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{{short description|Graph generated by a random process}}

{{for|'''the''' countably-infinite random graph|Rado graph}}

{{Network Science}}

In [[mathematics]], '''random graph''' is the general term to refer to [[probability distributionsdistribution]]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">[[Béla{{cite book|first=Béla|last=Bollobás]], ''|title=Random Graphs'', |edition=2nd|year=2001|publisher=Cambridge Edition,University 2001,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]] 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 network]]s need to be modeled&nbsp;– a large number ofmany 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''.

== Random graph modelsModels ==

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">[[Béla Bollobás]], ''Random Graphs'', 1985, Academic Press Inc., London Ltd.</ref> Different '''random graph models''' produce different [[probability distribution]]s on graphs. Most commonly studied is the one proposed by [[Edgar Gilbert]] but often called the [[Erdős–Rényi model]], denoted ''G''(''n'',''p''),. inIn whichit, 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">[[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'',''pM'') has <math>\tbinom{N}{M}</math> elements and every element occurs with probability <math>1/\tbinom{N}{M}</math>.<ref name = "Random Graphs2" /> The latter''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>, 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:

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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 of random graphs==

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 1−''<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 ''p<submath>cp_c</submath>'' 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" />

<ref>{{cite book |title= Complex Networks: Structure, Robustness and Function |authors= Reuven Cohen and [[Shlomo Havlin]] |year= 2010 |url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php |publisher= Cambridge University Press}}</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.

''(threshold functions, evolution of G~)''

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 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'' = 2''m''2m</math> is even.<ref name = "Random Graphs2" />

The degree sequence of a graph ''<math>G''</math> in ''G<supmath>G^n</supmath>'' 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, ''G<submath>MG_M</submath>'' is large enough to ensure that almost every ''G<submath>MG_M</submath>'' has minimum degree at least 1, then almost every ''G<submath>MG_M</submath>'' is connected and, if ''<math>n''</math> is even, almost every ''G<submath>MG_M</submath>'' 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 labelledlabeled graph with ''<math>n''</math> vertices and at least ''<math>cn''\log(''n'')</math> edges is [[Hamiltonian cycle|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. [[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|pages=046107|year=2003|doi=10.1103/PhysRevE.67.046107|pmid=12786436|bibcode=2003PhRvE..67d6107R|arxiv=cond-mat/0212469|s2cid=33054818 }}</ref>

== Triangles in Random Graphs ==

Given a random graph ''G(n,p_n)''. If <math>p_{n}</math> ≪ <math>1/n</math>, then almost every ''G(n,p_n)'' dose not contain a triangle. If <math>p_{n}</math> ≫ <math>1/n</math>, then almost every ''G(n,p_n)'' contains a triangle.

===The proof=Colouring ==

Given a random graph ''G'' of order ''n'' with the vertex ''V''(''G'') = {1, ..., ''n''}, by the [[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" />

Let ''X(G)'' be the number of triangles in ''G''. Let the set <math>{T_{1}, {T_{2}}, ..., {T_{{n \choose 3}}}}</math> be all possible triangles in G and let <math>X_{i}(G)</math> be a random variable such that <math>X_{i}(G)</math>=1 if ''G'' contains the triangle <math>T_{i}</math>, or <math>X_{i}(G)</math>=0 otherwise.

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. <ref name = "Chromatic Polynomials of Random Graphs">{{cite Frankjournal | last1 = Van Bussel, Christoph| first1 = Frank | last2 = Ehrlich, Denny| first2 = Christoph | last3 = Fliegner, Sebastian| first3 = Denny | last4 = Stolzenberg and| Marcfirst4 = Sebastian | last5 = Timme, | first5 = Marc | year = 2010 | title = Chromatic Polynomials of Random Graphs, | journal = J. Phys. A: Math. Theor. 43,| 175002volume (2010)= [http://dx.doi.org/10.1088/1751-8113/43/ | issue = 17/| page = 175002 | doi: = 10.1088/1751-8113/43/17/175002] | arxiv = 1709.06209 | bibcode = 2010JPhA...43q5002V | s2cid = 15723612 }}</ref>

== Triangles in Random Graphstrees ==

Note that

{{main|randomRandom tree}}

:#<math>X = \sum_{i} X_{i}</math>

A [[random tree]] is a [[tree (graph theory)|tree]] or [[Arborescence (graph theory)|arborescence]] that is formed by a [[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 [[SiméonPoisson Denis Poissondistribution|Poisson]]. Types of random trees include [[uniform spanning tree]], [[random minimalminimum spanning tree]], [[random binary tree]], [[treap]], [[rapidly exploring random tree]], [[Brownian tree]], and [[random forest]].

:#<math>X_{i}</math> and <math>X_{j}</math> are independent if <math>\left |VT_{i} \cap VT_{j}\right |</math> ≤ 1.

== Conditional random graphs ==

Let <math>p_{n}</math> ≪ <math>1/n</math>, then

:<math>Pr(X \ge 1) \le E(X) = {n \choose 3}p_{n}^{3} \le (\frac{p_{n}}{1/n})^{3} \to 0</math>, as <math>n \to \infty</math>.

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.

Let <math>p_{n}</math> ≫ <math>1/n</math>, then

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>.

:<math>\begin{align}

Pr(X=0) &\le Pr(|X-E(X)| \ge E(X)) \\

&\le \frac{Var(X)}{E(X)^2} \ \ (by\ Chebyshev's\ inequality) \\

&= \left [ \sum_{i}Var(X_{i}) + \sum_{|VT_{j} \cap VT_{k}|=2}Cov(X_{j},X_{k}) \right ] / E(X)^2 \\

&\le \left [ \sum_{i}E(X_{i}) + \sum_{|VT_{j} \cap VT_{k}|=2}E(X_{j}X_{k}) \right ] / EX(X)^2 \\

&= \frac{{n \choose 3}p_{n}^3 + 2{n \choose 4}{4 \choose 2}p_{n}^5}{{n \choose 3}^{2}p_{n}^6} \\

&\le \left (\frac{1/n}{p_{n}}\right )^3 + 18\left (\frac{1/n}{p_n}\right )\left (\frac{n-3}{(n-1)(n-2)}\right ) \\

&\to 0, \ as \ n \to \infty

\end{align}</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.

== Coloring of Random Graphs ==

Given a random graph ''G'' of order ''n'' with the vertex ''V''(''G'') = {1, ..., ''n''}, by the [[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]], 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"> Frank Van Bussel, Christoph Ehrlich, Denny Fliegner, Sebastian Stolzenberg and Marc Timme, Chromatic Polynomials of Random Graphs, J. Phys. A: Math. Theor. 43, 175002 (2010) [http://dx.doi.org/10.1088/1751-8113/43/17/175002 | doi:10.1088/1751-8113/43/17/175002]</ref>

== Random trees ==

{{main|random tree}}

A [[random tree]] is a [[tree (graph theory)|tree]] or [[Arborescence (graph theory)|arborescence]] that is formed by a [[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 [[Siméon Denis Poisson|Poisson]]. Types of random trees include [[uniform spanning tree]], [[random minimal spanning tree]], [[random binary tree]], [[treap]], [[rapidly exploring random tree]], [[Brownian tree]], and [[random forest]].

==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 [[Ray Solomonoff]] and [[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>

Random graphs were first defined by [[Paul Erd&#337;s]] and [[Alfréd Rényi]] in their 1959 paper "On Random Graphs"<ref name ="On Random Graphs">[[Paul Erd&#337;s|Erdős, P.]] [[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]</ref> and independently by Gilbert in his paper "Random graphs".<ref name = "Random graphs">{{citation |last= Gilbert |first= E. N. |authorlink=Edgar Gilbert|year=1959 |title=Random graphs |journal=Annals of Mathematical Statistics |volume= 30|pages=1141–1144|doi=10.1214/aoms/1177706098 }}.</ref>

RandomThe [[Erdős–Rényi model]] of random graphs werewas first defined by [[Paul Erd&#337;sErdős]] and [[Alfréd Rényi]] in their 1959 paper "On Random Graphs"<ref name ="On Random Graphs">[[Paul Erd&#337;sErdős|Erdős, P.]] [[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. |authorlinkauthor-link=Edgar Gilbert|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|Bose–Einstein condensation: a network theory approach]]}}

* [[{{annotated link|Cavity method]]}}

* [[{{annotated link|Complex networks]]}}

* {{annotated link|Dual-phase evolution}}

* [[Erdős–Rényi model]]

* [[Exponential{{annotated random graphlink|Erdős–Rényi model]]}}

* {{annotated link|Exponential random graph model}}

* [[{{annotated link|Graph theory]]}}

* [[Network science]]

* {{annotated link|Interdependent networks}}

* [[Percolation]]

* [[{{annotated link|Network science]]}}

* [[Semilinear response]]

* [[{{annotated link|Percolation]]}}

* {{annotated link|Percolation theory}}

*{{annotated link|Random graph theory of gelation}}

* {{annotated link|Regular graph}}

* {{annotated link|Scale free network}}

* [[{{annotated link|Semilinear response]]}}

* {{annotated link|Stochastic block model}}

*{{annotated link|Lancichinetti–Fortunato–Radicchi benchmark}}

==References==

{{reflistReflist}}

{{Stochastic processes}}

{{Authority control}}

{{DEFAULTSORT:Random Graph}}

[[Category:Graph theory]]

[[Category:Random graphs|*]]

[[nl:Complexe netwerken#Random netwerken]]