Percolation on dense graph sequences
In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ obtained by keeping each edge independently with probability $p_n$. W...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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2007
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| _version_ | 1826290039226630144 |
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| author | Bollobás, B Borgs, C Chayes, J Riordan, O |
| author_facet | Bollobás, B Borgs, C Chayes, J Riordan, O |
| author_sort | Bollobás, B |
| collection | OXFORD |
| description | In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ obtained by keeping each edge independently with probability $p_n$. We show that the appearance of a giant component in $G_n(p_n)$ has a sharp threshold at $p_n=1/\lambda_n$. In fact, we prove much more: if $(G_n)$ converges to an irreducible limit, then the density of the largest component of $G_n(c/n)$ tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollob\'as, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs. |
| first_indexed | 2024-03-07T02:38:05Z |
| format | Journal article |
| id | oxford-uuid:a97aa735-d20f-4a32-a744-f11a5363fd64 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T02:38:05Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:a97aa735-d20f-4a32-a744-f11a5363fd642022-03-27T03:08:44ZPercolation on dense graph sequencesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:a97aa735-d20f-4a32-a744-f11a5363fd64EnglishSymplectic Elements at Oxford2007Bollobás, BBorgs, CChayes, JRiordan, OIn this paper we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ obtained by keeping each edge independently with probability $p_n$. We show that the appearance of a giant component in $G_n(p_n)$ has a sharp threshold at $p_n=1/\lambda_n$. In fact, we prove much more: if $(G_n)$ converges to an irreducible limit, then the density of the largest component of $G_n(c/n)$ tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollob\'as, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs. |
| spellingShingle | Bollobás, B Borgs, C Chayes, J Riordan, O Percolation on dense graph sequences |
| title | Percolation on dense graph sequences |
| title_full | Percolation on dense graph sequences |
| title_fullStr | Percolation on dense graph sequences |
| title_full_unstemmed | Percolation on dense graph sequences |
| title_short | Percolation on dense graph sequences |
| title_sort | percolation on dense graph sequences |
| work_keys_str_mv | AT bollobasb percolationondensegraphsequences AT borgsc percolationondensegraphsequences AT chayesj percolationondensegraphsequences AT riordano percolationondensegraphsequences |