Spectral Gap of Random Hyperbolic Graphs and Related Parameters

TL;DR

This paper determines the spectral gap of the normalized Laplacian in random hyperbolic graphs, revealing its dependence on network parameters and providing insights into conductance and bisection properties.

Contribution

It precisely characterizes the spectral gap of the normalized Laplacian in random hyperbolic graphs, linking it to the model parameter and network diameter, and explores related conductance and bisection implications.

Findings

Spectral gap is  n^{-(2-1)}/D with high probability.

Upper bound on eigenvalue gap matches the lower bound up to polylogarithmic factors.

Conductance bounds are tight and relate to vertex subset volumes.

Abstract

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 $\Omega(n^{-(2\alpha-1)}/D)$, where $\frac12<\alpha<1$ 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 $n^{-(2\alpha-1)}(\log n)^{1+o(1)}$. 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 $\Omega(n^{-(2\alpha-1)\varepsilon+o(1)})$. Finally, we also show consequences of our result for the minimum and maximum bisection of the giant component.