Probability Seminar: Dieter Mitsche, Universite de Nice Sophia-Antipolis

 “On the spectral gap of random hyperbolic graphs”

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 Gugelmann et al. 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 an $n$-vertex random hyperbolic graph is at least $n^{-(2\alpha-1)}/(D\log n)^{1+o(1)}$, 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 $O(n^{\varepsilon})$ for $0<\varepsilon< 1$, 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.

Joint work with Marcos Kiwi.

Refreshments will be served at 3:45 in the 3rd floor lounge of Hanes Hall