Greedy routing and the algorithmic small-world phenomenon

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

doi:10.1016/j.jcss.2021.11.003

Cited by 4 publications

(4 citation statements)

References 43 publications

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“…Random graphs have a long research history in graph theory, probability, and adjacent areas in applied mathematics and theoretical computer science (TCS). Given that hyperbolic networks turned out to be the first popular ensemble of random graphs reproducing not only inhomogeneous degree distributions but also nonvanishing clustering and small-worldness observed in many real-world networks, this ensemble attracted significant research attention in mathematics and TCS, where many basic and advanced properties of random hyperbolic graphs have been (re)derived rigorously, see for instance [106,[125][126][127][128][129][130][131][132][133][134][135].…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…Such processes are the more efficient and robust, so that the network is the more navigable, the smaller the γ, and the larger the β, defining a navigable parameter range to which many real-world networks belong [19]. Networks in the hyperbolic model described above are nearly maximally efficient for such geometric navigation [18], which has recently been proven rigorously [106]. The main reason behind this phenomenon is the proximity of shortest paths in hyperbolic networks to the corresponding geodesics in the underlying hyperbolic geometry (Fig.…”

Section: Hyperbolic Geometry Of Latent Spacesmentioning

confidence: 91%

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Network geometry

et al. 2021

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Networks are natural geometric objects. Yet the discrete metric structure of shortest path lengths in a network, known as chemical distances, is definitely not the only reservoir of geometric distances that characterize many networks. The other forms of network-related geometries are the geometry of continuous latent spaces underlying many networks, and the effective geometry induced by dynamical processes in networks. A solid and rapidly growing amount of evidence shows that the three approaches are intimately related. Network geometry is immensely efficient in discovering hidden symmetries, such as scale-invariance, and other fundamental physical and mathematical properties of networks, as well as in a variety of practical applications, ranging from the understanding how the brain works, to routing in the Internet. Here, we review the most important theoretical and practical developments in network geometry in the last two decades, and offer perspectives on future research directions and challenges in this novel frontier in the study of complexity.

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Section: Discussion and Outlookmentioning

confidence: 99%

Section: Hyperbolic Geometry Of Latent Spacesmentioning

confidence: 91%

Network geometry

et al. 2021

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“…A continuum version of scale-free percolation on a finite domain (properly rescaled) is known a geometric inhomogeneous random graph; see [5,6,7].…”

Section: Related Workmentioning

confidence: 99%

Graph distances in scale-free percolation: the logarithmic case

Hao

Heydenreich

2022

J. Appl. Probab.

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Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on independent and identically distributed vertex weights and the Euclidean distance $|x-y|$ . Depending on the various parameters involved, we get a rich phase diagram. We study graph distance and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.

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“…Network geometry is currently rising as a compelling research area in physics 1 . The question of how network geometry influences network navigation is a crucial topic in science and engineering, and a recent review 1 (that might soon become an essential reference for the field of network geometry) reports a list of theoretical studies according to which hyperbolic networks are maximally efficient for geometric navigation 4,5 . The main reason behind this phenomenon is assumed the proximity of topological shortest paths (TSP) in the hyperbolic networks to the corresponding geodesics in the underlying hyperbolic geometry.…”

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confidence: 99%

Geometrical congruence, greedy navigability and myopic transfer in complex networks and brain connectomes

Cannistraci

Muscoloni

2022

Nat Commun

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We introduce in network geometry a measure of geometrical congruence (GC) to evaluate the extent a network topology follows an underlying geometry. This requires finding all topological shortest-paths for each nonadjacent node pair in the network: a nontrivial computational task. Hence, we propose an optimized algorithm that reduces 26 years of worst scenario computation to one week parallel computing. Analysing artificial networks with patent geometry we discover that, different from current belief, hyperbolic networks do not show in general high GC and efficient greedy navigability (GN) with respect to the geodesics. The myopic transfer which rules GN works best only when degree-distribution power-law exponent is strictly close to two. Analysing real networks—whose geometry is often latent—GC overcomes GN as marker to differentiate phenotypical states in macroscale structural-MRI brain connectomes, suggesting connectomes might have a latent neurobiological geometry accounting for more information than the visible tridimensional Euclidean.

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Greedy routing and the algorithmic small-world phenomenon

Cited by 4 publications

References 43 publications

Network geometry

Network geometry

Graph distances in scale-free percolation: the logarithmic case

Geometrical congruence, greedy navigability and myopic transfer in complex networks and brain connectomes

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