关于小世界树状随机图的双曲性

Q3 Mathematics Internet Mathematics Pub Date : 2012-01-09 DOI:10.1080/15427951.2013.828336
Wei Chen, Wenjie Fang, Guangda Hu, Michael W. Mahoney
{"title":"关于小世界树状随机图的双曲性","authors":"Wei Chen, Wenjie Fang, Guangda Hu, Michael W. Mahoney","doi":"10.1080/15427951.2013.828336","DOIUrl":null,"url":null,"abstract":"Hyperbolicity is a property of a graph that may be viewed as a “soft” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here we consider Gromov's notion of δ-hyperbolicity and establish several positive and negative results for small-world and treelike random graph models. First, we study the hyperbolicity of the class of Kleinberg small-world random graphs , where n is the number of vertices in the graph, d is the dimension of the underlying base grid B, and γ is the small-world parameter such that each node u in the graph connects to another node v in the graph with probability proportional to 1/dB (u, v)γ, with dB (u, v) the grid distance from u to v in the base grid B. We show that when γ=d, the parameter value allowing efficient decentralized routing in Kleinberg's small-world network,the hyperbolic δ is with probability 1−o(1) for every ϵ>0 independent of n. We see that hyperbolicity is not significantly improved in relation to graph diameter even when the long-range connections greatly improve decentralized navigation. We also show that for other values of γ, the hyperbolic δ is very close to the graph diameter, indicating poor hyperbolicity in these graphs as well. Next we study a class of treelike graphs called ringed trees that have constant hyperbolicity. We show that adding random links among the leaves in a manner similar to the small-world graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides one of the first significant analytic results on the hyperbolicity of a rich class of random graphs, which sheds light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2012-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2013.828336","citationCount":"82","resultStr":"{\"title\":\"On the Hyperbolicity of Small-World and Treelike Random Graphs\",\"authors\":\"Wei Chen, Wenjie Fang, Guangda Hu, Michael W. Mahoney\",\"doi\":\"10.1080/15427951.2013.828336\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Hyperbolicity is a property of a graph that may be viewed as a “soft” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here we consider Gromov's notion of δ-hyperbolicity and establish several positive and negative results for small-world and treelike random graph models. First, we study the hyperbolicity of the class of Kleinberg small-world random graphs , where n is the number of vertices in the graph, d is the dimension of the underlying base grid B, and γ is the small-world parameter such that each node u in the graph connects to another node v in the graph with probability proportional to 1/dB (u, v)γ, with dB (u, v) the grid distance from u to v in the base grid B. We show that when γ=d, the parameter value allowing efficient decentralized routing in Kleinberg's small-world network,the hyperbolic δ is with probability 1−o(1) for every ϵ>0 independent of n. We see that hyperbolicity is not significantly improved in relation to graph diameter even when the long-range connections greatly improve decentralized navigation. We also show that for other values of γ, the hyperbolic δ is very close to the graph diameter, indicating poor hyperbolicity in these graphs as well. Next we study a class of treelike graphs called ringed trees that have constant hyperbolicity. We show that adding random links among the leaves in a manner similar to the small-world graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides one of the first significant analytic results on the hyperbolicity of a rich class of random graphs, which sheds light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.\",\"PeriodicalId\":38105,\"journal\":{\"name\":\"Internet Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2012-01-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1080/15427951.2013.828336\",\"citationCount\":\"82\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Internet Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1080/15427951.2013.828336\",\"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/10.1080/15427951.2013.828336","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 82

摘要

双曲性是图的一种属性,可以看作是树的“软”版本,最近的经验和理论工作表明,在互联网和相关数据应用中出现的许多图都具有双曲性。本文考虑了Gromov的δ-双曲性概念,并建立了小世界和树状随机图模型的几个正负结果。首先,我们研究类的双曲率小世界jonkleinberg随机图,其中n是图中顶点的数量,d是底层基础网格的尺寸B,γ是小世界参数,这样每个节点u图中连接到另一个节点图中概率正比于1 / dB (u, v)γ与dB (u, v)网格距离u, v在网格基础我们表明,当γ= d,在Kleinberg的小世界网络中,允许高效分散路由的参数值,对于每个λ >0,双曲δ的概率为1−0(1),与n无关。我们看到,即使远程连接极大地改善了分散导航,双曲度也没有显著改善。我们还表明,对于γ的其他值,双曲δ非常接近图直径,表明这些图的双曲性也很差。接下来,我们研究一类具有常双曲性的树状图,称为环状树。我们表明,以类似于小世界图构造的方式在叶之间添加随机链接可能很容易破坏图的双曲性,除了使用基于叶之间环距离的指数衰减概率函数添加的一类随机边。我们的研究提供了关于一类丰富的随机图的双曲性的第一个重要的分析结果之一,它揭示了双曲性与随机图的可通航性之间的关系,以及随机图中双曲δ对噪声的敏感性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
On the Hyperbolicity of Small-World and Treelike Random Graphs
Hyperbolicity is a property of a graph that may be viewed as a “soft” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here we consider Gromov's notion of δ-hyperbolicity and establish several positive and negative results for small-world and treelike random graph models. First, we study the hyperbolicity of the class of Kleinberg small-world random graphs , where n is the number of vertices in the graph, d is the dimension of the underlying base grid B, and γ is the small-world parameter such that each node u in the graph connects to another node v in the graph with probability proportional to 1/dB (u, v)γ, with dB (u, v) the grid distance from u to v in the base grid B. We show that when γ=d, the parameter value allowing efficient decentralized routing in Kleinberg's small-world network,the hyperbolic δ is with probability 1−o(1) for every ϵ>0 independent of n. We see that hyperbolicity is not significantly improved in relation to graph diameter even when the long-range connections greatly improve decentralized navigation. We also show that for other values of γ, the hyperbolic δ is very close to the graph diameter, indicating poor hyperbolicity in these graphs as well. Next we study a class of treelike graphs called ringed trees that have constant hyperbolicity. We show that adding random links among the leaves in a manner similar to the small-world graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides one of the first significant analytic results on the hyperbolicity of a rich class of random graphs, which sheds light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Internet Mathematics
Internet Mathematics Mathematics-Applied Mathematics
自引率
0.00%
发文量
0
期刊最新文献
Graph search via star sampling with and without replacement Preferential Placement for Community Structure Formation A Multi-type Preferential Attachment Tree Editorial Board EOV A Theory of Network Security: Principles of Natural Selection and Combinatorics
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1