随机双曲图的谱隙及相关参数
Spectral Gap of Random Hyperbolic Graphs and Related Parameters
摘要：
随机双曲图被认为是社交网络的一个有前途的模型。研究了它们的一些基本参数。然而，它们中没有一个涉及它们的光谱。我们认为随机双曲图模型是由[GPP 12]形式化的，并从本质上确定了它们的归一化拉普拉斯算子的谱间隙。具体地说，我们建立了高概率的第二个最小特征值的规范化拉普拉斯的巨组件和$n$ -顶点随机双曲图是$Ω(n^{-(2α-1)}/D)$，其中$\frac12<α<1$是一个模型参数和$D$是网络直径（这是已知的至多是多对数在$n$）。我们还显示了一个匹配（多对数因子）的上界$n^{-(2α-1)}(\log n)^{1+o(1)}$。作为一个副产品，我们得出结论，电导上界的本征值间隙通过Cheeger的不等式得到本质上是紧的。我们还提供了一个更详细的图片上的电导上的边界上获得的顶点的集合，特别是显示，对于所有的子集，其体积是$O(n^{1-\varepsilon})$所获得的电导是高概率$Ω(n^{-(2α-1)\varepsilon+o(1)})$。最后，我们还显示了我们的结果的最小和最大的二分法的巨大的组成部分。
英文摘要：
Random hyperbolic graphs have been suggested as a promising model of social networks. A few of their fundamental parameters have been studied. However, none of them concerns their spectra. We consider the random hyperbolic graph model as formalized by [GPP12] and essentially determine the spectral gap of their normalized Laplacian. Specifically, we establish that with high probability the second smallest eigenvalue of the normalized Laplacian of the giant component of and $n$-vertex random hyperbolic graph is $Ω(n^{-(2α-1)}/D)$, where $\frac12<α<1$ is a model parameter and $D$ is the network diameter (which is known to be at most polylogarithmic in $n$). We also show a matching (up to a polylogarithmic factor) upper bound of $n^{-(2α-1)}(\log n)^{1+o(1)}$. As a byproduct we conclude that the conductance upper bound on the eigenvalue gap obtained via Cheeger's inequality is essentially tight. We also provide a more detailed picture of the collection of vertices on which the bound on the conductance is attained, in particular showing that for all subsets whose volume is $O(n^{1-\varepsilon})$ the obtained conductance is with high probability $Ω(n^{-(2α-1)\varepsilon+o(1)})$. Finally, we also show consequences of our result for the minimum and maximum bisection of the giant component.
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