Cliques in Hyperbolic Random Graphs

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton

doi:10.1007/s00453-017-0323-3

Cited by 27 publications

(37 citation statements)

References 36 publications

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“…connected, for α > 1/2 the graph is a.a.s. disconnected and when α = 1/2 the probability of being connected tends to a continuous, non-decreasing function of ν which is identically one for ν ≥ π and strictly less than one for ν < π. Friedrich and Krohmer [5] studied the size of the largest clique as well as the number of cliques of a given size. Boguña et al [9] and Bläsius et al [6] considered fitting the KPKVB model to data using maximum likelihood estimation.…”

Section: Kpkvb Modelmentioning

confidence: 99%

Clustering in a hyperbolic model of complex networks

Fountoulakis¹,

Hoorn²,

Müller³

et al. 2021

Electron. J. Probab.

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In this paper we consider the clustering coefficient, and clustering function in a random graph model proposed by In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most at 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, "short distances" and a nonvanishing 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, the parameter α controls the power law exponent of the degree sequence and ν the average degree.Here we show that the clustering coefficient tends in probability to a constant γ that we give explicitly as a closed form expression in terms of α, ν and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that c(k), the average clustering coefficient over all vertices of degree exactly k, tends in probability to a limit γ(k) which we give explicitly as a closed form expression in terms of α, ν and certain special functions. We are able to extend this last result also to sequences (kn)n where kn grows as a function of n. Our results show that γ(k) scales differently, as k grows, for different ranges of α. More precisely, there exists constants cα,ν depending on α and ν, such that as3 4 and γ(k) ∼ cα,ν • k −1 when α > 3 4 . These results contradict a claim of Krioukov et al., which stated that γ(k) should always scale with k −1 as we let k grow.

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Section: Kpkvb Modelmentioning

confidence: 99%

Clustering in a hyperbolic model of complex networks

Fountoulakis¹,

Hoorn²,

Müller³

et al. 2021

Electron. J. Probab.

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“…Bläsius et al [3] and Boguña et al [6] considered fitting the KPKVB model to data using maximum likelihood estimation. Kiwi and Mitsche [14] studied the spectral gap and related properties, and Bläsius et al [2] considered the treewidth and related parameters of the KPKVB model.…”

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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“…From the mathematical perspective, the focus lies on studying structural properties. The degree distribution and clustering [17], diameter [15,22], component structure [7,18], clique size [6], and separation properties [5] have been studied successfully.…”

Section: Homogeneous Heterogeneousmentioning

confidence: 99%

Efficient Shortest Paths in Scale-Free Networks with Underlying Hyperbolic Geometry

Bläsius

Freiberger

Friedrich

et al. 2022

ACM Trans. Algorithms

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A standard approach to accelerating shortest path algorithms on networks is the bidirectional search, which explores the graph from the start and the destination, simultaneously. In practice this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is \mathcal {\tilde{O}}(n^{2 - 1/\alpha } + n^{1/(2\alpha)} + \delta _{\max }) with high probability, where α ∈ (1/2, 1) controls the power-law exponent of the degree distribution, and δ max is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.

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Cliques in Hyperbolic Random Graphs

Cited by 27 publications

References 36 publications

Clustering in a hyperbolic model of complex networks

Clustering in a hyperbolic model of complex networks

The diameter of KPKVB random graphs

Efficient Shortest Paths in Scale-Free Networks with Underlying Hyperbolic Geometry

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