Optimal solutions for Rubik's Cube

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{{for|'''the''' random graph|Rado graph}}
{{Network Science}}

In [[mathematics]], '''random graph''' is the general term to refer to probability distributions over [[Graph (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 Bollobás]], ''Random Graphs'', 2nd Edition, 2001, 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 all areas in which [[complex network]]s need to be modeled – a large number of 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 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">[[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]], denoted ''G''(''n'',''p''), in which every possible edge occurs independently with probability 0 < ''p'' < 1. The probability of a random graph with ''m'' edges is ''pm''(1−''p'')''N''−''m''.<ref name = "Random Graphs3">[[Béla Bollobás]], ''Probabilistic Combinatorics and Its Applications'', 1991, Providence, RI: American Mathematical Society.</ref> A closely related model, the [[Erdős–Rényi model]] denoted ''G''(''n'',''M''), assigns equal probability to all graphs with exactly ''M'' edges. With the notation <math>N = \begin{bmatrix} n\\2 \end{bmatrix},</math> with 0 ≤ ''M'' ≤ ''N'', ''G''(''n'',''p'') has <math>\begin{bmatrix} N\\M \end{bmatrix}</math> elements and every element occurs with probability <math>\begin{bmatrix} N \\ M \\ \end{bmatrix}^{-1}.</math><ref name = "Random Graphs2" /> The latter 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:

<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]] [[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]] '''u''' • '''v''' of their respective vectors.

The [[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.

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

''(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, ''G''(''n'',''r''-reg) are the set of ''r''-regular graphs with ''r'' = ''r''(''n'') such that ''n'' and ''m'' are the natural numbers, 3 ≤ ''r'' < ''n'', and ''rn'' = 2''m'' is even.<ref name = "Random Graphs2" />

The degree sequene of a graph ''G'' in ''Gn'' 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, ''M'' in a random graph, ''GM'' is large enough to ensure that almost every ''GM'' has minimum degree at least 1, then almost every ''GM'' is connected and, if ''n'' is even, almost every ''GM'' 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 (''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 ''c'', almost every labelled graph with ''n'' vertices and at least ''cn''log(''n'') 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.

== 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" />

== 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==
Random graphs were first defined by [[Paul Erdős]] and [[Alfréd Rényi]] in their 1959 paper "On Random Graphs"<ref name ="On Random Graphs">[[Paul Erdős|Erdős, P.]] [[Alfréd Rényi|Rényi, A]] (1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–297 [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>

==See also==
* [[Bose–Einstein condensation: a network theory approach]]
* [[Cavity method]]
* [[Complex networks]]
* [[Erdős–Rényi model]]
* [[Exponential random graph model]]
* [[Graph theory]]
* [[Network science]]
* [[Percolation]]
* [[Semilinear response]]

==References==
{{reflist}}

{{Stochastic processes}}

{{DEFAULTSORT:Random Graph}}
[[Category:Graph theory]]
[[Category:Random graphs|*]]

[[nl:Complexe netwerken#Random netwerken]]

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Optimal solutions for Rubik's Cube