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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 phenomenon i ...
s over
graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
. Random graphs may be described simply by a probability distribution, or by a
random process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
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 conne ...
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 ...
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 Béla Bollobás FRS (born 3 August 1943) is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul E ...
, ''Random Graphs'', 1985, Academic Press Inc., London Ltd. Different random graph models produce different
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 phenomenon i ...
s on graphs. Most commonly studied is the one proposed by
Edgar Gilbert Edgar Nelson Gilbert (July 25, 1923 – June 15, 2013) was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories whose accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Ell ...
, 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 Béla Bollobás FRS (born 3 August 1943) is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul E ...
, ''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 Alfr ...
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 In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
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 numbers; ...
then there is,
up to Two Mathematical object, 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'' wi ...
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 is ...
, only a single graph with this property, namely the
Rado graph In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whe ...
. 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 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 algebra ...
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 graph A random ''r''-regular graph is a graph selected from \mathcal_, which denotes the probability space of all ''r''-regular graphs on n vertices, where 3 \le r 0 is a positive constant, and d is the least integer satisfying (r-1)^ \ge (2 + \epsilon ...
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 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 po ...
. 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 Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
. 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, disconnected ...
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 The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects fr ...
, 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 Szemerédi's regularity lemma is one of the most powerful tools in extremal graph theory, particularly in the study of large dense graphs. It states that the vertices of every large enough graph can be partitioned into a bounded number of parts so ...
, the existence of that property on almost all graphs. In
random regular graph A random ''r''-regular graph is a graph selected from \mathcal_, which denotes the probability space of all ''r''-regular graphs on n vertices, where 3 \le r 0 is a positive constant, and d is the least integer satisfying (r-1)^ \ge (2 + \epsilon ...
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 \le r < n, and rn = 2m is even. The degree sequence of a graph G in 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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 locally ...
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 The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to s ...
, 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 In mathematics and computer science, a random tree is a tree or arborescence that is formed by a stochastic process. Types of random trees include: *Uniform spanning tree, a spanning tree of a given graph in which each different tree is equally ...
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 probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
. 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 A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, s ...
,
random minimal spanning tree In mathematics, a random minimum spanning tree may be formed by assigning random weights from some distribution to the edges of an undirected graph, and then constructing the minimum spanning tree of the graph. When the given graph is a complete gr ...
,
random binary tree In computer science and probability theory, a random binary tree is a binary tree selected at random from some probability distribution on binary trees. Two different distributions are commonly used: binary trees formed by inserting nodes one at ...
,
treap In computer science, the treap and the randomized binary search tree are two closely related forms of binary search tree data structures that maintain a dynamic set of ordered keys and allow binary searches among the keys. After any sequence of in ...
, rapidly exploring random tree,
Brownian tree In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a special case from random real trees which may be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three a ...
, and random forest.


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
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 Alfr ...
''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 Helen Hall Jennings (September 20, 1905 – October 4, 1966) was a social psychologist and trailblazer in the field of social networks in the early 20th century. She developed quantitative research methods used to study sociometry, a quantitative m ...
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 rec ...
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 Ray Solomonoff (July 25, 1926 – December 7, 2009) was the inventor of algorithmic probability, his General Theory of Inductive Inference (also known as Universal Inductive Inference),Samuel Rathmanner and Marcus Hutter. A philosophical treatise ...
and
Anatol Rapoport Anatol Rapoport ( uk, Анатолій Борисович Рапопо́рт; russian: Анато́лий Бори́сович Рапопо́рт; May 22, 1911January 20, 2007) was an American mathematical psychologist. He contributed to genera ...
in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices. 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 Alfr ...
of random graphs was first defined by
Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
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 to ...
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"..


See also

* Bose–Einstein condensation: a network theory approach *
Cavity method The cavity method is a mathematical method presented by Marc Mézard, Giorgio Parisi and Miguel Angel Virasoro in 1987 to solve some mean field type models in statistical physics, specially adapted to disordered systems. The method has been used t ...
*
Complex networks Complex Networks is an American media and entertainment company for youth culture, based in New York City. It was founded as a bi-monthly magazine, ''Complex'', by fashion designer Marc (Ecko) Milecofsky. Complex Networks reports on popular a ...
* Dual-phase evolution *
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 Alfr ...
*
Exponential random graph model Exponential family random graph models (ERGMs) are a family of statistical models for analyzing data from social and other networks. Examples of networks examined using ERGM include knowledge networks, organizational networks, colleague networks, ...
*
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 conne ...
*
Interdependent networks The study of interdependent networks is a subfield of network science dealing with phenomena caused by the interactions between complex networks. Though there may be a wide variety of interactions between networks, ''dependency'' focuses on the ...
*
Network science Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors repre ...
*
Percolation Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applicatio ...
*
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, disconnected ...
*
Random graph theory of gelation Random graph theory of gelation is a mathematical theory for sol–gel processes. The theory is a collection of results that generalise the Flory–Stockmayer theory, and allow identification of the gel point, gel fraction, size distribution of pol ...
*
Regular graph In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree o ...
*
Scale free network A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction ''P''(''k'') of nodes in the network having ''k'' connections to other nodes goes for large values of ''k'' as : P(k) ...
*
Semilinear response Semi-linear response theory (SLRT) is an extension of linear response theory (LRT) for mesoscopic Mesoscopic physics is a subdiscipline of condensed matter physics that deals with materials of an intermediate size. These materials range in ...
* Stochastic block model * Lancichinetti–Fortunato–Radicchi benchmark


References

{{DEFAULTSORT:Random Graph Graph theory *