On the Largest Component of a Hyperbolic Model of Complex Networks

Bode, Michel; Fountoulakis, Nikolaos; Müller, Tobias

doi:10.37236/4958

Cited by 43 publications

(81 citation statements)

References 23 publications

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“…As the graph size tends to infinity, most models have a vanishing average clustering coefficient. Therefore, among other methods to increase clustering such as the hierarchical configuration model [52], spatial random graph models have been introduced, including the spatial preferential attachment model (SPA) [3], power-law radius geometric random graphs [46,77], the hyperbolic random graph (HRG) [18,19,63], and the models that we study in this paper, scale-free percolation on Z d (SFP) [32], their continuum analogue on R d [34] and the geometric inhomogeneous random graph (GIRG) [23], respectively. The SFP and GIRG can be seen as the spatial counterparts of the Norros-Reitu and the Chung-Lu graph, respectively.…”

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confidence: 99%

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Komjáthy

Lodewijks

2020

Stochastic Processes and their Applications

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In this paper we study weighted distances in scale-free spatial network models: hyperbolic random graphs, geometric inhomogeneous random graphs and scale-free percolation. In hyperbolic random graphs, n = Θ(e R/2 ) vertices are sampled independently from the hyperbolic disk with radius R and two vertices are connected either when they are within hyperbolic distance R, or independently with a probability depending on the hyperbolic distance. In geometric inhomogeneous random graphs, and in scale-free percolation, each vertex is given an independent weight and location from an underlying measured metric space and Z d , respectively, and two vertices are connected independently with a probability that is a function of their distance and their weights. We assign independent and identically distributed (i.i.d.) weights to the edges of the obtained random graphs, and investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical weighted distance. In scale-free percolation, we study the weighted distance from the origin of vertex-sequences with norm tending to infinity.In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent τ ∈ (2, 3) (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution, that is, the branching process produces infinitely many individuals in finite time. We show that in all three models, typical distances within the giant/infinite component converge in distribution, that is, no re-scaling is necessary. This solves an open question in [51].The main tools of our proof are to develop elaborate couplings of the models to infinite versions, to follow the shortest paths to infinity and then to connect these paths by using weight-dependent percolation on the graphs, that is, we delete edges attached to vertices with higher weight with higher probability. We realise the percolation using the edge-weights: only very short edges connected to high weight vertices are allowed to stay, hence establishing arbitrarily short upper bounds for connections.(3) High clustering. Local clustering is the formation of locally concentrated clusters in a network and it is measured by the average clustering coefficient: the proportion of triangles present at a vertex versus the total number of possible triangles present at said vertex, averaged over all vertices [76]. Many real-life networks show high clustering, e.g. biological networks such as food webs, technological networks as the World Wide Web, and social networks [30,64,69,70,72].The study of real-world networks could be further improved by the analysis of network models. In the last three decades, network science has become a significant field of study, both theoretically and experimentally. One of the first models that was proposed for studying real-world networks was the Watts-Strogatz model [76], that has the small-...

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confidence: 99%

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Komjáthy

Lodewijks

2020

Stochastic Processes and their Applications

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Section: Local Clusteringmentioning

confidence: 99%

“…Furthermore, the t i 's follow a power-law with exponent τ [10], so that the degrees have a power-law distribution as well. The t i 's can be interpreted as the weights in a hidden-variable model [10]. Because the degrees and the types of vertices have the same scaling, we investigate the probability that two neighbors of a vertex of type k connect.…”

Section: Local Clusteringmentioning

confidence: 99%

“…We can write this probability as P ( on types k, n α1 , n α2 ) = P (k ↔ n α1 ) P (k ↔ n α2 ) P (n α1 and n α2 neighbors connect) . (9) The probability that two vertices with types t i and t j connect satisfies by [10]…”

Section: Local Clusteringmentioning

confidence: 99%

“…the probability that they connect. Two vertices with types t i and t j and angular coordinates φ i and φ j connect if the relative angle between φ i and φ j , ∆θ, satisfies [10] ∆θ ≤ Θ (2νt i t j /n) .…”

Section: Local Clusteringmentioning

confidence: 99%

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Scale-free network clustering in hyperbolic and other random graphs

Stegehuis¹,

Hofstad²,

Leeuwaarden³

2019

J. Phys. A: Math. Theor.

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Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as resilience, navigability and small distances. We introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees. We apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale-free networks with latent geometries and clustering. We show that clustering in the hyperbolic model is non-vanishing and self-averaging, so that a single random graph sample is a good representation in the largenetwork limit. We also demonstrate the variational principle for some classical random graphs including the preferential attachment model and the configuration model.

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The probability of connectivity in a hyperbolic model of complex networks

Bode

Fountoulakis

Müller

2016

Random Struct. Alg.

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We consider a model for complex networks that was introduced by Krioukov et al. (Phys Rev E 82 (2010) 036106). In this model, N points are chosen randomly inside a disk on the hyperbolic plane according to a distorted version of the uniform distribution and any two of them are joined by an edge if they are within a certain hyperbolic distance. This model exhibits a power-law degree sequence, small distances and high clustering. The model is controlled by two parameters α and ν where, roughly speaking, α controls the exponent of the power-law and ν controls the average degree.In this paper we focus on the probability that the graph is connected. We show the following results. For α > 1 2 and ν arbitrary, the graph is disconnected with high probability. For α < 1 2 and ν arbitrary, the graph is connected with high probability. When α = 1 2 and ν is fixed then the probability of being connected tends to a constant f (ν) that depends only on ν, in a continuous manner. Curiously, f (ν) = 1 for ν ≥ π while it is strictly increasing, and in particular bounded away from zero and one, for 0 < ν < π.1 This means that the length of a curve γ : [0, 1] → D is given by 2 1 0 (γ 1 (t)) 2 +(γ 2 (t)) 2 1−γ 2 1 (t)−γ 2 1 (t) dt.Random Structures and Algorithms DOI 10.1002/rsa CONNECTIVITY IN A HYPERBOLIC MODEL OF COMPLEX NETWORKS there exists a t 0 such that the probability that a particle of type i ≥ t 0 has a child of type greater than i amongst its children is at most ε. That is:We now set t := 2t 0 . Then we have that z 1 ,z 2 ,···≥0, z t+1 +z t+2 +···>0by condition iii of the lemma. And, if t 0 ≤ i ≤ t then we have: z 1 ,z 2 ,···≥0, z t+1 +z t+2 +···>0 p(i; z 1 , z 2 , . . . ) ≤ z 1 ,z 2 ,···≥0, z i+1 +z i+2 +···>0

show abstract

On the Largest Component of a Hyperbolic Model of Complex Networks

Cited by 43 publications

References 23 publications

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Scale-free network clustering in hyperbolic and other random graphs

The probability of connectivity in a hyperbolic model of complex networks

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