{"title":"On the Distances Within Cliques in a Soft Random Geometric Graph","authors":"Ercan Sönmez, Clara Stegehuis","doi":"10.1007/s10955-024-03254-3","DOIUrl":null,"url":null,"abstract":"<p>We study the distances of vertices within cliques in a soft random geometric graph on a torus, where the vertices are points of a homogeneous Poisson point process, and far-away points are less likely to be connected than nearby points. We obtain the scaling of the maximal distance between any two points within a clique of size <i>k</i>. Moreover, we show that asymptotically in all cliques with large distances, there is only one remote point and all other points are nearby. Furthermore, we prove that a re-scaled version of the maximal <i>k</i>-clique distance converges in distribution to a Fréchet distribution. Thereby, we describe the order of magnitude according to which the largest distance between two points in a clique decreases with the clique size.</p>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s10955-024-03254-3","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We study the distances of vertices within cliques in a soft random geometric graph on a torus, where the vertices are points of a homogeneous Poisson point process, and far-away points are less likely to be connected than nearby points. We obtain the scaling of the maximal distance between any two points within a clique of size k. Moreover, we show that asymptotically in all cliques with large distances, there is only one remote point and all other points are nearby. Furthermore, we prove that a re-scaled version of the maximal k-clique distance converges in distribution to a Fréchet distribution. Thereby, we describe the order of magnitude according to which the largest distance between two points in a clique decreases with the clique size.
我们研究了环上软随机几何图中小集团内顶点的距离,其中顶点是同质泊松点过程中的点,远处的点比近处的点更不可能相连。我们得到了大小为 k 的小群内任意两点间最大距离的缩放。此外,我们还证明了在所有大距离的小群中,渐近地只有一个远处的点,其他所有点都在附近。此外,我们还证明了最大 k 小块距离的重新缩放版本在分布上收敛于弗雷谢特分布。因此,我们描述了小集团中两点间最大距离随小集团大小而减小的数量级。
期刊介绍:
The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.