Achlioptas process phase transitions are continuous
It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are chosen uniformly at random, but only one pair is joined, name...
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| Format: | Journal article |
| Language: | English |
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2011
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| _version_ | 1797050659696017408 |
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| author | Riordan, O Warnke, L |
| author_facet | Riordan, O Warnke, L |
| author_sort | Riordan, O |
| collection | OXFORD |
| description | It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are chosen uniformly at random, but only one pair is joined, namely, one minimizing the product of the sizes of the components to be joined. Making explicit an earlier belief of Achlioptas and others, in 2009, Achlioptas, D'Souza and Spencer [Science 323 (2009) 1453-1455] conjectured that there exists a $\delta>0$ (in fact, $\delta\ge1/2$) such that with high probability the order of the largest component "jumps" from $o(n)$ to at least $\delta n$ in $o(n)$ steps of the process, a phenomenon known as "explosive percolation." We give a simple proof that this is not the case. Our result applies to all "Achlioptas processes," and more generally to any process where a fixed number of independent random vertices are chosen at each step, and (at least) one edge between these vertices is added to the current graph, according to any (online) rule. We also prove the existence and continuity of the limit of the rescaled size of the giant component in a class of such processes, settling a number of conjectures. Intriguing questions remain, however, especially for the product rule described above. |
| first_indexed | 2024-03-06T18:08:27Z |
| format | Journal article |
| id | oxford-uuid:02349557-2fb9-4fbf-821e-5c62d74f7913 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T18:08:27Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:02349557-2fb9-4fbf-821e-5c62d74f79132022-03-26T08:39:24ZAchlioptas process phase transitions are continuousJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:02349557-2fb9-4fbf-821e-5c62d74f7913EnglishSymplectic Elements at Oxford2011Riordan, OWarnke, LIt is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are chosen uniformly at random, but only one pair is joined, namely, one minimizing the product of the sizes of the components to be joined. Making explicit an earlier belief of Achlioptas and others, in 2009, Achlioptas, D'Souza and Spencer [Science 323 (2009) 1453-1455] conjectured that there exists a $\delta>0$ (in fact, $\delta\ge1/2$) such that with high probability the order of the largest component "jumps" from $o(n)$ to at least $\delta n$ in $o(n)$ steps of the process, a phenomenon known as "explosive percolation." We give a simple proof that this is not the case. Our result applies to all "Achlioptas processes," and more generally to any process where a fixed number of independent random vertices are chosen at each step, and (at least) one edge between these vertices is added to the current graph, according to any (online) rule. We also prove the existence and continuity of the limit of the rescaled size of the giant component in a class of such processes, settling a number of conjectures. Intriguing questions remain, however, especially for the product rule described above. |
| spellingShingle | Riordan, O Warnke, L Achlioptas process phase transitions are continuous |
| title | Achlioptas process phase transitions are continuous |
| title_full | Achlioptas process phase transitions are continuous |
| title_fullStr | Achlioptas process phase transitions are continuous |
| title_full_unstemmed | Achlioptas process phase transitions are continuous |
| title_short | Achlioptas process phase transitions are continuous |
| title_sort | achlioptas process phase transitions are continuous |
| work_keys_str_mv | AT riordano achlioptasprocessphasetransitionsarecontinuous AT warnkel achlioptasprocessphasetransitionsarecontinuous |