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.

Contents

Random graph models

Properties of random graphs

History

References

See also

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, called ''G(n,p)'', in which each of the $n\; choose\; 2$ possible edges occurs independently with probability ''p''. A closely related model, ''G(n,M)'' assigns equal probability to all graphs with exactly ''M'' edges. Both models can be viewed as snapshots at a particular time of the 'random graph process' $ilde\{G\}\_n$, 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 (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'. Associated with each vertex of a random dot-product graph is 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.

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.
''(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 implies, 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

★ Erdos-Renyi model

★ http://www.math.cornell.edu/~durrett/RGD/RGD.html