In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. 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. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a 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 but often called the Erdős–Rényi model, denoted G(n,p). In it, every possible edge occurs independently with probability 0 < p < 1. The probability of obtaining any one particular random graph with m edges is p m ( 1 − p ) N − m {\displaystyle p^{m}(1-p)^{N-m}} with the notation N = ( n 2 ) {\displaystyle N={\tbinom {n}{2}}} . A closely related model, also called the Erdős–Rényi model and denoted G(n,M), assigns equal probability to all graphs with exactly M edges. With 0 ≤ M ≤ N, G(n,M) has ( N M ) {\displaystyle {\tbinom {N}{M}}} elements and every element occurs with probability 1 / ( N M ) {\displaystyle 1/{\tbinom {N}{M}}} . The G(n,M) model can be viewed as a snapshot at a particular time (M) of the random graph process G ~ n {\displaystyle {\tilde {G}}_{n}} , 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 < 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 1 , … , a n , b 1 , … , b m ∈ V {\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{m}\in V} , there is a vertex c in V that is adjacent to each of a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} and is not adjacent to any of b 1 , … , b m {\displaystyle b_{1},\ldots ,b_{m}} . 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 p i , j {\displaystyle p_{i,j}} that a given edge e i , j {\displaystyle 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 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. 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 if, or at least estimate the probability that, a property may occur.
… excerpt ends here. Continue reading the full article.
