A random graph of moderate density

IF 0.5 4区 数学 Q4 STATISTICS & PROBABILITY Electronic Communications in Probability Pub Date : 2022-01-01 DOI:10.1214/21-ecp444
Á. Backhausz, T. F. Móri
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Abstract

We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a fixed vertex (with normalization n 1 / 6 ). We also analyse the proportion of vertices with degree greater than a fixed multiple of the average degree, and the maximal degree.
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中等密度的随机图
我们分析了一个随机增长图模型,其中平均度渐近等于一个常数乘以顶点数量的平方根,并且聚类系数相当小。在每一步中,我们随机均匀地选择两个顶点,检查它们是否连接,然后我们要么添加一条新边,要么删除一条边,再向图中添加一个二阶新顶点。这种对连接所选顶点状态的依赖性使得n步后顶点的总数是随机的。我们证明了这个量的渐近正态性,也证明了固定顶点的阶(归一化为1/6)。我们还分析了阶数大于平均阶数和最大阶数的固定倍数的顶点的比例。
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来源期刊
Electronic Communications in Probability
Electronic Communications in Probability 工程技术-统计学与概率论
CiteScore
1.00
自引率
0.00%
发文量
38
审稿时长
6-12 weeks
期刊介绍: The Electronic Communications in Probability (ECP) publishes short research articles in probability theory. Its sister journal, the Electronic Journal of Probability (EJP), publishes full-length articles in probability theory. Short papers, those less than 12 pages, should be submitted to ECP first. EJP and ECP share the same editorial board, but with different Editors in Chief.
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