Random graph

For the random graph, see Rado graph.

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In mathematics, a random graph is a graph that is generated by some random process.^[1] The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.

1 Random graph models
2 Terminology
3 Properties of random graphs
4 Coloring of Random Graphs
5 Random trees
6 History
7 See also
8 References

Random graph models [edit]

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. ^[2]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 a random graph with m edges is ${p}^m$ ${(1-p)}^{N-m}$ .^[3] 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 $\mathbf N = \begin{bmatrix} n\\2 \end{bmatrix}\,$ , with 0≤M≤N, G(n,p) has $\mathbf \begin{bmatrix} N\\M \end{bmatrix}\,$ elements and every element occurs with probability $\begin{bmatrix} N \\ M \\ \end{bmatrix}^{-1}$ . ^[2]The fastest known algorithm for generating the former model is proposed by Nobari et al. in.^[4] The latter model can be viewed as a snapshot at a particular time (M) of the random graph process $\tilde{G}_n$ , 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 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 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 $p_{i,j}$ that a given edge $e_{i,j}$ exists for a specified time period. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs.

For $M \simeq pN$ , 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, or at least estimate the probability a property may occur. ^[3].

Terminology [edit]

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.^[3].

Properties of random graphs [edit]

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 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 $p_c=1/\langle k\rangle$ below which the network becomes fragmented while above $p_c$ a giant connected component exists ^[1] ^[5] ^[6] ^[7] ^[8] .^[9]

(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 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≤r<n, and rn=2m is even.^[2]

The degree sequene of a graph $G \in G^{n}$ depends only on the number of edges in the sets ${V}_n^{(2)}$ = {ij:1 ≤ j ≤ n, i≠j} ⊂ ${V}^{(2)}$ , i=1, ..., n.^[2]

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. ^[2]

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. With the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian.

Coloring of Random Graphs [edit]

Given a random graph G of order n with the vertex V(G)={1,2,...,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).^[2]

Random trees [edit]

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.

History [edit]

Random graphs were first defined by Paul Erdős and Alfréd Rényi in their 1959 paper "On Random Graphs"^[8] and independently by Gilbert in his paper "Random graphs".^[9]

References [edit]

^ ^a ^b Béla Bollobás, Random Graphs, 2nd Edition, 2001, Cambridge University Press
^ ^a ^b ^c ^d ^e ^f Béla Bollobás, Random Graphs, 1985, Academic Press Inc., London Ltd.
^ ^a ^b ^c Béla Bollobás, Probabilistic Combinatorics and Its Applications, 1991, Providence, RI: American Mathematical Society.
^ Nobari, Sadegh; Lu, Xuesong; Karras, Panagiotis; Bressan, Stéphane (2011), "Fast random graph generation", Proceedings of the 14th International Conference on Extending Database Technology (Uppsala, Sweden: ACM) (11): 331–342, doi:10.1145/1951365.1951406, ISBN 978-1-4503-0528-0 Unknown parameter |keywords= ignored (help); Unknown parameter |address= ignored (help).
^ ^a ^b 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
^ Newman, M. E. J. (2010). Networks: An Introduction. Oxford.
^ Reuven Cohen and Shlomo Havlin (2010). Complex Networks: Structure, Robustness and Function. Cambridge University Press.
^ ^a ^b Erdős, P. Rényi, A (1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–297 [1]
^ ^a ^b Gilbert, E. N. (1959), "Random graphs", Annals of Mathematical Statistics 30: 1141–1144, doi:10.1214/aoms/1177706098 .

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