{"title":"多类型优先依恋树","authors":"Sebastian Rosengren","doi":"10.24166/im.05.2018","DOIUrl":null,"url":null,"abstract":"A multi-type preferential attachment tree is introduced, and studied using general multi-type branching processes. For the $p$-type case we derive a framework for studying the tree where a type $i$ vertex generates new type $j$ vertices with rate $w_{ij}(n_1,n_2,\\ldots, n_p)$ where $n_k$ is the number of type $k$ vertices previously generated by the type $i$ vertex, and $w_{ij}$ is a non-negative function from $\\mathbb{N}^p$ to $\\mathbb{R}$. The framework is then used to derive results for trees with more specific attachment rates. \nIn the case with linear preferential attachment---where type $i$ vertices generate new type $j$ vertices with rate $w_{ij}(n_1,n_2,\\ldots, n_p)=\\gamma_{ij}(n_1+n_2+\\dots +n_p)+\\beta_{ij}$, where $\\gamma_{ij}$ and $\\beta_{ij}$ are positive constants---we show that under mild regularity conditions on the parameters $\\{\\gamma_{ij}\\}, \\{\\beta_{ij}\\}$ the asymptotic degree distribution of a vertex is a power law distribution. The asymptotic composition of the vertex population is also studied.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2017-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"A Multi-type Preferential Attachment Tree\",\"authors\":\"Sebastian Rosengren\",\"doi\":\"10.24166/im.05.2018\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A multi-type preferential attachment tree is introduced, and studied using general multi-type branching processes. For the $p$-type case we derive a framework for studying the tree where a type $i$ vertex generates new type $j$ vertices with rate $w_{ij}(n_1,n_2,\\\\ldots, n_p)$ where $n_k$ is the number of type $k$ vertices previously generated by the type $i$ vertex, and $w_{ij}$ is a non-negative function from $\\\\mathbb{N}^p$ to $\\\\mathbb{R}$. The framework is then used to derive results for trees with more specific attachment rates. \\nIn the case with linear preferential attachment---where type $i$ vertices generate new type $j$ vertices with rate $w_{ij}(n_1,n_2,\\\\ldots, n_p)=\\\\gamma_{ij}(n_1+n_2+\\\\dots +n_p)+\\\\beta_{ij}$, where $\\\\gamma_{ij}$ and $\\\\beta_{ij}$ are positive constants---we show that under mild regularity conditions on the parameters $\\\\{\\\\gamma_{ij}\\\\}, \\\\{\\\\beta_{ij}\\\\}$ the asymptotic degree distribution of a vertex is a power law distribution. The asymptotic composition of the vertex population is also studied.\",\"PeriodicalId\":38105,\"journal\":{\"name\":\"Internet Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2017-04-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Internet Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.24166/im.05.2018\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Internet Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.24166/im.05.2018","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
A multi-type preferential attachment tree is introduced, and studied using general multi-type branching processes. For the $p$-type case we derive a framework for studying the tree where a type $i$ vertex generates new type $j$ vertices with rate $w_{ij}(n_1,n_2,\ldots, n_p)$ where $n_k$ is the number of type $k$ vertices previously generated by the type $i$ vertex, and $w_{ij}$ is a non-negative function from $\mathbb{N}^p$ to $\mathbb{R}$. The framework is then used to derive results for trees with more specific attachment rates.
In the case with linear preferential attachment---where type $i$ vertices generate new type $j$ vertices with rate $w_{ij}(n_1,n_2,\ldots, n_p)=\gamma_{ij}(n_1+n_2+\dots +n_p)+\beta_{ij}$, where $\gamma_{ij}$ and $\beta_{ij}$ are positive constants---we show that under mild regularity conditions on the parameters $\{\gamma_{ij}\}, \{\beta_{ij}\}$ the asymptotic degree distribution of a vertex is a power law distribution. The asymptotic composition of the vertex population is also studied.