Clustering in a hyperbolic model of complex networks

Fountoulakis, Nikolaos; Hoorn, Pim van der; Müller, Tobias; Schepers, Markus

doi:10.1214/21-ejp583

Cited by 35 publications

(29 citation statements)

References 30 publications

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“…More precise results about the scaling of the local clustering coefficient in terms of the degrees of the vertices were obtained by Stegehuis et al [27]. More recently in [9], convergence in probability of the clustering coefficient to an explicitly determined constant was derived.…”

Section: Typical Properties Of the Kpkbv Modelmentioning

confidence: 90%

The modularity of random graphs on the hyperbolic plane

Chellig¹,

Fountoulakis²,

Skerman³

2021

Preprint

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Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together with relatively few edges. In this paper, we consider this quantity on a recent probabilistic model of complex networks introduced by Krioukov et al. (Phys. Rev. E 2010).This model views a complex network as an expression of hidden hierarchies, encapsulated by an underlying hyperbolic space. For certain parameters, this model was proved to have typical features that are observed in complex networks such as power law degree distribution, bounded average degree, clustering coefficient that is asymptotically bounded away from zero, and ultra-small typical distances. In the present work, we investigate its modularity and we show that, in this regime, it converges to 1 in probability.

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Section: Typical Properties Of the Kpkbv Modelmentioning

confidence: 90%

The modularity of random graphs on the hyperbolic plane

Chellig¹,

Fountoulakis²,

Skerman³

2021

Preprint

Self Cite

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“…More precise results including a law of large numbers for the largest component in these networks were established in [17]. Further results on the static version of this model include results on the diameter [23,19,34], on the spectral gap [25], on typical distances [1], on the clustering coefficient [12,18], on bootstrap percolation [26] and on the contact process [29].…”

Section: Related Workmentioning

confidence: 99%

Tail bounds for detection times in mobile hyperbolic graphs

Kiwi¹,

Linker²,

Mitsche³

2022

Preprint

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Motivated by Krioukov et al.'s model of random hyperbolic graphs [28] for realworld networks, and inspired by the analysis of a dynamic model of graphs in Euclidean space by Peres et al. [37], we introduce a dynamic model of hyperbolic graphs in which vertices are allowed to move according to a Brownian motion maintaining the distribution of vertices in hyperbolic space invariant. For different parameters of the speed of angular and radial motion, we analyze tail bounds for detection times of a fixed target and obtain a complete picture, for very different regimes, of how and when the target is detected: as a function of the time passed, we characterize the subset of the hyperbolic space where particles typically detecting the target are initially located. Our analysis shows that our dynamic model exhibits a phase transition as a function of the relation of angular and radial speed.We overcome several substantial technical difficulties not present in Euclidean space, and provide a complete picture on tail bounds. On the way, moreover, we obtain results for a class of one-dimensional continuous processes with drift and reflecting barrier, concerning the time they spend within a certain interval. We also derive improved bounds for the tail of independent sums of Pareto random variables.

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“…Recently, precise results about convergence of clustering coefficients, and scaling of the clustering function as k grows to infinity, for Hyperbolic Random Graphs has been obtained in [15]. Also, it was shown in [11] that under suitable conditions, the CSFP model has non-zero clustering in the limit.…”

Section: Consequences Of Local Convergence: Clusteringmentioning

confidence: 99%

Local limits of spatial inhomogeneous random graphs

Hofstad¹,

Hoorn²,

Maitra³

2021

Preprint

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Consider a set of n vertices, where each vertex has a location in R d that is sampled uniformly from the unit cube in R d , and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations, and the vertex weights.Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on R d with vertex locations given by a homogeneous Poisson point process, having weights which are i.i.d. copies of limiting vertex weights. Our setup covers many sparse geometric random graph models from the literature, including Geometric Inhomogeneous Random Graphs (GIRGs), Hyperbolic Random Graphs, Continuum Scale-Free Percolation and Weight-dependent Random Connection Models.We prove that the limiting degree distribution is mixed Poisson, and the typical degree sequence is uniformly integrable, and obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a by-product of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

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Clustering in a hyperbolic model of complex networks

Cited by 35 publications

References 30 publications

The modularity of random graphs on the hyperbolic plane

The modularity of random graphs on the hyperbolic plane

Tail bounds for detection times in mobile hyperbolic graphs

Local limits of spatial inhomogeneous random graphs

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