PlanetMath: random graph (infinite)

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random graph (infinite)

(Definition)

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

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.

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

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 and , 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 and no points in .

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

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

iff it is joined to all the members of

and non of the members of

. Then the probability that $b_{i}$ is not correctly joined is $(1-x^{\vert U\vert}(1-x)^{\vert V\vert})$ which is some real number

strictly between 0 and

. The probability that none of the first

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].

Bibliography

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

"random graph (infinite)" is owned by bbukh. [ full author list (2) | owner history (1) ]

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Cross-references: infinite, class, universal, homogeneous, limit, argument, back and forth, isomorphic, countable, enumeration, proof, point, subsets, disjoint, finite, property, generate, iff, join, graph, vertices, strictly, real number, sequences, generating
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This is version 5 of random graph (infinite), born on 2003-04-01, modified 2006-08-16.
Object id is 4137, canonical name is RandomGraph.
Accessed 4550 times total.

Classification:

AMS MSC:	03C15 (Mathematical logic and foundations :: Model theory :: Denumerable structures)
	03C30 (Mathematical logic and foundations :: Model theory :: Other model constructions)
	03C52 (Mathematical logic and foundations :: Model theory :: Properties of classes of models)
	05C30 (Combinatorics :: Graph theory :: Enumeration of graphs and maps)

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