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Almost sure convergence of vertex degree densities in the vertex splitting model
We study the limiting degree distribution of the vertex splitting model introduced in Ref. [ 3 ] . This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new verti...
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| Published in: | Stochastic models 2016-10, Vol.32 (4), p.575-592 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c370t-b245a35021c3d53b00850b0c987db28256fb5e5a574b189b81412dabe5c443953 |
| container_end_page | 592 |
| container_issue | 4 |
| container_start_page | 575 |
| container_title | Stochastic models |
| container_volume | 32 |
| creator | Stefansson, Sigurdur O. Thörnblad, Erik |
| description | We study the limiting degree distribution of the vertex splitting model introduced in Ref.
[
3
]
. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this, we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature. |
| doi_str_mv | 10.1080/15326349.2016.1182029 |
| format | article |
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[
3
]
. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this, we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.</description><identifier>ISSN: 1532-6349</identifier><identifier>ISSN: 1532-4214</identifier><identifier>EISSN: 1532-4214</identifier><identifier>DOI: 10.1080/15326349.2016.1182029</identifier><language>eng</language><publisher>Philadelphia: Taylor & Francis</publisher><subject>05C05 ; 05C80 ; 05C80; 05C05 ; Almost sure convergence ; Convergence ; degree densities ; random trees ; vertex splitting</subject><ispartof>Stochastic models, 2016-10, Vol.32 (4), p.575-592</ispartof><rights>2016 Taylor & Francis 2016</rights><rights>2016 Taylor & Francis</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c370t-b245a35021c3d53b00850b0c987db28256fb5e5a574b189b81412dabe5c443953</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,776,780,881,27903,27904</link.rule.ids><backlink>$$Uhttps://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-303159$$DView record from Swedish Publication Index$$Hfree_for_read</backlink></links><search><creatorcontrib>Stefansson, Sigurdur O.</creatorcontrib><creatorcontrib>Thörnblad, Erik</creatorcontrib><title>Almost sure convergence of vertex degree densities in the vertex splitting model</title><title>Stochastic models</title><description>We study the limiting degree distribution of the vertex splitting model introduced in Ref.
[
3
]
. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this, we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.</description><subject>05C05</subject><subject>05C80</subject><subject>05C80; 05C05</subject><subject>Almost sure convergence</subject><subject>Convergence</subject><subject>degree densities</subject><subject>random trees</subject><subject>vertex splitting</subject><issn>1532-6349</issn><issn>1532-4214</issn><issn>1532-4214</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp9kEtPwzAQhCMEEqXwE5AicaXFz8a5UZWnVAkOwNWyk01wlcTBdij99yRKy5HTjna_Ga0mii4xmmMk0A3mlCwoS-cE4cUcY0EQSY-iybCfMYLZ8V4P0Gl05v0GIcwTISbR67KqrQ-x7xzEmW2-wZXQZBDbIu51gJ84h9IB9KPxJhjwsWni8AmHs28rE4Jpyri2OVTn0UmhKg8X-zmN3h_u31ZPs_XL4_NquZ5lNEFhpgnjinJEcEZzTjVCgiONslQkuSaC8EWhOXDFE6axSLXADJNcaeAZYzTldBpdj7l-C22nZetMrdxOWmXknflYSutK2XWSIop52uNXI946-9WBD3JjO9f0H0osEBGM4oT0FB-pzFnvHRR_sRjJoWt56FoOXct9173vdvSZprCuVlvrqlwGtausK5xqMuMl_T_iF_BahQU</recordid><startdate>20161001</startdate><enddate>20161001</enddate><creator>Stefansson, Sigurdur O.</creator><creator>Thörnblad, Erik</creator><general>Taylor & Francis</general><general>Taylor & Francis Ltd</general><scope>AAYXX</scope><scope>CITATION</scope><scope>ADTPV</scope><scope>AOWAS</scope><scope>DF2</scope></search><sort><creationdate>20161001</creationdate><title>Almost sure convergence of vertex degree densities in the vertex splitting model</title><author>Stefansson, Sigurdur O. ; Thörnblad, Erik</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c370t-b245a35021c3d53b00850b0c987db28256fb5e5a574b189b81412dabe5c443953</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>05C05</topic><topic>05C80</topic><topic>05C80; 05C05</topic><topic>Almost sure convergence</topic><topic>Convergence</topic><topic>degree densities</topic><topic>random trees</topic><topic>vertex splitting</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Stefansson, Sigurdur O.</creatorcontrib><creatorcontrib>Thörnblad, Erik</creatorcontrib><collection>CrossRef</collection><collection>SwePub</collection><collection>SwePub Articles</collection><collection>SWEPUB Uppsala universitet</collection><jtitle>Stochastic models</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Stefansson, Sigurdur O.</au><au>Thörnblad, Erik</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Almost sure convergence of vertex degree densities in the vertex splitting model</atitle><jtitle>Stochastic models</jtitle><date>2016-10-01</date><risdate>2016</risdate><volume>32</volume><issue>4</issue><spage>575</spage><epage>592</epage><pages>575-592</pages><issn>1532-6349</issn><issn>1532-4214</issn><eissn>1532-4214</eissn><abstract>We study the limiting degree distribution of the vertex splitting model introduced in Ref.
[
3
]
. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this, we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.</abstract><cop>Philadelphia</cop><pub>Taylor & Francis</pub><doi>10.1080/15326349.2016.1182029</doi><tpages>18</tpages><oa>free_for_read</oa></addata></record> |
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| identifier | ISSN: 1532-6349 |
| ispartof | Stochastic models, 2016-10, Vol.32 (4), p.575-592 |
| issn | 1532-6349 1532-4214 1532-4214 |
| language | eng |
| recordid | cdi_proquest_journals_1802843172 |
| source | EBSCOhost Business Source Ultimate; Taylor and Francis Science and Technology Collection |
| subjects | 05C05 05C80 05C80 05C05 Almost sure convergence Convergence degree densities random trees vertex splitting |
| title | Almost sure convergence of vertex degree densities in the vertex splitting model |
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