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{{short description|Graph generated by a random process}}
{{for|the countably-infinite random graph|Rado graph}}
{{Network Science}}

In [[Mathematics|mathematics]], '''random graph''' is the general term to refer to [[Probability distribution|probability distribution]]s over [[Graph (discrete mathematics)|graphs]].  Random graphs may be described simply by a probability distribution, or by a random process which generates them.<ref name = "Random Graphs">{{cite book|first=Béla|last=Bollobás|title=Random Graphs|edition=2nd|year=2001|publisher=Cambridge University Press}}</ref><ref name = "Introduction to Random graphs">{{cite book|first1=Alan|last1=Frieze|first2=Michal|last2=Karonski|title=Introduction to Random Graphs|year=2015|publisher=Cambridge University Press}}</ref> The theory of random graphs lies at the intersection between [[Graph theory|graph theory]] and [[Probability theory|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 network|complex network]]s need to be modeled&nbsp;– 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 model|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.<ref name = "Random Graphs2">[[Biography:Béla Bollobás|Béla Bollobás]], ''Random Graphs'', 1985, Academic Press Inc., London Ltd.</ref> Different '''random graph models''' produce different [[Probability distribution|probability distribution]]s on graphs. Most commonly studied is the one proposed by [[Biography:Edgar Gilbert|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 <math>p^m (1-p)^{N-m}</math> with the notation <math>N = \tbinom{n}{2}</math>.<ref name = "Random Graphs3">[[Biography:Béla Bollobás|Béla Bollobás]], ''Probabilistic Combinatorics and Its Applications'', 1991, Providence, RI: American Mathematical Society.</ref>

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 <math>\tbinom{N}{M}</math> elements and every element occurs with probability <math>1/\tbinom{N}{M}</math>.<ref name = "Random Graphs2" />  The ''G''(''n'',''M'') model can be viewed as a snapshot at a particular time (''M'') of the '''random graph process''' <math>\tilde{G}_n</math>, a [[Stochastic process|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|almost surely]] has the following property:

<blockquote>Given any ''n'' + ''m'' elements <math>a_1,\ldots, a_n,b_1,\ldots, b_m \in V</math>, there is a vertex ''c'' in ''V'' that is adjacent to each of <math>a_1,\ldots, a_n</math> and is not adjacent to any of <math>b_1,\ldots, b_m</math>.</blockquote>

It turns out that if the vertex set is countable then there is, [[Up to|up to]] [[Graph isomorphism|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|dot product]] '''u''' • '''v''' of their respective vectors.

The [[Social:Network probability matrix|network probability matrix]] models random graphs through edge probabilities, which represent the probability <math>p_{i,j}</math> that a given edge <math>e_{i,j}</math> 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.<ref name ="Handbook ">[[Biography:Béla Bollobás|Bollobas, B.]] and Riordan, O.M. "Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed., 2003</ref>

[[Random regular graph]]s 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|random variable]]. The study of this model is to determine if, or at least estimate the probability that, a property may occur.<ref name = "Random Graphs3" />

==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.<ref name = "Random Graphs3" />

==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 <math>n</math> and <math>p</math> what the probability is that <math>G(n,p)</math> is [[Connection (mathematics)|connected]].  In studying such questions, researchers often concentrate on the asymptotic behavior of random graphs&mdash;the values that various probabilities converge to as <math>n</math> 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 <math>n</math> nodes and an average degree <math>\langle k\rangle</math>. Next we remove randomly a fraction <math>1-p</math> of nodes and leave only a fraction <math>p</math>. There exists a critical percolation threshold <math>p_c=\tfrac{1}{\langle k\rangle}</math> below which the network becomes fragmented while above <math>p_c</math> a giant connected component exists.<ref name = "Random Graphs" /><ref name ="Handbook " /><ref name = "Random graphs" /><ref>{{cite book  |title=Networks: An Introduction |last= Newman |first=M. E. J. |year= 2010 |publisher=  Oxford}}</ref><ref name ="On Random Graphs" />

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

Random graphs are widely used in the [[Probabilistic method|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 graph|random regular graph]]s, <math>G(n,r-reg)</math> are the set of <math>r</math>-regular graphs with <math>r = r(n)</math> such that <math>n</math> and <math>m</math> are the natural numbers, <math>3 \le r  < n</math>, and <math>rn = 2m</math> is even.<ref name = "Random Graphs2" />

The degree sequence of a graph <math>G</math> in <math>G^n</math> depends only on the number of edges in the sets<ref name = "Random Graphs2" />
:<math>V_n^{(2)} = \left \{ij \ : \ 1 \leq j \leq n, i \neq j \right \} \subset V^{(2)}, \qquad  i=1, \cdots, n.</math>

If edges, <math>M</math> in a random graph, <math>G_M</math> is large enough to ensure that almost every <math>G_M</math> has minimum degree at least 1, then almost every <math>G_M</math> is connected and, if <math>n</math> is even, almost every <math>G_M</math> has a perfect matching. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.<ref name = "Random Graphs2" />

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 <math>\tfrac{n}{4}\log(n)</math> 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 <math>c</math>, almost every labeled graph with <math>n</math> vertices and at least <math>cn\log(n)</math> edges is Hamiltonian. 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. [[Biography:Alireza Mashaghi|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.<ref>{{cite journal|first1=A.|last1=Ramezanpour|first2=V.|last2=Karimipour|first3=A.|last3=Mashaghi|title=Generating correlated networks from uncorrelated ones|journal=Phys. Rev. E|volume=67|issue=46107|article-number=046107|year=2003|doi=10.1103/PhysRevE.67.046107|pmid=12786436|bibcode=2003PhRvE..67d6107R|arxiv=cond-mat/0212469|s2cid=33054818 }}</ref>

== Colouring ==
Given a random graph ''G'' of order ''n'' with the vertex ''V''(''G'') = {1, ..., ''n''}, by the [[Greedy algorithm|greedy 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.).<ref name = "Random Graphs2" />
The number of proper colorings of random graphs given a number of ''q'' colors, called its [[Chromatic polynomial|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.<ref name = "Chromatic Polynomials of Random Graphs">{{cite journal | last1 = Van Bussel | first1 = Frank | last2 = Ehrlich | first2 = Christoph | last3 = Fliegner | first3 = Denny | last4 = Stolzenberg | first4 = Sebastian | last5 = Timme | first5 = Marc | year = 2010 | title = Chromatic Polynomials of Random Graphs | journal = J. Phys. A: Math. Theor. | volume = 43 | issue = 17| article-number = 175002 | doi = 10.1088/1751-8113/43/17/175002 | arxiv = 1709.06209 | bibcode = 2010JPhA...43q5002V | s2cid = 15723612 }}</ref>

== Random trees ==
{{main|Random tree}}
A [[Random tree|random tree]] is a [[Tree (graph theory)|tree]] or [[Arborescence (graph theory)|arborescence]] that is formed by a [[Stochastic process|stochastic process]]. 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 distribution|Poisson]]. Types of random trees include [[Uniform spanning tree|uniform spanning tree]], [[Random minimum spanning tree|random minimum spanning tree]], [[Random binary tree|random binary tree]], [[Treap|treap]], [[Rapidly exploring random tree|rapidly exploring random tree]], [[Brownian tree]], and [[Random forest|random forest]].

== Conditional random graphs ==

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

Special cases are ''conditionally uniform random graphs'', where <math>P</math> 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 <math>\mathcal{P}(G)</math>. In this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties.

==History==
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.<ref>{{cite journal |last1=Moreno |first1=Jacob L |last2=Jennings |first2=Helen Hall |title=Statistics of Social Configurations |journal=Sociometry |date=Jan 1938 |volume=1 |issue=3/4 |pages=342–374 |doi=10.2307/2785588|jstor=2785588 |url=https://hal.science/hal-03963403/file/morenojennings1938groupefmr.pdf }}</ref>  Another use, under the name "random net", was by [[Biography:Ray Solomonoff|Ray Solomonoff]] and [[Biography:Anatol Rapoport|Anatol Rapoport]] in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.<ref>{{cite journal |last1=Solomonoff |first1=Ray |last2=Rapoport |first2=Anatol |title=Connectivity of random nets |journal=Bulletin of Mathematical Biophysics |date=June 1951 |volume=13 |issue=2 |pages=107–117 |doi=10.1007/BF02478357}}</ref>

The [[Erdős–Rényi model]] of random graphs was first defined by [[Biography:Paul Erdős|Paul Erdős]] and [[Biography:Alfréd Rényi|Alfréd Rényi]] in their 1959 paper "On Random Graphs"<ref name ="On Random Graphs">[[Biography:Paul Erdős|Erdős, P.]] [[Biography:Alfréd Rényi|Rényi, A]] (1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p.&nbsp;290&ndash;297 [http://www.renyi.hu/~p_erdos/1959-11.pdf] {{Webarchive|url=https://web.archive.org/web/20200807021117/https://www.renyi.hu/~p_erdos/1959-11.pdf |date=2020-08-07 }}</ref> and independently by Gilbert in his paper "Random graphs".<ref name = "Random graphs">{{citation |last= Gilbert |first= E. N. |year=1959 |title=Random graphs |journal=Annals of Mathematical Statistics |volume= 30|issue= 4 |pages=1141–1144|doi=10.1214/aoms/1177706098 |doi-access=free }}.</ref>

==See also==
* {{annotated link|Physics:Cavity method}}
* {{annotated link|Dual-phase evolution}}
* {{annotated link|Erdős–Rényi model}}
* {{annotated link|Graph theory}}
* {{annotated link|Interdependent networks}}
* {{annotated link|Network science}}
* {{annotated link|Percolation}}
* {{annotated link|Percolation theory}}
*{{annotated link|Chemistry:Random graph theory of gelation}}
* {{annotated link|Regular graph}}
* {{annotated link|Physics:Semilinear response}}
* {{annotated link|Stochastic block model}}
*{{annotated link|Lancichinetti–Fortunato–Radicchi benchmark}}

==References==
{{Reflist}}

{{Stochastic processes}}

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[[Category:Graph theory]]
[[Category:Random graphs|*]]

{{Sourceattribution|Random graph}}