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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.^[1][2] 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.^[3] 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 obtaining any one particular random graph with m edges is ${\displaystyle p^{m}(1-p)^{N-m}}$ with the notation ${\displaystyle N={\tbinom {n}{2}}}$.^[4]

A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges. With 0 ≤ M ≤ N, G(n,M) has ${\displaystyle {\tbinom {N}{M}}}$ elements and every element occurs with probability ${\displaystyle 1/{\tbinom {N}{M}}}$.^[3] The latter model can be viewed as a snapshot at a particular time (M) of the random graph process ${\displaystyle {\tilde {G}}_{n}}$, 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.

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 ${\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{m}\in V}$, there is a vertex c in V that is adjacent to each of ${\displaystyle a_{1},\ldots ,a_{n}}$ and is not adjacent to any of ${\displaystyle b_{1},\ldots ,b_{m}}$.

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 ${\displaystyle p_{i,j}}$ that a given edge ${\displaystyle e_{i,j}}$ 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.^[5]

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.^[4]

Terminology

The term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that the error probabilities tend to zero.^[4]

Properties

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 ${\displaystyle n}$ and ${\displaystyle p}$ what the probability is that ${\displaystyle G(n,p)}$ is connected. In studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as ${\displaystyle n}$ grows very large. Percolation theory characterizes the connectedness of random graphs, especially infinitely large ones.

Percolation is related to the robustness of the graph (called also network). Given a random graph of ${\displaystyle n}$ nodes and an average degree ${\displaystyle \langle k\rangle }$. Next we remove randomly a fraction ${\displaystyle 1-p}$ of nodes and leave only a fraction ${\displaystyle p}$. There exists a critical percolation threshold ${\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}$ below which the network becomes fragmented while above ${\displaystyle p_{c}}$ a giant connected component exists.^{[1][5][6][7][8]}

Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of ${\displaystyle 1-p}$ of nodes from the network is removed. It was shown that for random graph with Poisson distribution of degrees ${\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}$ exactly as for random removal.

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 Szemerédi regularity lemma, the existence of that property on almost all graphs.

In random regular graphs, ${\displaystyle G(n,r-reg)}$ are the set of ${\displaystyle r}$-regular graphs with ${\displaystyle r=r(n)}$ such that ${\displaystyle n}$ and ${\displaystyle m}$ are the natural numbers, ${\displaystyle 3\leq r<n}$, and ${\displaystyle rn=2m}$ is even.^[3]

The degree sequence of a graph ${\displaystyle G}$ in ${\displaystyle G^{n}}$ depends only on the number of edges in the sets^[3]

${\displaystyle V_{n}^{(2)}=\left\{ij\ :\ 1\leq j\leq n,i\neq j\right\}\subset V^{(2)},\qquad i=1,\cdots ,n.}$

If edges, ${\displaystyle M}$ in a random graph, ${\displaystyle G_{M}}$ is large enough to ensure that almost every ${\displaystyle G_{M}}$ has minimum degree at least 1, then almost every ${\displaystyle G_{}}$Zdroj:https://en.wikipedia.org?pojem=Random_graphs
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