Random graph

In mathematics, a random graph is a graph that is generated by some random process. 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 of the random graph process $tilde\{G\}\_n$, 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.

For any graph G=(V, E), the set E of the edges of G may be understood as a binary relation on V. This is the adjacency relation of G, in which vertices a and b are related precisely if $\{a,b\}in\; E$, so ab is an edge of G. Conversely, every symmetric relation $R$ on V gives rise to (and is the edge set of) a graph on $V$.

We can also construct an object G called an infinite random graph on an infinite set $V$ of vertices. The edge set of G, seen as a binary relation R on V satisfies the following properties:

i) R is irreflexive, ii) R is symmetric, and iii) Given any $n+m$ elements $a\_1,ldots,\; a\_n,b\_1,ldots,\; b\_m\; in\; V$, there is a vertex $cin\; V$ that is adjacent to each of $a\_1,ldots,\; a\_n$ and is not adjacent to any of $b\_1,ldots,\; b\_m$.

It turns out that if $V$ is countable then there is, to within isomorphism, only a single infinite random graph, namely the Rado graph (put differently, any two countable random graphs are isomorphic). This is an example of an $omega$-categorical theory.

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 f(u) • f(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.

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 limit 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" in Publ. Math. Debrecen 6, p. 290–297.

References

• Béla Bollobás, Random Graphs, 2nd Edition, 2001, Cambridge University Press
• Christine Nickel, Random Dot Product Graphs: A Model for Social Networks, Ph.D. Thesis, The Johns Hopkins University, 2007.

See also

• Percolation
• Bose-Einstein condensation: a network theory approach