复杂网络双曲模型中的聚类分析
Clustering in a hyperbolic model of complex networks
摘要：
本文研究了Krioukov等人提出的随机图模型中的聚类系数和聚类函数。2010年在这个模型中，节点在双曲平面内的圆盘内随机选择，如果两个节点彼此之间至多有一定的双曲距离，则两个节点是连通的。它已被证明，该模型具有各种性质的复杂网络，如幂律度分布，短距离和非零聚类系数。在这里，我们表明，聚类系数倾向于概率为一个常数$γ$，我们明确给出作为一个封闭的形式表达的$α, ν$和某些特殊的功能。这改进了Gugelmann等人的早期工作，他证明了聚类系数以高概率保持远离零的有界性，但留下了收敛到极限常数的问题。同样，我们能够表明，$c(k)$，平均聚类系数在所有顶点的程度完全$k$，往往在概率上的限制$γ(k)$，我们明确给出作为一个封闭的形式表达的$α, ν$和某些特殊的功能。我们能够将最后的结果也扩展到序列$(k_n)_n$，其中$k_n$作为$n$的函数而增长。我们的研究结果表明，$γ(k)$的规模不同，$k$的增长，$α$的不同范围。更准确地说，存在依赖于$α$和$ν$的常数$c_{α,ν}$，如$k \to \infty$，如果是$\frac{1}{2} < α< \frac{3}{4}$则为$γ(k) \sim c_{α,ν} \cdot k^{2 - 4α}$，如果是$α=\frac{3}{4}$则为$γ(k) \sim c_{α,ν} \cdot \log(k) \cdot k^{-1} $，如果是$α> \frac{3}{4}$则为$γ(k) \sim c_{α,ν} \cdot k^{-1}$。这些结果与Krioukov等人的主张相矛盾，这说明当我们让$k$增长时，极限值$γ(k)$应始终与$k^{-1}$成比例。
英文摘要：
In this paper we consider the clustering coefficient and clustering function in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been shown that this model has various properties associated with complex networks, e.g. power-law degree distribution, short distances and non-vanishing clustering coefficient. Here we show that the clustering coefficient tends in probability to a constant $γ$ that we give explicitly as a closed form expression in terms of $α, ν$ and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that $c(k)$, the average clustering coefficient over all vertices of degree exactly $k$, tends in probability to a limit $γ(k)$ which we give explicitly as a closed form expression in terms of $α, ν$ and certain special functions. We are able to extend this last result also to sequences $(k_n)_n$ where $k_n$ grows as a function of $n$. Our results show that $γ(k)$ scales differently, as $k$ grows, for different ranges of $α$. More precisely, there exists constants $c_{α,ν}$ depending on $α$ and $ν$, such that as $k \to \infty$, $γ(k) \sim c_{α,ν} \cdot k^{2 - 4α}$ if $\frac{1}{2} < α< \frac{3}{4}$, $γ(k) \sim c_{α,ν} \cdot \log(k) \cdot k^{-1} $ if $α=\frac{3}{4}$ and $γ(k) \sim c_{α,ν} \cdot k^{-1}$ when $α> \frac{3}{4}$. These results contradict a claim of Krioukov et al., which stated that the limiting values $γ(k)$ should always scale with $k^{-1}$ as we let $k$ grow.
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