Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas.
The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise.
Most commonly studied is the one proposed by Edgar Gilbert but often called the Erdős–Rényi model, denoted G(n,p).
The probability of obtaining any one particular random graph with m edges is
[3] The G(n,M) model can be viewed as a snapshot at a particular time (M) of the random graph process
If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < p < 1, 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
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.
This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure.
For M ≃ pN, where N is the maximal number of edges possible, the two most widely used models, G(n,M) and G(n,p), are almost interchangeable.
The study of this model is to determine if, or at least estimate the probability that, a property may occur.
In studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as
Percolation theory characterizes the connectedness of random graphs, especially infinitely large ones.
Percolation is related to the robustness of the graph (called also network).
It was shown that for random graph with Poisson distribution of degrees
edges and with probability close to 1 ensures that the graph has a complete matching, with exception of at most one vertex.
With the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian.
[3] The number of proper colorings of random graphs given a number of q colors, called its chromatic polynomial, remains unknown so far.
The scaling of zeros of the chromatic polynomial of random graphs with parameters n and the number of edges m or the connection probability p has been studied empirically using an algorithm based on symbolic pattern matching.
In a large range of random graphs of order n and size M(n) the distribution of the number of tree components of order k is asymptotically Poisson.
Consider a given random graph model defined on the probability space
be a real valued function which assigns to each graph in
, conditional random graphs are models in which the probability measure
Special cases are conditionally uniform random graphs, where
assigns equal probability to all the graphs having specified properties.
They can be seen as a generalization of the Erdős–Rényi model G(n,M), when the conditioning information is not necessarily the number of edges M, but whatever other arbitrary graph property
In this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties.
The earliest use of a random graph model was by Helen Hall Jennings and Jacob Moreno in 1938 where a "chance sociogram" (a directed Erdős-Rényi model) was considered in studying comparing the fraction of reciprocated links in their network data with the random model.
[11] Another use, under the name "random net", was by Ray Solomonoff and Anatol Rapoport in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.