一种新的自优化节点随机图模型:连通性和直径

Q3 Mathematics Internet Mathematics Pub Date : 2015-08-03 DOI:10.1080/15427951.2015.1022626
R. La, Maya Kabkab
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引用次数: 3

摘要

我们引入了一种新的随机图模型。在我们的模型n, n≥2中,顶点通过考虑(估计的)边的收益或效用来选择潜在边的子集。更准确地说,每个顶点选择k, k≥1,它希望建立的关联边,并且当且仅当两个顶点之间的无向边都选择了边时,图中才存在两个顶点之间的无向边。首先,我们研究了图连通性所需的最小k随n增加的缩放规律,并证明它是Θ(log (n))。其次,我们研究了随机图的直径,并证明在k上的某些条件下,直径有高概率接近于log (n)/log (log (n))。此外,作为我们研究结果的副产品,我们表明,对于所有足够大的n,如果k > β -百科log (n),其中β -百科≈2.4626,存在一个连接的Erds-Rnyi随机图,该随机图以高概率嵌入到我们的随机图中。
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A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter
We introduce a new random graph model. In our model, n, n ≥ 2, vertices choose a subset of potential edges by considering the (estimated) benefits or utilities of the edges. More precisely, each vertex selects k, k ≥ 1, incident edges it wishes to set up, and an undirected edge between two vertices is present in the graph if and only if both of the end vertices choose the edge. First, we examine the scaling law of the smallest k needed for graph connectivity with increasing n and prove that it is Θ(log (n)). Second, we study the diameter of the random graph and demonstrate that, under certain conditions on k, the diameter is close to log (n)/log (log (n)) with high probability. In addition, as a byproduct of our findings, we show that, for all sufficiently large n, if k > β⋆log (n), where β⋆ ≈ 2.4626, there exists a connected Erds–Rnyi random graph that is embedded in our random graph, with high probability.
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Internet Mathematics
Internet Mathematics Mathematics-Applied Mathematics
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