Spectral gap of random hyperbolic graphs and related parameters

Kiwi, Marcos; Mitsche, Dieter

doi:10.1214/17-aap1323

Cited by 27 publications

(28 citation statements)

References 17 publications

(27 reference statements)

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“…Boguña et al [9] and Bläsius et al [6] considered fitting the KPKVB model to data using maximum likelihood estimation. Kiwi and Mitsche [23] studied the spectral gap and related properties, and Bläsius et al [4] considered the tree-width and related parameters of the KPKVB model. Recently Owada and Yogeshwaran [33] considered subgraph counts, and in particular established a central limit theorem for the number of copies of a fixed tree T in G(n; α, ν), subject to some restrictions on the parameter α.…”

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. Abdullah et al [1] considered typical distances in the graph.…”

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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“…The model is able to do so while being a maximum-entropy model, meaning that the model makes the minimum number of assumptions to explain observations (29), so it is the most parsimonious option. Likewise, the S 1 /H 2 model is particularly interesting because a body of analytic results for the most relevant topological properties have already been derived, including degree distribution (13,27,30), clustering (27,30,31), diameter (32)(33)(34), percolation (35,36), self-similarity (13), or spectral properties (37). The model has been extended to weighted networks (38) and multiplexes (39)(40)(41) and has been used to model real networks from many different domains, from metabolic networks (42) or the brain (43,44) to the WTW (19) and the Internet (28).…”

Section: Gbgmentioning

confidence: 99%

Scaling up real networks by geometric branching growth

Zheng

García-Pérez

Boguñá

et al. 2021

Proc. Natl. Acad. Sci. U.S.A.

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Real networks often grow through the sequential addition of new nodes that connect to older ones in the graph. However, many real systems evolve through the branching of fundamental units, whether those be scientific fields, countries, or species. Here, we provide empirical evidence for self-similar growth of network structure in the evolution of real systems—the journal-citation network and the world trade web—and present the geometric branching growth model, which predicts this evolution and explains the symmetries observed. The model produces multiscale unfolding of a network in a sequence of scaled-up replicas preserving network features, including clustering and community structure, at all scales. Practical applications in real instances include the tuning of network size for best response to external influence and finite-size scaling to assess critical behavior under random link failures.

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Spectral gap of random hyperbolic graphs and related parameters

Cited by 27 publications

References 17 publications

Clustering in a hyperbolic model of complex networks

Clustering in a hyperbolic model of complex networks

The diameter of KPKVB random graphs

Scaling up real networks by geometric branching growth

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