Greedy Routing and the Algorithmic Small-World Phenomenon

Bringmann, Karl; Keusch, Ralph; Lengler, Johannes; Maus, Yannic; Molla, Anisur Rahaman

doi:10.1145/3087801.3087829

Cited by 21 publications

(19 citation statements)

References 75 publications

(204 reference statements)

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“…However, general studies such as [15] are limited to properties that do not depend on the specific underlying geometry. Very recently, GIRGs turned out to be accesible for studying processes such as bootstrap percolation [36] and greedy routing [16].…”

Section: Introductionmentioning

confidence: 99%

Geometric inhomogeneous random graphs

Bringmann

Keusch

Lengler

2019

Theoretical Computer Science

Self Cite

171

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Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for realworld networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs.With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we use a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies.We prove the following fundamental structural and algorithmic results on GIRGs.(1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor O( √ n).(2) We establish that GIRGs have clustering coefficients in Ω(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits. * We choose a toroidal ground space for the technical simplicity that comes with its symmetry and in order to obtain hyperbolic random graphs as a special case. The results of this paper stay true if T d is replaced, say, by the d-dimensional unitcube [0, 1] d . † A major difference between hyperbolic random graphs and our generalisation is that we ignore constant factors in the edge probabilities puv. This allows to greatly simplify the edge probability expressions, thus reducing the technical overhead. W [20,21]. Note that the term min{1, .} is necessary, as the product w u w v may be larger than W. Classically, the Θ simply hides a factor 1, but by ‡ We say that an event holds with high probability (whp) if it holds with probability 1 − n −ω(1) .

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Section: Introductionmentioning

confidence: 99%

Geometric inhomogeneous random graphs

Bringmann

Keusch

Lengler

2019

Theoretical Computer Science

Self Cite

171

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“…Compared to (2), any distribution in the class (4) has the same power-law tail exponent γ, but it can have drastically different shapes for finite degrees. The exact shape of (k) is of much less significance than the value of the tail exponent γ, because it is γ, and not (k), that is solely definitive for a number of important structural and dynamical properties of networks in the limit of large network size [5][6][7][8][35][36][37][38][39][40][41]. As the simplest example, the value of γ determines how many moments of the degree distribution remain bounded in the large-graph limit, affecting many important network properties.…”

Section: Introductionmentioning

confidence: 99%

Scale-free networks well done

Voitalov

Hoorn

Hofstad

et al. 2019

Phys. Rev. Research

138

149

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We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying distributions that are widely used in statistics and other fields. This definition allows the distribution to deviate from a pure power law arbitrarily but without affecting the power-law tail exponent. We then identify three estimators of these exponents that are proven to be statistically consistent-that is, converging to the true value of the exponent for any regularly varying distribution-and that satisfy some additional niceness requirements. In contrast to estimators that are currently popular in network science, the estimators considered here are based on fundamental results in extreme value theory, and so are the proofs of their consistency. Finally, we apply these estimators to a representative collection of synthetic and real-world data. According to their estimates, real-world scale-free networks are definitely not as rare as one would conclude based on the popular but unrealistic assumption that real-world data comes from power laws of pristine purity, void of noise and deviations.

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“…The significance of Observation 3.1 stems from the broad class of networks to which it applies. Greedy routing, for example, can often be very efficient in networks which come with an appropriate geometric embedding, see for example [BK17]. The attraction of Observation 3.1, however, is that it can be applied in scenarios where there is no given geometric embedding (and where it is not even clear how greedy routing would be defined).…”

Section: Finding Short Paths In Idemetric Networkmentioning

confidence: 99%

The idemetric property: when most distances are (almost) the same

et al. 2019

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We introduce the idemetric property, which formalises the idea that most nodes in a graph have similar distances between them, and which turns out to be quite standard amongst small-world network models. Modulo reasonable sparsity assumptions, we are then able to show that a strong form of idemetricity is actually equivalent to a very weak expander condition (PUMP). This provides a direct way of providing short proofs that small-world network models such as the Watts-Strogatz model are strongly idemetric (for a wide range of parameters), and also provides further evidence that being idemetric is a common property.We then consider how satisfaction of the idemetric property is relevant to algorithm design. For idemetric graphs we observe, for example, that a single breadth-first search provides a solution to the all-pairs shortest paths problem, so long as one is prepared to accept paths which are of stretch close to 2 with high probability. Since we are able to show that Kleinberg's model is idemetric, these results contrast nicely with the well known negative results of Kleinberg concerning efficient decentralised algorithms for finding short paths: for precisely the same model as Kleinberg's negative results hold, we are able to show that very efficient (and decentralised) algorithms exist if one allows for reasonable preprocessing. For deterministic distributed routing algorithms we are also able to obtain results proving that less routing information is required for idemetric graphs than in the worst case in order to achieve stretch less than 3 with high probability: while Ω(n 2 ) routing information is required in the worst case for stretch strictly less than 3 on almost all pairs, for idemetric graphs the total routing information required is O(nlog(n)).

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Greedy Routing and the Algorithmic Small-World Phenomenon

Cited by 21 publications

References 75 publications

Geometric inhomogeneous random graphs

Geometric inhomogeneous random graphs

Scale-free networks well done

The idemetric property: when most distances are (almost) the same

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