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In mathematics, random graph is the general term to refer to

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

s over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them.graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

and

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

. 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 In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real ...

s 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. Béla Bollobás, ''Random Graphs'', 1985, Academic Press Inc., London Ltd. Different random graph models produce different

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 obtaining ''any one particular'' random graph with ''m'' edges is

p^m (1-p)^

with the notation

N = \tbinom

. Béla Bollobás, ''Probabilistic Combinatorics and Its Applications'', 1991, Providence, RI: American Mathematical Society. A closely related model, the

Erdős–Rényi model In the mathematical field of graph theory, the Erdős–Rényi model is either of two closely related models for generating random graphs or the evolution of a random network. They are named after Hungarian mathematicians Paul Erdős and Alf ...

denoted ''G''(''n'',''M''), assigns equal probability to all graphs with exactly ''M'' edges. With 0 ≤ ''M'' ≤ ''N'', ''G''(''n'',''M'') has

\tbinom

elements and every element occurs with probability

1/\tbinom

. The latter model can be viewed as a snapshot at a particular time (''M'') of the random graph process

\tilde_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 In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0 ...

has the following property:

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

It turns out that if the vertex set is

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...

then there is,

up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

, 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

. The probability of an edge ''uv'' between any vertices ''u'' and ''v'' is some function of the

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...

u • v of their respective vectors. The

network probability matrix The network probability matrix describes the probability structure of a network based on the historical presence or absence of edges in a network. For example, individuals in a social network are not connected to other individuals with uniform ...

models random graphs through edge probabilities, which represent the probability

p_

that a given edge

e_

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. 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 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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

. The study of this model is to determine if, or at least estimate the probability that, a property may occur.

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.

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

n

and

p

what the probability is that

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

n

grows very large.

Percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnecte ...

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

\langle k\rangle

. Next we remove randomly a fraction

1-p

of nodes and leave only a fraction

p

. There exists a critical percolation threshold

p_c=\tfrac

below which the network becomes fragmented while above

p_c

a giant connected component exists. Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of

1-p

of nodes from the network is removed. It was shown that for random graph with Poisson distribution of degrees

p_c=\tfrac

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,

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 \le r  < n

, and

rn = 2m

is even. The degree sequence of a graph

G

G^n

depends only on the number of edges in the sets :

V_n^ = \left \ \subset V^, \qquad  i=1, \cdots, n.

If edges,

M

in a random graph,

G_M

is large enough to ensure that almost every

G_M

has minimum degree at least 1, then almost every

G_M

is connected and, if

n

is even, almost every

G_M

has a perfect matching. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected. 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

\tfrac\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 labeled graph with

n

vertices and at least

cn\log(n)

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. 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.

Coloring

Given a random graph ''G'' of order ''n'' with the vertex ''V''(''G'') = , by the

greedy algorithm A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locall ...

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.). 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.

Random trees

A random tree is a

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

or 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 Poisson. Types of random trees include uniform spanning tree, random minimal spanning tree, random binary tree, treap,

rapidly exploring random tree A rapidly exploring random tree (RRT) is an algorithm designed to efficiently search nonconvex, high-dimensional spaces by randomly building a space-filling tree. The tree is constructed incrementally from samples drawn randomly from the search s ...

, Brownian tree, and

random forest Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that operates by constructing a multitude of decision trees at training time. For classification tasks, the output of th ...

Conditional random graphs

Consider a given random graph model defined on the probability space

(\Omega, \mathcal, P)

and let

\mathcal(G) : \Omega \rightarrow R^

be a real valued function which assigns to each graph in

\Omega

a vector of ''m'' properties. For a fixed

\mathbf \in R^

, ''conditional random graphs'' are models in which the probability measure

P

assigns zero probability to all graphs such that '

\mathcal(G) \neq \mathbf

. Special cases are ''conditionally uniform random graphs'', where

P

assigns equal probability to all the graphs having specified properties. They can be seen as a generalization of the

''G''(''n'',''M''), when the conditioning information is not necessarily the number of edges ''M'', but whatever other arbitrary graph property

\mathcal(G)

. 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 Jacob Levy Moreno (born Iacob Levy; May 18, 1889 – May 14, 1974) was a Romanian-American psychiatrist, psychosociologist, and educator, the founder of psychodrama, and the foremost pioneer of group psychotherapy. During his lifetime, he was r ...

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. 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. The

of random graphs was first defined by Paul Erdős and

Alfréd Rényi Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician known for his work in probability theory, though he also made contributions in combinatorics, graph theory, and number theory. Life Rényi was born in Budapest ...

in their 1959 paper "On Random Graphs" Erdős, P. Rényi, A (1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–29

/ref> and independently by Gilbert in his paper "Random graphs"..

References

{{DEFAULTSORT:Random Graph Graph theory *