The cut metric, random graphs, and branching processes
In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an app...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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2009
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| _version_ | 1826303284872216576 |
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| author | Bollobas, B Janson, S Riordan, O |
| author_facet | Bollobas, B Janson, S Riordan, O |
| author_sort | Bollobas, B |
| collection | OXFORD |
| description | In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model we introduced recently, as well as related results of Bollob\'as, Borgs, Chayes and Riordan, all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering. |
| first_indexed | 2024-03-07T06:00:20Z |
| format | Journal article |
| id | oxford-uuid:ebf81da9-a7bd-4a1f-8c27-fc1030e6638b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:00:20Z |
| publishDate | 2009 |
| record_format | dspace |
| spelling | oxford-uuid:ebf81da9-a7bd-4a1f-8c27-fc1030e6638b2022-03-27T11:13:50ZThe cut metric, random graphs, and branching processesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ebf81da9-a7bd-4a1f-8c27-fc1030e6638bEnglishSymplectic Elements at Oxford2009Bollobas, BJanson, SRiordan, OIn this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model we introduced recently, as well as related results of Bollob\'as, Borgs, Chayes and Riordan, all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering. |
| spellingShingle | Bollobas, B Janson, S Riordan, O The cut metric, random graphs, and branching processes |
| title | The cut metric, random graphs, and branching processes |
| title_full | The cut metric, random graphs, and branching processes |
| title_fullStr | The cut metric, random graphs, and branching processes |
| title_full_unstemmed | The cut metric, random graphs, and branching processes |
| title_short | The cut metric, random graphs, and branching processes |
| title_sort | cut metric random graphs and branching processes |
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