[Paper Review] A Phase Transition for the Metric Distortion of Percolation on the Hypercube

Omer Angel, Itaï Benjamini|ArXiv.org|Jun 25, 2003

Stochastic processes and statistical mechanics5 references4 citations

TL;DR

This paper establishes a phase transition in metric distortion for Bernoulli bond percolation on the hypercube $H_n$ at $\alpha = 1/2$ when $p = n^{-\alpha}$. For $\alpha < 1/2$, constant distortion is achievable with high probability; for $\alpha > 1/2$, distortion grows polynomially in $n$, due to the giant component becoming locally treelike and losing geodesic cycles. The result reveals a sharp threshold in the geometry of percolated hypercubes.

ABSTRACT

Let H_n be the hypercube {0,1}^n, and let H_{n,p} denote the same graph with Bernoulli bond percolation with parameter p=n^-α. It is shown that at α=1/2 there is a phase transition for the metric distortion between H_n and H_{n,p}. For α&lt;1/2, asymptotically there is a map from H_n to H_{n,p} with constant distortion (depending only on α). For α&gt;1/2 the distortion tends to infinity as a power of n. We indicate the similarity to the existence of a non-uniqueness phase in the context of infinite nonamenable graphs.

Motivation & Objective

• To analyze how metric distortion between the hypercube $H_n$ and its percolated version $H_{n,p}$ changes with the percolation parameter $p = n^{-\alpha}$.
• To identify a sharp phase transition in distortion behavior at $\alpha = 1/2$.
• To determine whether the distortion remains bounded or grows with $n$ depending on $\alpha$.
• To connect the geometric behavior of percolated hypercubes to broader phenomena in nonamenable graphs, such as non-uniqueness of infinite clusters.

Proposed method

• Define metric distortion using $D(f) = D_+(f)/D_-(f)$, where $D_+(f)$ measures expansion and $D_-(f)$ measures contraction of distances under map $f$.
• Use the identity map as a baseline and show that for $\alpha < 1/2$, most vertex pairs maintain bounded distance ratios in $H_{n,p}$, implying constant distortion.
• For $\alpha > 1/2$, prove that the giant component becomes locally treelike, lacking geodesic cycles, which forces distortion to grow.
• Bound the probability that a vertex lies within distance $\delta$ of a simple cycle of length $2l \in [2n^\beta, 2n^\gamma]$ using combinatorial counting and $p^{2l} = n^{-2\alpha l}$.
• Apply a double counting argument: if distortion is bounded by $n^\beta$, then the image of $f$ must contain at least $2^n n^{-n^\beta}$ vertices, all near cycles of controlled length.
• Show that the expected number of such vertices is sub-exponential, leading to a contradiction with the required size, proving distortion must exceed $n^\beta$ with high probability.

Experimental results

Research questions

• RQ1 Is there a phase transition in metric distortion at $\alpha = 1/2$ when $p = n^{-\alpha}$?
• RQ2 What happens to distortion when $p = a n^{-1/2}$—does it exhibit a critical window around $\alpha = 1/2$?
• RQ3 How does the distortion scale for $\alpha > 1/2$? Is the lower bound $n^\beta$ with $\beta < (2\alpha - 1)/4$ tight?
• RQ4 For $\alpha < 1/2$, is the distortion uniformly bounded, and can the bound be improved to $1 + \lfloor \alpha / (1/2 - \alpha) \rfloor$?
• RQ5 What is the minimal latency of an embedding of $H_n$ into $H_{n,p}$ when $\alpha < 1/2$?

Key findings

• For $\alpha < 1/2$, there exists a constant $c = c(\alpha)$ such that $\mathbb{P}(D(H_n, H_{n,p}) < c) \to 1$ as $n \to \infty$.
• For $\alpha > 1/2$, there exists $\beta = \beta(\alpha) > 0$ such that $\mathbb{P}(D(H_n, H_{n,p}) < n^\beta) \to 0$ as $n \to \infty$.
• The distortion grows polynomially in $n$ for $\alpha > 1/2$, with the exponent $\beta$ satisfying $\beta < (2\alpha - 1)/4$.
• The probability that a vertex lies within distance $\delta$ of a simple cycle of length in $[2n^\beta, 2n^\gamma]$ is at most $n^{\delta + 1 + n^\beta(\beta + 1 - 2\alpha)}$.
• The expected number of vertices within distance $2n^{2\beta}$ of a cycle of length in $[n^{\gamma - \beta}, 2n^{\gamma + \beta}]$ is sub-exponential, contradicting the existence of low-distortion maps.
• The proof implies that the distortion is super-polynomial in $n$ for $\alpha > 1/2$, and latency is likely super-polynomial when $\alpha > 1/2$.