论文标题
集群在复杂网络的双曲线模型中
Clustering in a hyperbolic model of complex networks
论文作者
论文摘要
在本文中，我们考虑了Krioukov等人在2010年提出的随机图模型中的聚类系数和聚类函数。在该模型中，如果在双曲机平面中的磁盘中随机选择节点，并且如果它们最多在彼此之间具有一定的双重距离，则可以连接两个节点。已经表明，该模型具有与复杂网络相关的各种属性，例如幂律程度分布，短距离和非变化聚类系数。在这里，我们表明，聚类系数趋于概率为常数$γ$，我们将其明确作为封闭形式表达式以$α，ν$和某些特殊功能表示。这改善了Gugelmann等人的早期工作，后者证明了聚类系数以高概率远离零的界限，但仍将收敛问题与限制常数保持一致。同样，我们能够证明$ c（k）$是所有准确的$ k $的顶点的平均聚类系数，往往限制$γ（k）$，我们以$α，ν$和某些特殊功能而明确地作为封闭形式表达式表达式表达式。我们能够将此最后的结果扩展到序列$（k_n）_n $，其中$ k_n $随着$ n $的函数而生长。我们的结果表明，$γ（k）$的规模不同，因为$ k $的增长，$ k $的范围为$α$。更准确地说，存在常数$ c_ {α，ν} $，具体取决于$α$和$ν$，使得为$ k \ to \ to \ infty $，$γ（k）\ sim c_ c_ {α，α，ν} \ cdot k^{2-4α} $ \ frac {3} {4} $，$γ（k）\ sim c_ {α，ν} \ cdot \ cdot \ log（k）\ cdot k^{ - 1} $如果$α= \ frac {3} {3} \ frac {3} {4} $。这些结果与Krioukov等人的主张相矛盾，该主张指出，限制值$γ（k）$应始终用$ k^{ - 1} $扩展，因为我们让$ k $生长。
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.
