## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 2 (2018), 941-989.

### Spectral gap of random hyperbolic graphs and related parameters

Marcos Kiwi and Dieter Mitsche

#### 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 [*Automata, Languages, and Programming—39th International Colloquium—ICALP Part II*. (2012) 573–585 Springer], 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 $\Omega(n^{-(2\alpha-1)}/D)$, where $\frac{1}{2}<\alpha<1$ 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 2 (2018), 941-989.

**Dates**

Received: July 2016

Revised: June 2017

First available in Project Euclid: 11 April 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1523433629

**Digital Object Identifier**

doi:10.1214/17-AAP1323

**Mathematical Reviews number (MathSciNet)**

MR3784493

**Zentralblatt MATH identifier**

06897948

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 05C80: Random graphs [See also 60B20]

**Keywords**

Random hyperbolic graphs spectral gap conductance

#### Citation

Kiwi, Marcos; Mitsche, Dieter. Spectral gap of random hyperbolic graphs and related parameters. Ann. Appl. Probab. 28 (2018), no. 2, 941--989. doi:10.1214/17-AAP1323. https://projecteuclid.org/euclid.aoap/1523433629