## The Annals of Applied Probability

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

#### 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
Revised: June 2017
First available in Project Euclid: 11 April 2018

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

#### 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

#### References

• [1] Abdullah, M. A., Bode, M. and Fountoulakis, N. (2017). Typical distances in a geometric model for complex networks. Internet Math. DOI:10.24166/im.13.2017.
• [2] Alon, N. and Spencer, J. H. (2008). The Probabilistic Method, 3rd ed. Wiley, Hoboken, NJ.
• [3] Bode, M., Fountoulakis, N. and Müller, T. (2015). On the largest component of a hyperbolic model of complex networks. Electron. J. Combin. 22 P3.24.
• [4] Bode, M., Fountoulakis, N. and Müller, T. (2016). The probability of connectivity in a hyperbolic model of complex networks. Random Structures Algorithms 49 65–94.
• [5] Boguñá, M., Papadopoulos, F. and Krioukov, D. (2010). Sustaining the Internet with hyperbolic mapping. Nat. Commun. 1 62.
• [6] Bringmann, K., Keusch, R. and Lengler, J. Average distance in a general class of scale-free networks with underlying geometry. Available at http://arxiv.org/pdf/1602.05712v1.pdf.
• [7] Bringmann, K., Keusch, R. and Lengler, J. Sampling Geometric Inhomogeneous Random Graphs in Linear Time. Preprint. Available at http://arxiv.org/pdf/1511.00576v3.pdf.
• [8] Candellero, E. and Fountoulakis, N. (2013). Clustering in random geometric graphs on the hyperbolic plane. Preprint. Available at http://arxiv.org/abs/1309.0459.
• [9] Chung, F. R. K. (1997). Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92. Amer. Math. Soc., Providence, RI.
• [10] Deijfen, M., van der Hofstad, R. and Hooghiemstra, G. (2013). Scale-free percolation. Ann. Inst. Henri Poincaré Probab. Stat. 49 817–838.
• [11] Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36–61.
• [12] Fountoulakis, N. (2012). On the evolution of random graphs on spaces with negative curvature. ArXiv E-prints.
• [13] Fountoulakis, N. and Müller, T. (2018). Law of large numbers for the largest component in a hyperbolic model of complex networks. Ann. Appl. Probab. 28 607–650.
• [14] Friedrich, T. and Krohmer, A. (2015). On the diameter of hyperbolic random graphs. In Automata, Languages, and Programming. Part II. Lecture Notes in Computer Science 9135 614–625. Springer, Heidelberg.
• [15] Gugelmann, L., Panagiotou, K. and Peter, U. (2012). Random hyperbolic graphs: Degree sequence and clustering. In Automata, Languages, and Programming—39th International Colloquium—ICALP Part II. LNCS 7392 573–585. Springer, Berlin.
• [16] Kiwi, M. and Mitsche, D. (2015). A bound for the diameter of random hyperbolic graphs. In Proceedings of the 12th Workshop on Analytic Algorithmics and Combinatorics—ANALCO 26–39. SIAM, Philadelphia, PA.
• [17] Koch, C. and Lengler, J. (2016). Bootstrap percolation on geometric inhomogeneous random graphs. In Automata, Languages, and Programming—43rd International Colloquium—ICALP. LIPIcs 55 127:1–127:15.
• [18] Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. and Boguñá, M. (2010). Hyperbolic geometry of complex networks. Phys. Rev. E 82 036106.
• [19] Mohaisen, A., Yun, A. and Kim, Y. (2010). Measuring the mixing time of social graphs. In Proceedings of the 10th ACM SIGCOMM Conference on Internet Measurement—ICM 383–389. ACM, New York.
• [20] Sinclair, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351–370.