In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them.[1] [2] 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 networks 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.
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.[3] Different random graph models produce different probability distributions 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
pm(1-p)N-m
N=\tbinom{n}{2}
A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges. With 0 ≤ M ≤ N, G(n,M) has
\tbinom{N}{M}
1/\tbinom{N}{M}
\tilde{G}n
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:
Given any n + m elements, there is a vertex c in V that is adjacent to each ofa1,\ldots,an,b1,\ldots,bm\inV
and is not adjacent to any ofa1,\ldots,an
.b1,\ldots,bm
It turns out that if the vertex set is countable then there is, up to 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
pi,j
ei,j
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.[5]
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. The study of this model is to determine if, or at least estimate the probability that, a property may occur.
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.
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
p
G(n,p)
n
Percolation is related to the robustness of the graph (called also network). Given a random graph of
n
\langlek\rangle
1-p
p
pc=\tfrac{1}{\langlek\rangle}
pc
Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of
1-p
pc=\tfrac{1}{\langlek\rangle}
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)
r
r=r(n)
n
m
3\ler<n
rn=2m
The degree sequence of a graph
G
Gn
| (2) | |
| V | |
| n |
=\left\{ij : 1\leqj\leqn,i ≠ j\right\}\subsetV(2), i=1, … ,n.
If edges,
M
GM
GM
GM
n
GM
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{n}{4}log(n)
For some constant
c
n
cnlog(n)
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.[7]
Given a random graph G of order n with the vertex V(G) =, 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.).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.[8]
See main article: Random tree. A random tree is a tree 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, Brownian tree, and random forest.
Consider a given random graph model defined on the probability space
(\Omega,l{F},P)
l{P}(G):\Omega → Rm
\Omega
p\inRm
P
l{P}(G) ≠ p
Special cases are conditionally uniform random graphs, where
P
l{P}(G)
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.[9] 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.[10]
The Erdős–Rényi model of random graphs was first defined by Paul Erdős and Alfréd Rényi in their 1959 paper "On Random Graphs"[11] and independently by Gilbert in his paper "Random graphs".[12]