On the Diameter of Hyperbolic Random Graphs

Friedrich, Tobias; Krohmer, Anton

doi:10.1137/17m1123961

Cited by 42 publications

(46 citation statements)

References 26 publications

(50 reference statements)

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“…Proof. Again, the results stated in [FK15] are stated with smaller probability, but a close inspection of them shows that they hold w.e.p. The original results are stated in the uniform model, but again, they hold in the Poissonized model as well.…”

Section: Preliminariesmentioning

confidence: 83%

Spectral gap of random hyperbolic graphs and related parameters

Kiwi¹,

Mitsche²

2018

Ann. Appl. Probab.

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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 an n-vertex random hyperbolic graph is at least Ω(n −(2α−1) /D), where 1 2 < 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 ε ) for 0 < ε < 1 the obtained conductance is with high probability Ω(n −(2α−1)ε+o (1) ). Finally, we also show consequences of our result for the minimum and maximum bisection of the giant component.

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Section: Preliminariesmentioning

confidence: 83%

Spectral gap of random hyperbolic graphs and related parameters

Kiwi¹,

Mitsche²

2018

Ann. Appl. Probab.

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“…In particular, hyperbolic random graphs are a promising model, as Boguñá et al [9] computed a (heuristic) maximum likelihood fit of the internet graph into the hyperbolic random graph model and demonstrated its quality by showing that greedy routing in the underlying geometry of the fit finds near-optimal shortest paths. Further properties that have been studied on hyperbolic random graphs, mostly agreeing with empirical findings on real-world networks, are scale-freeness and clustering coefficient [31,17], existence of a giant component [7], diameter [35,30], average distance [1], separators and treewidth [6], bootstrap percolation [17], and clique number [29]. Algorithmic aspects include sampling algorithms [44] and compression schemes [43].…”

Section: Introductionmentioning

confidence: 85%

Geometric inhomogeneous random graphs

Bringmann

Keusch

Lengler

2019

Theoretical Computer Science

100

178

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Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for realworld networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs.With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we use a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies.We prove the following fundamental structural and algorithmic results on GIRGs.(1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor O( √ n).(2) We establish that GIRGs have clustering coefficients in Ω(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits. * We choose a toroidal ground space for the technical simplicity that comes with its symmetry and in order to obtain hyperbolic random graphs as a special case. The results of this paper stay true if T d is replaced, say, by the d-dimensional unitcube [0, 1] d . † A major difference between hyperbolic random graphs and our generalisation is that we ignore constant factors in the edge probabilities puv. This allows to greatly simplify the edge probability expressions, thus reducing the technical overhead. W [20,21]. Note that the term min{1, .} is necessary, as the product w u w v may be larger than W. Classically, the Θ simply hides a factor 1, but by ‡ We say that an event holds with high probability (whp) if it holds with probability 1 − n −ω(1) .

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“…It has been shown previously by Kiwi and Mitsche [13] that, for α ∈ ( 1 2 , 1), the largest component of G(N; α, ν) has a diameter that is O((log N) 8/(1−α)(2−α) ) a.a.s. This was subsequently improved by Friedrich and Krohmer [9] to O((log N) 1/(1−α) ). Friedrich and Krohmer [9] also gave an a.a.s.…”

Section: Introductionmentioning

confidence: 99%

“…This was subsequently improved by Friedrich and Krohmer [9] to O((log N) 1/(1−α) ). Friedrich and Krohmer [9] also gave an a.a.s. lower bound of (log N).…”

Section: Introductionmentioning

confidence: 99%

The diameter of KPKVB random graphs

Müller

Staps

2019

Adv. Appl. Probab.

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We consider a random graph model that was recently proposed as a model for complex networks by Krioukov et al. [15]. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes N , which we think of as going to infinity, and α, ν > 0, which we think of as constant. Roughly speaking α controls the power law exponent of the degree sequence and ν the average degree.Earlier work of Kiwi and Mitsche [14] has shown that when α < 1 (which corresponds to the exponent of the power law degree sequence being < 3) then the diameter of the largest component is a.a.s. at most polylogarithmic in N . Friedrich and Krohmer [9] have shown it is a.a.s. Ω(log N ) and they improved the exponent of the polynomial in log N in the upper bound. Here we show the maximum diameter over all components is a.a.s. O(log N ) thus giving a bound that is tight up to a multiplicative constant.

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On the Diameter of Hyperbolic Random Graphs

Cited by 42 publications

References 26 publications

Spectral gap of random hyperbolic graphs and related parameters

Spectral gap of random hyperbolic graphs and related parameters

Geometric inhomogeneous random graphs

The diameter of KPKVB random graphs

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