Inmathematics,random graph is the general term to refer toprobability distributions overgraphs. Random graphs may be described simply by a probability distribution, or by arandom process which generates them.^[1][2] The theory of random graphs lies at the intersection betweengraph theory andprobability theory. From a mathematical perspective, random graphs are used to answer questions about the properties oftypical graphs. Its practical applications are found in all areas in whichcomplex 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 theErdős–Rényi random graph model. In other contexts, any graph model may be referred to as arandom graph.

A random graph is obtained by starting with a set ofn 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] Differentrandom graph models produce differentprobability distributions on graphs. Most commonly studied is the one proposed byEdgar Gilbert but often called theErdős–Rényi model, denotedG(n,p). In it, every possible edge occurs independently with probability 0 <p < 1. The probability of obtainingany one particular random graph withm edges is${\displaystyle p^m(1-p{)}^{N-m}}$ with the notation${\displaystyle N=(\genfrac{}{}{0ex}{}{n}{2})}$.^[4]

A closely related model, also called the Erdős–Rényi model and denotedG(n,M), assigns equal probability to all graphs with exactlyM edges. With 0 ≤M ≤N,G(n,M) has${\displaystyle (\genfrac{}{}{0ex}{}{N}{M})}$ elements and every element occurs with probability${\displaystyle 1/(\genfrac{}{}{0ex}{}{N}{M})}$.^[3] TheG(n,M) model can be viewed as a snapshot at a particular time (M) of therandom graph process${\displaystyle {\tilde{G}}_n}$, astochastic process that starts withn 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 objectG called aninfinite random graph. Except in the trivial cases whenp is 0 or 1, such aGalmost surely has the following property:

Given anyn +m elements${\displaystyle a_1,\dots ,a_n,b_1,\dots ,b_m\in V}$, there is a vertexc inV that is adjacent to each of${\displaystyle a_1,\dots ,a_n}$ and is not adjacent to any of${\displaystyle b_1,\dots ,b_m}$.

It turns out that if the vertex set iscountable then there is,up toisomorphism, only a single graph with this property, namely theRado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply therandom 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 therandom dot-product model. A random dot-product graph associates with each vertex areal vector. The probability of an edgeuv between any verticesu andv is some function of thedot productu •v of their respective vectors.

Thenetwork 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.

ForM ≃pN, whereN is the maximal number of edges possible, the two most widely used models,G(n,M) andG(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 arandom variable. The study of this model is to determine if, or at least estimate the probability that, a property may occur.^[4]

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

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${\displaystyle G(n,p)}$ isconnected. 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${\displaystyle ⟨k⟩}$. Next we remove randomly a fraction${\displaystyle 1-p}$ of nodes and leave only a fraction${\displaystyle p}$. There exists a critical percolation threshold${\displaystyle p_c=\frac{1}{⟨k⟩}}$ 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${\displaystyle p_c=\frac{1}{⟨k⟩}}$ exactly as for random removal.

Random graphs are widely used in theprobabilistic 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 theSzemerédi regularity lemma, the existence of that property on almost all graphs.

Inrandom 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,, 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]

${\displaystyle V_n^{(2)}=\left\{ij:1\le j\le n,i\ne j\right\}\subset V^{(2)},i=1,\cdots ,n.}$

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${\displaystyle G_M}$ is connected and, if${\displaystyle n}$ is even, almost every${\displaystyle G_M}$ has a perfect matching. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.^[3]

Almost every graph process on an even number of vertices with the edge raising the minimum degree to 1 or a random graph with slightly more than${\displaystyle \frac{n}{4}\log (n)}$ edges and with probability close to 1 ensures that the graph has a complete matching, with exception of at most one vertex.

For some constant${\displaystyle c}$, almost every labeled graph with${\displaystyle n}$ vertices and at least${\displaystyle cn\log (n)}$ edges isHamiltonian. With the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian.

Properties of random graph may change or remain invariant under graph transformations.Mashaghi A. et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.^[9]

Given a random graphG of ordern with the vertexV(G) = {1, ...,n}, by thegreedy algorithm on the number of colors, the vertices can be colored with colors 1, 2, ... (vertex 1 is colored 1, vertex 2 is colored 1 if it is not adjacent to vertex 1, otherwise it is colored 2, etc.).^[3]The number of proper colorings of random graphs given a number ofq colors, called itschromatic polynomial, remains unknown so far. The scaling of zeros of the chromatic polynomial of random graphs with parametersn and the number of edgesm or the connection probabilityp has been studied empirically using an algorithm based on symbolic pattern matching.^[10]

Arandom tree is atree orarborescence that is formed by astochastic process. In a large range of random graphs of ordern and sizeM(n) the distribution of the number of tree components of orderk is asymptoticallyPoisson. Types of random trees includeuniform spanning tree,random minimum spanning tree,random binary tree,treap,rapidly exploring random tree,Brownian tree, andrandom forest.

Consider a given random graph model defined on the probability space${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$ and let${\displaystyle \mathcal{P}(G):\mathrm{\Omega }\to R^m}$ be a real valued function which assigns to each graph in${\displaystyle \mathrm{\Omega }}$ a vector ofm properties. For a fixed${\displaystyle \mathbf{p}\in R^m}$,conditional random graphs are models in which the probability measure${\displaystyle P}$ assigns zero probability to all graphs such that${\displaystyle \mathcal{P}(G)\ne \mathbf{p}}$.

Special cases areconditionally uniform random graphs, where${\displaystyle P}$ assigns equal probability to all the graphs having specified properties. They can be seen as a generalization of theErdős–Rényi modelG(n,M), when the conditioning information is not necessarily the number of edgesM, but whatever other arbitrary graph property${\displaystyle \mathcal{P}(G)}$. 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 byHelen Hall Jennings andJacob 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 byRay Solomonoff andAnatol Rapoport in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.^[12]

TheErdős–Rényi model of random graphs was first defined byPaul Erdős andAlfréd Rényi in their 1959 paper "On Random Graphs"^[8] and independently by Gilbert in his paper "Random graphs".^[6]

• Bose–Einstein condensation: a network theory approach – Model in network sciencePages displaying short descriptions of redirect targets
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• Dual-phase evolution – Process that drives self-organization within complex adaptive systems
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• Graph theory – Area of discrete mathematics
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• Percolation – Filtration of fluids through porous materials
• Percolation theory – Mathematical theory on behavior of connected clusters in a random graph
• Random graph theory of gelation – Mathematical theory for sol–gel processes
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• Semilinear response – Extension of linear response theory in mesoscopic regimes
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• Random graphs
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