random graph (infinite)

Suppose we have some method $M$ of generating sequences of letters from $\{p,q\}$ so that at each generation the probability of obtaining $p$ is $x$ , a real number strictly between $0$ and $1$ .

Let $\{a_{i}:i<\omega\}$ be a set of vertices. For each $i<\omega$ , $i\geq 1$ we construct a graph $G_{i}$ on the vertices $a_{1},\ldots,a_{i}$ recursively.

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$G_{1}$ is the unique graph on one vertex.
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For $i>1$ we must describe for any $j<k\leq i$ when $a_{j}$ and $a_{k}$ are joined.
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  If $k<i$ then join $a_{j}$ and $a_{k}$ in $G_{i}$ iff $a_{j}$ and $a_{k}$ are joined in $G_{i-1}$
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  If $k=i$ then generate a letter $l(j,k)$ with $M$ . Join $a_{j}$ to $a_{k}$ iff $l(j,k)=p$ .

Now let $\Gamma$ be the graph on $\{a_{i}:i<\omega\}$ so that for any $n,m<\omega$ , $a_{n}$ is joined to $a_{m}$ in $\Gamma$ iff it is in some $G_{i}$ .

Then we call $\Gamma$ a random graph. Consider the following property which we shall call f-saturation:

Given any finite disjoint $U$ and $V$ , subsets of $\{a_{i}:i<\omega\}$ there is some $a_{n}\in\{a_{i}:i<\omega\}\setminus(U\cup V)$ so that $a_{n}$ is joined to every point of $U$ and no points in $V$ .

Proposition 1

A random graph has f-saturation with probability $1$ .

Proof: Let $b_{1},b_{2},\ldots,b_{n},\ldots$ be an enumeration of $\{a_{i}:i<\omega\}\setminus(U\cup V)$ . We say that $b_{i}$ is correctly joined to $(U,V)$ iff it is joined to all the members of $U$ and non of the members of $V$ . Then the probability that $b_{i}$ is not correctly joined is $(1-x^{|U|}(1-x)^{|V|})$ which is some real number $y$ strictly between $0$ and $1$ . The probability that none of the first $m$ are correctly joined is $y^{m}$ and the probability that none of the $b_{i}$ s are correctly joined is $\textrm{lim}_{n\to\infty}y^{n}=0$ . Thus one of the $b_{i}$ s is correctly joined.

Proposition 2

Any two countable graphs with f-saturation are isomorphic.

Proof: This is via a back and forth argument. The property of f-saturation is exactly what is needed.

Thus although the system of generation of a random graph looked as though it could deliver many potentially different graphs, this is not the case. Thus we talk about the random graph.

The random graph can also be constructed as a Fraisse limit of all finite graphs, and in many other ways. It is homogeneous and universal for the class of all countable graphs.

The theorem that almost every two infinite random graphs are isomorphic was first proved in [1].

References

1 Paul Erdős and Alfréd Rényi. Asymmetric graphs. Acta Math. Acad. Sci. Hung., 14:295–315, 1963.

Title	random graph (infinite)
Canonical name	RandomGraphinfinite
Date of creation	2013-03-22 13:32:18
Last modified on	2013-03-22 13:32:18
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	8
Author	bbukh (348)
Entry type	Definition
Classification	msc 03C52
Classification	msc 03C30
Classification	msc 03C15
Classification	msc 05C30
Related topic	ExampleOfUniversalStructure
Related topic	Homogeneous4
Defines	random graph