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 according to their spatial dimension: "popular" nodes (those with many links) are particularly likely to be linked to other popular nodes.

Percolation theory on the random geometric graph (the study of its global connectivity) is sometimes called the Gilbert disk model after the work of Edgar Gilbert, who introduced these graphs and percolation in them in a 1961 paper. A real-world application of RGGs is the modeling of ad hoc networks. Furthermore, they are used to perform benchmarks for graph algorithms.

Definition

In the following, let  denote an undirected Graph with a set of vertices and a set of edges . The set sizes are denoted by and . Additionally, if not noted otherwise, the metric space with the euclidean distance the euclidean distance of x and y is defined as

1. .

A random geometric graph (RGG) is an undirected geometric graph with nodes randomly sampled from the uniform distribution of the underlying space . Two vertices are connected if, and only if, their distance is less than a previously specified parameter , excluding any loops. Thus, the parameters and fully characterize a RGG.

Algorithms

Naive algorithm

. The samples are generated by using a random number generator, one RNG for every dimension.

Pseudocode

V := generateSamples(n)  // Generates n samples in the unit cube.
for each p ∈ V do
    for each q ∈ V\{p} do
        if distance(p, q) ≤ r then
            addConnection(p, q) // Add the edge (p, q) to the edge data structure.
        end if
    end for
end for

As this algorithm is not scalable (every vertex needs information of every other vertex), Holtgrewe et al. and Funke et al. have introduced new algorithms for this problem.

Distributed algorithms

Holtgrewe et al.

This algorithm, which was proposed by Holtgrewe et al., was the first distributed RGG generator algorithm for dimension 2. vertices, which are then distributed to their respective owners. Then the vertices are sorted by the cell number they fall into, for example with Quicksort denotes the time for an all-to-all communication with messages of length bits to is the time taken for a point-to-point communication for a message of length bits.

Since this algorithm is not communication free, Funke et al. proposed a scalable distributed RGG generator for higher dimensions, which works without any communication between the processing units.

Funke et al.

The approach used in this algorithm is similar to the approach in Holtgrewe: Partition the unit cube into equal sized chunks with side length of at least . So in = 2 this will be squares, in chunks, for which it generates the vertices. To achieve a communication free process, each processor then generates the same vertices in the adjacent chunks by exploiting pseudorandomization of seeded hash functions. This way, each processor calculates the same vertices and there is no need for exchanging vertex information.

, without any cost for communication between processing units.

Properties

Isolated vertices and connectivity

..

.

Hamiltonicity

also provides information about the existence of a Hamiltonian cycle (Hamiltonian Path)., then the RGG has asymptotically almost surely a Hamiltonian cycle.

Clustering coefficient

The clustering coefficient of RGGs only depends on the dimension of the underlying space . The clustering coefficient is

.

Generalized random geometric graphs

In 1988, Bernard Waxman we have the standard RGG. Intuitively these types 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. 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 for the disk model. Often the properties of these networks such as betweenness centrality and connectivity 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 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.

References