Random-Connection Model

A random-connection model (RCM) is a graph-theoretic model of continuum percolation theory characterized by the existence of a stationary point process X and a non-increasing function g:R^+->[0,1] which together determine a methodology for drawing edges between various vertex points in R^d for some d.

In this case, the function g is said to be a connection function and the RCM is said to be driven by X. The model itself is denoted by (X,g).

A generalization of the ordinary nearest-neighbor bond percolation of Z^d, the random-connection model of continuum percolation should be regarded as a contrasting alternative to the Boolean and Boolean-Poisson models, two models in which the second condition above is replaced with a random variable rho. More precisely, the Boolean and Boolean-Poisson models demonstrate percolation by way of constructing d-dimensional closed balls centered at points of X and assigning random radii determined by rho; contrarily, an RCM uses notions from graph theory, viewing points from X as vertices and inserting edges between pairs x_1,x_2 in X with probability g(|x_1-x_2|), independently of all other pairs of points in X, where |·| denotes the typical Euclidean distance in R^d. Despite their differences, however, the two methods are similar in that sophisticated machinery is necessary for a mathematically precise presentation of either; the formal construction for an RCM is as follows.

Let X be a point process defined on a probability space (Omega_1,F_1,P_1). Next, define the space Omega_2 to be the product


where the product is taken over all unordered pairs of binary cubes and define associated to Omega_2 the usual product measure P_2 so that all the marginal probabilities are given by Lebesgue measure on [0,1]. Finally, define Omega=Omega_1×Omega_2 and equip Omega with the product measure P=P_1 square P_2. Under this construction, an RCM is a measurable mapping from Omega into N×Omega_2 defined by


where here, N denotes the set of all counting measures on the sigma-algebra B^d of Borel sets in R^d which assign finite measure to all bounded Borel sets and which assign values of at most 1 to points.

One then transitions to percolation by first noting that each point x in X is contained in a unique binary cube K(n,z(n,x)) of order n and that, for each x in X, there is a unique smallest number n_0=n_0(x) such that K(n_0,z(n_0,x)) contains no other points of X P_1-almost surely. With this in mind, consider for any two points x,y in X(omega_1) the binary cubes




respectively, whereby the points x and y are connected if and only if


where omega_2({(n,z),(m,z^')}) is the notation used to denote an element omega_2 in Omega_2. Using this construction, one gets a collection of vertices in d-dimensional Euclidean space which are connected according to a the above-described random process.

Some terminology utilized in random-connection models: The edge between two points x_1 and x_2 is denoted by the unordered pair {x_1,x_2} and x_1,x_2 are called end vertices of {x_1,x_2}. Two points x,y in X are said to be connected if there exists a finite sequence


such that the edge {x_i,x_(i+1)} is inserted for all i=1,2,...,n-1, and in the typical graph-theoretic way, one then defines a component as a set S of points which is maximal with respect to the property that any two points a,b in S are connected to one another. The occupied component of the origin is denoted W and is considered to be this model's analogue of the notion of occupied components from Boolean models. There is no analogue of that model's notion of vacant components.


The above figure illustrates a realization of a random-connection model, illustrating some of the terminology related to thereto. In this figure, the component W=W({0}) has the form W={0,x,y,z}, the edges of which have the form {0,x}, {x,y}, and {y,z}.

See also

AB Percolation, Bernoulli Percolation Model, Bond Percolation, Boolean Model, Boolean-Poisson Model, Bootstrap Percolation, Cayley Tree, Cluster, Cluster Perimeter, Continuum Percolation Theory, Dependent Percolation, Discrete Percolation Theory, Disk Model, First-Passage Percolation, Germ-Grain Model, Inhomogeneous Percolation Model, Lattice Animal, Long-Range Percolation Model, Mixed Percolation Model, Oriented Percolation Model, Percolation, Percolation Theory, Percolation Threshold, Polyomino, Random-Cluster Model, Random Walk, s-Cluster, s-Run, Site Percolation

This entry contributed by Christopher Stover

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Meester, R. and Roy, R. Continuum Percolation. New York: Cambridge University Press, 2008.

Cite this as:

Stover, Christopher. "Random-Connection Model." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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