Random Graph Models
A random graph is obtained by starting with a set of n isolated vertices and adding successive edges between them at random. The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise. Different random graph models produce different probability distributions on graphs. Most commonly studied is the one proposed by Edgar Gilbert, denoted G(n,p), in which every possible edge occurs independently with probability 0 < p < 1. The probability of a random graph with m edges is . A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges. With the notation 
, with 0≤M≤N, G(n,p) has 
elements and every element occurs with probability 
. The fastest known algorithm for generating the former model is proposed by Nobari et al. in. 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 0
- Given any 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 Gilbert's random graph model, 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 that a given edge exists for a specified time period. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs.
For, where is the maximal number of edges possible, the two most widely used models, G(n,M) and G(n,p), are almost interchangeable.
Random regular graphs form a special case, with properties that may differ from random graphs in general.
Once we have a model of random graphs, every function on graphs, becomes a random variable. The study of this model is to determine, or at least estimate the probability a property may occur. .
Read more about this topic: Random Graphs
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