[Paper Review] Limiting probabilities of first order properties of random sparse graphs and hypergraphs

Alberto Larrauri, Tobias Müller+1 more|arXiv (Cornell University)|Aug 20, 2020

Limits and Structures in Graph Theory12 references4 citations

TL;DR

This paper investigates the limiting probabilities of first-order (FO) logical properties in sparse random graphs and hypergraphs, showing that for random graphs $ G(n, c/n) $, the closure of limiting FO probabilities becomes the full interval $[0,1]$ when $ c \geq c_0 \approx 0.93 $, but contains gaps when $ c < c_0 $. The critical threshold $ c_0 $ arises from a phase transition in the structure of hypergraph cycle counts, with analogous results for $ d $-uniform hypergraphs where $ c_0 $ depends on $ d $.

ABSTRACT

Let $G_n$ be the binomial random graph $G(n,p=c/n)$ in the sparse regime, which as is well-known undergoes a phase transition at $c=1$. Lynch (Random Structures Algorithms, 1992) showed that for every first order sentence $\\phi$, the limiting probability that $G_n$ satisfies $\\phi$ as $n\ o\\infty$ exists, and moreover it is an analytic function of $c$. In this paper we consider the closure $\\overline{L_c}$ in $[0,1]$ of the set $L_c$ of all limiting probabilities of first order sentences in $G_n$. We show that there exists a critical value $c_0 \\approx0.93$ such that $\\overline{L_c}= [0,1]$ when $c \\ge c_0$, whereas $\\overline{L_c}$ misses at least one subinterval when $c&lt;c_0$. We extend these results to random $d$-uniform sparse hypergraphs, where the probability of a hyperedge is given by $p=c/n^{d-1}$.

Motivation & Objective

• To characterize the topological structure of the set of limiting first-order (FO) probabilities in sparse random graphs $ G(n, c/n) $.
• To determine whether the closure $ \overline{L_c} $ of these limiting probabilities is the full interval $[0,1]$ or contains gaps.
• To extend the analysis to $ d $-uniform random hypergraphs with edge probability $ p = c/n^{d-1} $, identifying a similar phase transition.
• To identify the critical threshold $ c_0 $ at which the transition from gap-containing $ \overline{L_c} $ to full interval $[0,1]$ occurs.
• To understand how the structure of $ \overline{L_c} $ evolves as $ c \to 0 $, particularly the growth in the number of intervals.

Proposed method

• Uses Brun's sieve to establish convergence of subgraph counts to independent Poisson distributions in the sparse regime.
• Analyzes the expected number of cycles and unicyclic hypergraphs using asymptotic enumeration and generating functions.
• Applies the configuration model and hypergraph cycle counting to derive expressions for the limiting probabilities $ p_c(\phi) $ of FO sentences.
• Derives the critical threshold $ c_0 $ by solving the equation $ \exp\left(\frac{c}{2} + \frac{c^2}{4}\right)\sqrt{1 - c} = \frac{1}{2} $ for graphs and a similar equation for hypergraphs.
• Employs recursive inequalities to show that $ p_i \leq \sum_{j > i} p_j $ for large $ i $, proving $ \overline{L_c} $ is a finite union of intervals.
• Uses case analysis on hypergraph size and automorphism counts to bound the sum of probabilities of larger hypergraphs and compare with smaller ones.

Experimental results

Research questions

• RQ1 At what critical value $ c_0 $ does the closure $ \overline{L_c} $ of limiting FO probabilities in $ G(n, c/n) $ transition from having gaps to covering the full interval $[0,1]$?
• RQ2 How does the structure of $ \overline{L_c} $ change as $ c $ decreases below $ c_0 $, particularly in terms of the number of intervals?
• RQ3 What is the analogous critical threshold $ c_0 $ for $ d $-uniform random hypergraphs with edge probability $ p = c/n^{d-1} $?
• RQ4 Why does the critical threshold differ between graphs and hypergraphs, and how does the cycle structure (e.g., 2-cycles vs. 3-cycles) affect this?
• RQ5 How does the number of intervals in $ \overline{L_c} $ grow as $ c \to 0 $, and what determines the ordering of fragment probabilities?

Key findings

• For $ c \geq c_0 \approx 0.93 $, the closure $ \overline{L_c} $ of limiting FO probabilities in $ G(n, c/n) $ is the full interval $[0,1]$.
• For $ c < c_0 $, $ \overline{L_c} $ contains at least one gap, i.e., a subinterval of $[0,1]$ not intersecting $ \overline{L_c} $.
• The critical threshold $ c_0 $ is the unique positive solution to $ \exp\left(\frac{c}{2} + \frac{c^2}{4}\right)\sqrt{1 - c} = \frac{1}{2} $.
• For $ d $-uniform hypergraphs, the critical threshold $ c_0 $ is the unique positive solution to $ \exp\left(\frac{c}{2(d-2)!}\right)\sqrt{1 - \frac{c}{2(d-2)!}} = \frac{1}{2} $.
• When $ c \geq c_0 $, $ \overline{L_c} = [0,1] $, and when $ c < c_0 $, $ \overline{L_c} $ is a finite union of intervals with at least one gap.
• As $ c \to 0 $, the number of intervals in $ \overline{L_c} $ grows without bound, indicating increasingly complex structure in the limiting probability set.