Connectivity of Soft Random Geometric Graphs over Annuli

Giles, Alexander P.; Georgiou, Orestis; Dettmann, Carl P.

doi:10.1007/s10955-015-1436-1

Cited by 31 publications

(35 citation statements)

References 26 publications

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“…Finally, a circular or spherical boundary and Gaussian connection function were considered in Ref. [42]. In all cases the results agree with the more general expressions herein.…”

Section: Comparison With Previous Results and Numericssupporting

confidence: 78%

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Random geometric graphs with general connection functions

Dettmann

Georgiou

2016

Phys. Rev. E

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In the original (1961) Gilbert model of random geometric graphs, nodes are placed according to a Poisson point process, and links formed between those within a fixed range. Motivated by wireless ad hoc networks "soft" or "probabilistic" connection models have recently been introduced, involving a "connection function" H (r) that gives the probability that two nodes at distance r are linked (directly connect). In many applications (not only wireless networks), it is desirable that the graph is connected; that is, every node is linked to every other node in a multihop fashion. Here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions. It turns out that only a few quantities such as moments of the connection function appear. Good agreement is found with special cases from previous studies and with numerical simulations.

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“…Finally, a circular or spherical boundary and Gaussian connection function were considered in Ref. [42]. In all cases the results agree with the more general expressions herein.…”

Section: Comparison With Previous Results and Numericssupporting

confidence: 78%

“…Nonconvex domains with a line-of-sight condition: Examples have included keyhole geometries with [25] or without [26] reflections, circular or spherical obstacles [42], and fractal domains [43]. In the latter, remarkably, it is found that P f c decreases toward zero in the limit of high density.…”

Section: Discussionmentioning

confidence: 99%

Random geometric graphs with general connection functions

Dettmann

Georgiou

2016

Phys. Rev. E

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“…This is discussed in detail on [11]. In fact, a significant amount of recent work on random geometric networks has highlighted the importance of the enclosing boundary [31,73].…”

Section: Betweenness Centralitymentioning

confidence: 99%

Shape of shortest paths in random spatial networks

2019

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In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order L χ , while the transversal fluctuations, known as wandering, grow as L ξ . We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ = 3/5 and χ = 1/5, or ξ = 7/10 and χ = 2/5, depending only on coarse details of the specific connectivity laws used. Also, the travel time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbour random geometric graphs, where via Monte Carlo simulations we find ξ = 0.60 ± 0.01 and χ = 0.20 ± 0.01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β-skeletons and the Delaunay triangulation and are characterized by the values ξ = 0.70 ± 0.01 and χ = 0.40 ± 0.01, with a nearly theoretically maximal wandering exponent. We also show numerically that the KPZ relation χ = 2ξ − 1 is satisfied for all these models. These results shed some light on the Euclidean first passage process, but also raise some theoretical questions about the scaling laws and the derivation of the exponent values, and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space. Contents 10 VI. Conclusions 12References 12 arXiv:1906.04314v1 [cond-mat.stat-mech]

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“…For instance, Ref. [Gil16] studied the case of circular and spherical obstacles. The effect of large obstacles is similar to that of curvature in the domain boundary; small obstacles have a differing mathematical formulation and have a small effect on connectivity unless they are very numerous.…”

Section: Obstacles and Reflectionsmentioning

confidence: 99%

Spatial networks with wireless applications

Dettmann

Georgiou

Pratt

2018

Comptes Rendus. Physique

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Many networks have nodes located in physical space, with links more common between closely spaced pairs of nodes. For example, the nodes could be wireless devices and links communication channels in a wireless mesh network. We describe recent work involving such networks, considering effects due to the geometry (convex, non-convex, and fractal), node distribution, distance-dependent link probability, mobility, directivity and interference.

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Connectivity of Soft Random Geometric Graphs over Annuli

Cited by 31 publications

References 26 publications

Random geometric graphs with general connection functions

Random geometric graphs with general connection functions

Shape of shortest paths in random spatial networks

Spatial networks with wireless applications

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