Random geometric graph

Example of Random Geometric Graph on a flat 2-d closed square [0, 1] with N=256 vertices and connectivity threshold r=0.1.

In graph theory, a random geometric graph (RGG) is the mathematically simplest spatial network, namely an undirected graph constructed by randomly placing N nodes in some metric space (according to a specified probability distribution) and connecting two nodes by a link if and only if their distance is in a given range, e.g. smaller than a certain neighborhood radius, r.

Random geometric graphs resemble real human social networks in a number of ways. For instance, they spontaneously demonstrate community structure - clusters of nodes with high modularity. Other random graph generation algorithms, such as those generated using the Erdős–Rényi model or Barabási–Albert (BA) model do not create this type of structure. Additionally, random geometric graphs display degree assortativity: "popular" nodes (those with many links) are particularly likely to be linked to other popular nodes.

A real-world application of RGGs is the modeling of ad hoc networks.[1]

Connectivity in Random Geometric Graphs

The connectivity properties in RGGs have been well studied since their introduction by Edgar Gilbert in 1961 [2], which stated wireless communication networks as a suggested application; nodes (vertices) represent wireless devices distributed in the plane according to some point process and connect if their euclidean distance is less than some critical transmission range r0. Since then the connectivity properties of RGGs have been widely studied both on the plane [3] [4] and in other spaces such as hyperbolic space [5][6].

Gilbert’s initial paper provided a lower bound (by creating an associated branching process) and upper bound (by using results from bond percolation) for the critical transmission range needed for a giant component in the plane. Penrose [7] showed for a uniform Poisson point process that the main obstacle to full connectivity is isolated nodes as the number of nodes N →∞; a result which has since been used as an approximation for full connectivity. Furthermore, assuming a homogeneous Poisson point process, the probability a node is isolated is a ”local” event and thus the distribution of isolated nodes can be modelled by a limiting Poisson process [8]. For the 1-dimensional case the main obstacle for full connectivity is no loner isolated nodes since the network is likely to split into two clusters of arbitrary sizes.

Generalised Random Geometric Graphs

In 1988 Waxman [9] generalised the standard RGG by introducing a probabilistic connection function as opposed to the deterministic one suggested by Gilbert. The example introduced by Waxman was a stretched exponential where two nodes i and j connect with probability given by Hij = βexp(-​rij⁄r0), where ij is the euclidean separation and β, r0 are parameters determined by the system. This type of RGG with probabilistic connection function is often referred to a soft Random Geometric Graph, which now has two sources of randomness; the location of nodes (vertices) and the formation of links(edges). This connection function has been generalised further in the literature Hij = βexp(-(​rij⁄r0)η) which is often used to study wireless networks without interference. The parameter η represents how the signal decays with distance, when η = 2 is free space, η > 2 models a more cluttered environment like a town ( = 6 models cities like New York) whilst η < 2 models highly reflective environments. We notice that for η =1 is the Waxman model, whilst as η→∞ and β = 1 we have the standard RGG. Intuitively these type of connection functions model how the probability of a link being made decays with distance.

Overview of some results for Soft RGG

In the high density limit for a network with exponential connection function the number of isolated nodes is Poisson distributed, and the resulting network contains a unique giant component and isolated nodes only [10]. Therefore by ensuring there are no isolated nodes, in the dense regime, the network is a.a.s fully connected; similar to the results shown in [11] for the disk model. Often the properties of these networks such as betweenness centrality [12] and connectivity [13] are studied in the limit as the density → ∞ which often means border effects become negligible. However, in real life where networks are finite, although can still be extremely dense, border effects will impact on full connectivity; in fact [14] showed that for full connectivity, with an exponential connection function, is greatly impacted by boundary effects as nodes near the corner/face of a domain are less likely to connect compared with those in the bulk. As a result full connectivity can be expressed as a sum of the contributions from the bulk and the geometries boundaries. A more general analysis of the connection functions in wireless networks has shown that the probability of full connectivity can be well approximated expressed by a few moments of the connection function and the regions geometry [15].

Examples

  • In 1 dimension, one can study RGGs on a line of unit length (open boundary condition) or on a circle of unit circumference.
  • In 2 dimensions, an RGG can be constructed by choosing a flat unit square [0, 1] (see figure) or a torus of unit circumferences [0, 1)2 as the embedding space.

The simplest choice for the node distribution is to sprinkle them uniformly and independently in the embedding space.

References

  1. ↑ Nekovee, Maziar (28 June 2007). "Worm epidemics in wireless ad hoc networks". New Journal of Physics. 9 (6): 189–189. arXiv:0707.2293. Bibcode:2007NJPh....9..189N. doi:10.1088/1367-2630/9/6/189.
  2. ↑ Gilbert, Edgar (1961). "Random Plane Networks". Journal of the Society for Industrial and Applied Mathematics. 9.4: 533–543. doi:10.1137/0109045.
  3. ↑ Penrose, Mathew D (1997). "The longest edge of the random minimal spanning tree". The annals of applied probability: 340361.
  4. ↑ Penrose, Mathew D (1999). "On k‐connectivity for a geometric random graph". Random Structures & Algorithms. 15.2: 145–164. doi:10.1002/(sici)1098-2418(199909)15:2<145::aid-rsa2>3.0.co;2-g.
  5. ↑ Krioukov, D; Papadopoulos, F; Kitsak, M; Vahdat, A; Boguna, M (2010). "Hyperbolic geometry of complex networks". Physical Review E. 82 (3). arXiv:1006.5169. Bibcode:2010PhRvE..82c6106K. doi:10.1103/physreve.82.036106.
  6. ↑ Kleinberg, R (2007). "Geographic routing using hyperbolic space". 26th IEEE International Conference on Computer Communications.: 1902–1909.
  7. ↑ Penrose, Mathew D (1997). "The longest edge of the random minimal spanning tree". The annals of applied probability: 340361.
  8. ↑ Penrose, M (2003). Random geometric graphs. Oxford University Press.
  9. ↑ Waxman, B.M (1988). "Routing of multipoint connections,". IEEE journal on selected areas in communications. 6 (9): 1617–1622, . doi:10.1109/49.12889.
  10. ↑ Mao, G; Anderson, B.D (2013). "Connectivity of large wireless networks under a general connection model". IEEE Transactions on Information. 59 (3): 1761–1772. doi:10.1109/tit.2012.2228894.
  11. ↑ Penrose, Mathew D (1997). "The longest edge of the random minimal spanning tree". The annals of applied probability: 340361.
  12. ↑ Giles, A.P; Georgiou, O; Dettmann, C.P (2015). "Betweenness centrality in dense random geometric networks". IEEE International Conference on Communications.
  13. ↑ Mao, G; Anderson, B.D (2013). "Connectivity of large wireless networks under a general connection model". IEEE Transactions on Information. 59 (3): 1761–1772. doi:10.1109/tit.2012.2228894.
  14. ↑ Coon, J; Dettmann, C P; Georgiou, O (2012). "Full connectivity: corners, edges and faces". Journal of Statistical Physics. 147 (4): 758–778. arXiv:1201.3123. Bibcode:2012JSP...147..758C. doi:10.1007/s10955-012-0493-y.
  15. ↑ Dettmann, C.P; Georgiou, O (2016). "Random geometric graphs with general connection functions". Physical Review E,. 93 (3). arXiv:1411.3617. Bibcode:2016PhRvE..93c2313D. doi:10.1103/physreve.93.032313.
  • Penrose, Mathew: Random Geometric Graphs (Oxford Studies in Probability, 5), 2003.
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