Emergence of a Giant Component in Random Site Subgraphs of a d-Dimensional Hamming Torus

TL;DR

This paper analyzes the phase transition for the emergence of a giant component in random site subgraphs of a d-dimensional Hamming torus, revealing thresholds depending on vertex removal probability and graph parameters.

Contribution

It provides a detailed characterization of the giant component emergence and connectivity thresholds in site percolation on the Hamming torus, including explicit formulas and surprising distinctions from edge percolation.

Findings

Existence of a critical  for giant component emergence depending on  > 0.

Giant component size scales as (1-q)  (_1 _d) n^{d-1} above the threshold.

Connectivity threshold depends on the parameter c relative to (d-1)/(_1 + ... + _d).

Abstract

The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1-p. We show that if p=\lambda / n, then there exists \lambda_c > 0, which is the positive root of a degree d polynomial whose coefficients depend on a_1, ..., a_d, such that for \lambda < \lambda_c the largest component has O(log n) vertices (a.a.s. as n \to \infty), and for \lambda > \lambda_c the largest component has (1-q) \lambda (\prod_i a_i) n^{d-1} + o(n^{d-1}) vertices and the second largest component has O(log n) vertices (a.a.s.). An implicit formula for q < 1 is also given. Surprisingly, the value of \lambda_c that we find is distinct from the critical value for the emergence of a giant component in the random edge subgraph of the Hamming torus. Additionally, we show that if p = c log n / n, then when c < (d-1) / (\sum a_i) the site subgraph of the Hamming torus is not connected, and when c > (d-1) / (\sum a_i) the subgraph is connected (a.a.s.). We also show that the subgraph is connected precisely when it contains no isolated vertices.