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Color-avoiding percolation and branching processes

Part of: Graph theory Markov processes

Published online by Cambridge University Press:  08 March 2024

Panna Tímea Fekete ,
Roland Molontay ,
Balázs Ráth  and
Kitti Varga
Panna Tímea Fekete*
Affiliation:
HUN-REN Alfréd Rényi Institute of Mathematics; Eötvös Loránd University
Roland Molontay*
Affiliation:
Budapest University of Technology and Economics; HUN-REN–BME Stochastics Research Group
Balázs Ráth*
Affiliation:
Budapest University of Technology and Economics; HUN-REN–BME Stochastics Research Group; HUN-REN Alfréd Rényi Institute of Mathematics
Kitti Varga*
Affiliation:
Budapest University of Technology and Economics; HUN-REN–ELTE Egerváry Research Group
*
*Postal address: Reáltanoda u. 14, H-1053 Budapest, Hungary. Email: fekete.panna.timea@renyi.hu
**Postal address: Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary.
**Postal address: Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary.
*****Postal address: Department of Computer Science and Information Theory, Faculty of Electrical Engineering and Informatics, Budapest University of Technology and Economics, Pázmány Péter stny. 1/C, H-1117 Budapest, Hungary. Email: vkitti@math.bme.hu
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Abstract

We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős–Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if, after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size, as well as the fraction of vertices contained in the giant color-avoidin g connected component. It is known that these quantities converge, and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the fraction of vertices contained in the giant color-avoiding connected component, and we give a simpler asymptotic expression for it in the barely supercritical regime. In addition, in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.

Keywords

Erdős–Rényi random graphgiant componentgenerating function

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 05C80: Random graphs
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 25
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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