TY - JOUR

T1 - Percolation on irregular high-dimensional product graphs

AU - Diskin, Sahar

AU - Erde, Joshua

AU - Kang, Mihyun

AU - Krivelevich, Michael

N1 - Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press.

PY - 2023

Y1 - 2023

N2 - We consider bond percolation on high-dimensional product graphs G = □ i=1t G(i), where □ denotes the Cartesian product. We call the G(i) the base graphs and the product graph G the host graph. Very recently, Lichev (J. Graph Theory, 99(4):651-670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph Gp undergoes a phase transition when p is around 1|d, where d is the average degree of the host graph. In the supercritical regime, we strengthen Lichev's result by showing that the giant component is in fact unique, with all other components of order o(|G|), and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev (J. Graph Theory, 99(4):651-670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdos-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. arXiv:2209.03722, 2022).

AB - We consider bond percolation on high-dimensional product graphs G = □ i=1t G(i), where □ denotes the Cartesian product. We call the G(i) the base graphs and the product graph G the host graph. Very recently, Lichev (J. Graph Theory, 99(4):651-670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph Gp undergoes a phase transition when p is around 1|d, where d is the average degree of the host graph. In the supercritical regime, we strengthen Lichev's result by showing that the giant component is in fact unique, with all other components of order o(|G|), and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev (J. Graph Theory, 99(4):651-670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdos-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. arXiv:2209.03722, 2022).

KW - Bond percolation

KW - component sizes

KW - random subgraphs

UR - http://www.scopus.com/inward/record.url?scp=85181445986&partnerID=8YFLogxK

U2 - 10.1017/S0963548323000469

DO - 10.1017/S0963548323000469

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AN - SCOPUS:85181445986

SN - 0963-5483

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

ER -