Cliques in hyperbolic random graphs

Friedrich, Tobias; Krohmer, Anton

doi:10.1109/infocom.2015.7218533

Cited by 28 publications

(19 citation statements)

References 22 publications

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“…The IPv4 topology (layer 1) consists of N 1 = 37563 nodes (ASs), and has a power law degree distribution with exponent γ 1 = 2.1, average node degreek 1 = 5.06, and average clusteringc 1 = 0.63 (T 1 = 0.5). 5 The IPv6 topology (layer 2) consists of N 2 = 5162 nodes, has a power law degree distribution with exponent γ 2 = 2.1, average node degreek 2 = 5.21, and average clusteringc 2 = 0.55 (T 2 = 0.5). There are 4819 common nodes in the two Appendix C: k 2(r) in the uncorrelated and maximally correlated cases…”

Section: Appendix A: Real-world Multiplex Network Datamentioning

confidence: 99%

“…It has been shown that random geometric graphs in hyperbolic spaces are adequate models for complex networks, as they naturally and simultaneously possess many of their common structural and dynamical characteristics, including heterogeneous distributions of node degrees, strong clustering, and preferential attachment, cf. [1][2][3][4][5][6][7][8][9]. Specifically, the H 2 model [1,2] constructs networks by randomly distributing nodes on a hyperbolic disc of radius R, such that each node i has the polar coordinates, or hidden variables, r i , θ i , and connecting each pair of nodes with a probability that decreases with their hyperbolic distance.…”

Section: Introductionmentioning

confidence: 99%

“…(r, r , ∆θ)d∆θ ≈k 2 (r), (C3) (r, r , ∆θ)d∆θ ≈ Ck 2 (r), (C4)wherek 2 (r) in Eq (25),. C = c 2 T 1 sin T 2 π/(c 1 T 2 sin T 1 π) and c i , i = 1, 2 in Eq (5)…”

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confidence: 99%

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Link persistence and conditional distances in multiplex networks

Papadopoulos

Kleineberg

2019

Phys. Rev. E

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Recent progress towards unraveling the hidden geometric organization of real multiplexes revealed significant correlations across the hyperbolic node coordinates in different network layers, which facilitated applications like trans-layer link prediction and mutual navigation. But are geometric correlations alone sufficient to explain the topological relation between the layers of real systems? Here we provide the negative answer to this question. We show that connections in real systems tend to persist from one layer to another irrespectively of their hyperbolic distances. This suggests that in addition to purely geometric aspects the explicit link formation process in one layer impacts the topology of other layers. Based on this finding, we present a simple modification to the recently developed Geometric Multiplex Model to account for this effect, and show that the extended model can reproduce the behavior observed in real systems. We also find that link persistence is significant in all considered multiplexes and can explain their layers' high edge overlap, which cannot be explained by coordinate correlations alone. Furthermore, by taking both link persistence and hyperbolic distance correlations into account we can improve trans-layer link prediction. These findings guide the development of multiplex embedding methods, suggesting that such methods should be accounting for both coordinate correlations and link persistence across layers.A first step towards this direction is the finding that if the layers comprising real multiplexes are independently embedded into hyperbolic spaces, their coordinates exhibit significant correlations [14]. This finding motivated new applications, like multidimensional community de-1 2β r , gives the power law degree distribution, P (k) ≈ ρ(r(k))|r (k)| ∝ k −γ , γ = 1 + 1/β > 2. Without loss of generality, we assume here a hyperbolic plane of curvature K = −1. See [2] for further details.

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Section: Appendix A: Real-world Multiplex Network Datamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Link persistence and conditional distances in multiplex networks

Papadopoulos

Kleineberg

2019

Phys. Rev. E

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“…For the typical number of graphlets, we obtain a similar result, by adding the constraint α i ≤ 1 τ −1 to (47). This results in the optimization problem (like its equivalent version for motifs in (45)) B g,t (H) = max P |S 1 | − |S 2 | − 2E S1 + E S1,S3 + E S1,1 − E S2,1 τ − 1 ,…”

Section: Supplementary Note 4 Graphletsmentioning

confidence: 68%

Variational principle for scale-free network motifs

Stegehuis¹,

Hofstad²,

Leeuwaarden³

2019

Sci Rep

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For scale-free networks with degrees following a power law with an exponent τ ∈ (2, 3), the structures of motifs (small subgraphs) are not yet well understood. We introduce a method designed to identify the dominant structure of any given motif as the solution of an optimization problem. The unique optimizer describes the degrees of the vertices that together span the most likely motif, resulting in explicit asymptotic formulas for the motif count and its fluctuations. We then classify all motifs into two categories: motifs with small and large fluctuations.

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“…For 2 < β < 3, the hyperbolic random graph has a giant component of size Ω(n) [3,4], similar to other scale-free networks like Chung-Lu [10]. Other studied properties include the clique number [14], bootstrap percolation [9]; as well as algorithms for efficient generation of hyperbolic random graphs [24] and efficient embedding of real networks in the hyperbolic plane [21].…”

Section: Introductionmentioning

confidence: 99%

On the Diameter of Hyperbolic Random Graphs

Friedrich¹,

Krohmer²

2018

SIAM J. Discrete Math.

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Large real-world networks are typically scale-free. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing performs close to optimal (Boguná, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation pushed the interest in hyperbolic networks as a natural model for scale-free networks. Hyperbolic random graphs follow a powerlaw degree distribution with controllable exponent β and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573-585, 2012).

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Cliques in hyperbolic random graphs

Cited by 28 publications

References 22 publications

Link persistence and conditional distances in multiplex networks

Link persistence and conditional distances in multiplex networks

Variational principle for scale-free network motifs

On the Diameter of Hyperbolic Random Graphs

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