Electronic Communications in Probability

Connectivity threshold for random subgraphs of the Hamming graph

Lorenzo Federico, Remco van der Hofstad, and Tim Hulshof

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We study the connectivity of random subgraphs of the $d$-dimensional Hamming graph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$ vertices. We sample the random subgraph with an i.i.d. Bernoulli bond percolation on $H(d,n)$ with parameter $p$. We identify the window of the transition: when $ np- \log n \to - \infty $ the probability that the graph is connected tends to $0$, while when $ np- \log n \to + \infty $ it converges to $1$. We also investigate the connectivity probability inside the critical window, namely when $ np- \log n \to t \in \mathbb{R} $. We find that the threshold does not depend on $d$, unlike the phase transition of the giant connected component of the Hamming graph (see [1]). Within the critical window, the connectivity probability does depend on $d$. We determine how.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 27, 8 pp.

Received: 13 August 2015
Accepted: 23 February 2016
First available in Project Euclid: 14 March 2016

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Zentralblatt MATH identifier

Primary: 05C40: Connectivity 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

connectivity threshold percolation random graph critical window

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Federico, Lorenzo; van der Hofstad, Remco; Hulshof, Tim. Connectivity threshold for random subgraphs of the Hamming graph. Electron. Commun. Probab. 21 (2016), paper no. 27, 8 pp. doi:10.1214/16-ECP4479. https://projecteuclid.org/euclid.ecp/1457978024

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