Random graph

In mathematics, a random graph is a graph that is generated by some random process^[1]. The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.

Random graph models

A random graph is obtained by starting with a set of n vertices and adding edges between them at random. Different random graph models produce different probability distributions on graphs. Most commonly studied is the Erdős–Rényi model, denoted G(n,p), in which every possible edge occurs independently with probability p. A closely related model, denoted G(n,M), assigns equal probability to all graphs with exactly M edges. The latter model can be viewed as a snapshot at a particular time (M) of the random graph process , which is a stochastic process that starts with n vertices and no edges and at each step adds one new edge chosen uniformly from the set of missing edges.

If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability p, then we get an object G called an infinite random graph. Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property:

Given any n + m elements , there is a vertex that is adjacent to each of and is not adjacent to any of .

It turns out that if the vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.

Another model, which generalizes the Erdős–Rényi graphs, is the random dot-product model. A random dot-product graph associates with each vertex a real vector. The probability of an edge uv between any vertices u and v is some function of the dot product u • v of their respective vectors.

The network probability matrix models random graphs through edge probabilities, which represent the probability p_i,j that a given edge e_i,j exists for a specified time period. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs.

Random regular graphs form a special case, with properties that may differ from random graphs in general.

Properties of random graphs

The theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution. For example, we might ask for a given value of n and p what the probability is that G(n,p) is connected. In studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as n grows very large. Percolation theory characterizes the connectedness of random graphs, especially infinitely large ones.

(threshold functions, evolution of G~)

Random graphs are widely used in the probabilistic method, where one tries to prove the existence of graphs with certain properties. The existence of a property on a random graph can often imply, via the famous Szemerédi regularity lemma, the existence of that property on almost all graphs.

History

Random graphs were first defined by Paul Erdős and Alfréd Rényi in their 1959 paper "On Random Graphs".^[2]

See also

• Percolation
• Bose–Einstein condensation: a network theory approach
• Cavity method

References