random graph (infinite)

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

<

ω

}

be a set of vertices. For each

i

<

ω

,

i

≥

1

we

construct

a graph

G

i

on the vertices

a

1

,

…

,

a

i

recursively.

•

G

1

is the unique graph on one vertex.

•

For

i

>

1

we must describe for any

j

<

k

≤

i

when

a

j

and

a

k

are joined.

–

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

–

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

Γ

be the graph on

{

a

i

:

i

<

ω

}

so that for any

n

,

m

<

ω

,

a

n

is joined to

a

m

in

Γ

iff it is in some

G

i

.

Then we call

Γ

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

<

ω

}

there is some

a

n

∈

{

a

i

:

i

<

ω

}

∖

(

U

∪

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

,

…

,

b

n

,

…

be an enumeration of

{

a

i

:

i

<

ω

}

∖

(

U

∪

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

lim

n

→

∞

⁢

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

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