{"title":"通过具有扩展颜色的波利亚瓮生成优先附着图","authors":"Somya Singh, Fady Alajaji, Bahman Gharesifard","doi":"10.1017/nws.2024.3","DOIUrl":null,"url":null,"abstract":"We introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Similar to the Barabási-Albert model, the generated graph grows in size by one vertex at each time instance; in contrast however, each vertex of the graph is uniquely characterized by a color, which is represented by a ball color in the Pólya urn. More specifically at each time step, we draw a ball from the urn and return it to the urn along with a number of reinforcing balls of the same color; we also add another ball of a new color to the urn. We then construct an edge between the new vertex (corresponding to the new color) and the existing vertex whose color ball is drawn. Using color-coded vertices in conjunction with the time-varying reinforcing parameter allows for vertices added (born) later in the process to potentially attain a high degree in a way that is not captured in the Barabási-Albert model. We study the degree count of the vertices by analyzing the draw vectors of the underlying stochastic process. In particular, we establish the probability distribution of the random variable counting the number of draws of a given color which determines the degree of the vertex corresponding to that color in the graph. We further provide simulation results presenting a comparison between our model and the Barabási-Albert network.","PeriodicalId":51827,"journal":{"name":"Network Science","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generating preferential attachment graphs via a Pólya urn with expanding colors\",\"authors\":\"Somya Singh, Fady Alajaji, Bahman Gharesifard\",\"doi\":\"10.1017/nws.2024.3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Similar to the Barabási-Albert model, the generated graph grows in size by one vertex at each time instance; in contrast however, each vertex of the graph is uniquely characterized by a color, which is represented by a ball color in the Pólya urn. More specifically at each time step, we draw a ball from the urn and return it to the urn along with a number of reinforcing balls of the same color; we also add another ball of a new color to the urn. We then construct an edge between the new vertex (corresponding to the new color) and the existing vertex whose color ball is drawn. Using color-coded vertices in conjunction with the time-varying reinforcing parameter allows for vertices added (born) later in the process to potentially attain a high degree in a way that is not captured in the Barabási-Albert model. We study the degree count of the vertices by analyzing the draw vectors of the underlying stochastic process. In particular, we establish the probability distribution of the random variable counting the number of draws of a given color which determines the degree of the vertex corresponding to that color in the graph. We further provide simulation results presenting a comparison between our model and the Barabási-Albert network.\",\"PeriodicalId\":51827,\"journal\":{\"name\":\"Network Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-04-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Network Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/nws.2024.3\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"SOCIAL SCIENCES, INTERDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Network Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/nws.2024.3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"SOCIAL SCIENCES, INTERDISCIPLINARY","Score":null,"Total":0}
Generating preferential attachment graphs via a Pólya urn with expanding colors
We introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Similar to the Barabási-Albert model, the generated graph grows in size by one vertex at each time instance; in contrast however, each vertex of the graph is uniquely characterized by a color, which is represented by a ball color in the Pólya urn. More specifically at each time step, we draw a ball from the urn and return it to the urn along with a number of reinforcing balls of the same color; we also add another ball of a new color to the urn. We then construct an edge between the new vertex (corresponding to the new color) and the existing vertex whose color ball is drawn. Using color-coded vertices in conjunction with the time-varying reinforcing parameter allows for vertices added (born) later in the process to potentially attain a high degree in a way that is not captured in the Barabási-Albert model. We study the degree count of the vertices by analyzing the draw vectors of the underlying stochastic process. In particular, we establish the probability distribution of the random variable counting the number of draws of a given color which determines the degree of the vertex corresponding to that color in the graph. We further provide simulation results presenting a comparison between our model and the Barabási-Albert network.
期刊介绍:
Network Science is an important journal for an important discipline - one using the network paradigm, focusing on actors and relational linkages, to inform research, methodology, and applications from many fields across the natural, social, engineering and informational sciences. Given growing understanding of the interconnectedness and globalization of the world, network methods are an increasingly recognized way to research aspects of modern society along with the individuals, organizations, and other actors within it. The discipline is ready for a comprehensive journal, open to papers from all relevant areas. Network Science is a defining work, shaping this discipline. The journal welcomes contributions from researchers in all areas working on network theory, methods, and data.