mathcal{K}(\Theta_p)$ behave specifically in the Double Jump regime $p \sim C \log n / n^d$?
• RQ4 Can the convergence of probabilities be described via a Borel probability measure on a topological space of completions, and how does this generalize zero-one laws?
• RQ5 What is the role of $(r,s)$-patterns of edges and cycles in characterizing the limiting behavior of hypergraph properties?

Key findings

• For all logarithmo-exponential $p \ll n^{-d+\epsilon}$, the probabilities of all elementary properties converge to limits in $[0,1]$, establishing convergence laws.
• The space of completions $\mathcal{K}(\Theta_p)$ is compact, metrizable, and totally disconnected for all such $p$, with structure depending on the range of $p$.
• When $p \sim \lambda / n^d$, the space $\mathcal{K}(\Theta_p)$ is a Cantor space, indicating uncountably many completions and non-trivial convergence behavior.
• In the Double Jump regime $p \sim C \log n / n^d$, the space $\mathcal{K}(\Theta_p)$ is a Cantor space, and the convergence law holds with no isolated points.
• The limiting probabilities of elementary properties are governed by a Borel probability measure on $\mathcal{K}(\Theta_p)$, generalizing the concept of complete sets of completions.
• The distribution of $(r,s)$-patterns of edges and cycles in balls of radius $r$ converges to independent Poisson distributions with means proportional to $\lambda^l / l! \cdot p_\Delta$, ensuring asymptotic independence.