Efficiently Constructible Huge Graphs That Preserve First Order Properties of Random Graphs

Abstract

Abstract. We construct efficiently computable sequences of randomlooking graphs that preserve properties of the canonical random graphs G(2 n , p(n)). We focus on first-order graph properties, namely properties that can be expressed by a formula φ in the language where variables stand for vertices and the only relations are equality and adjacency (e.g. having an isolated vertex is a first-order property ∃x∀y(¬edge(x, y))). Random graphs are known to have remarkable structure w.r.t. first order properties, as indicated by the following 0/1 law: for a variety of choices of p(n), any fixed first-order property φ holds for G(2 n , p(n)) with probability tending either to 0 or to 1 as n grows to infinity.We first observe that similar 0/1 laws are satisfied by G(2 n , p(n)) even w.r.t. sequences of formulas {φn} n∈N with bounded quantifier depth, depth(φn) ≤ n lg(1/p(n)) . We also demonstrate that 0/1 laws do not hold for random graphs w.r.t. properties of significantly larger quantifier depth. For most choices of p(n), we present efficient constructions of huge graphs with edge density nearly p(n) that emulate G(2 n , p(n)) by satisfying Θ( n lg(1/p(n)) )-0/1 laws. We show both probabilistic constructions (which also have other properties such as K-wise independence and being computationally indistinguishable from G(N, p(n)) ), and deterministic constructions where for each graph size we provide a specific graph that captures the properties of G(2 n , p(n)) for slightly smaller quantifier depths.

On the complexity of powering in finite fields

“…Multiplicative characters of finite fields display highly pseudorandom properties, and yet are tractable for analysis (through tools like the Weil-bound for character sums). These pseudorandom properties of characters have been used in the construction of some remarkable pseudorandom graphs (the Paley graphs, see [8,15]), as well as other combinatorial objects likebiased sets [1,2]. It is therefore natural to ask whether pseudorandom constructions based on multiplicative characters can be executed in AC 0 (⊕).…”

Logical complexity of graphs: A survey

“…Therefore, the standard version of the 0-1 law implies that R(k, n) → 0 as n → ∞ for any fixed k. Naor, Nussboim, and Tromer [58] showed that R(log n−2 log log n, n) → 0. Another result in [58] states that one can choose p(n) = 1/2 + o(1) and k(n) = (2+o (1)) log n such that the probability that G n,p has a k(n)-clique is bounded away from 0 and 1. Thus for this probability p(n) the 0-1 law does not hold with respect to formulas of depth k(n).…”

“…We now prove Part 1. Like the proof in [58] we use the extension property, but we argue in a slightly different way. Proof of Part 1.…”

The logic of random regular graphs

“…Another approach was taken by Naor et al in [36] where they let the quantifier depth to go to infinity with n and considered the maximal quantifier depth for which the Zero-One law for G(n, p) with constant p still holds. This gives rise to a few more questions.…”