Models and probability spaces. Monday, September 21
A model of a probability space (Ω, F = 2Ω , P) is an stochastic process or “experiment”
that would produce objects from the sample space, and the probability that a certain
object a ∈ Ω is the result of the process is equal to its probability measure P(a).
One example of a probability space we saw earlier is the The random graph G(n, p).
Formal definition: Ω is the set of all graphs with vertex set {v1 , . . . , vn }. The σ-field
is 2Ω , and the probability measure of any particular graph G ∈ Ω equals:
n
P({G}) = pe(G) (1 − p)( 2 )−e(G) ,
where e(G) denotes the number of edges of G.
The probability of other sets of F can be calculated from this using the axioms.
This is generally the case if F = 2Ω . In this case, we often omit the curly brackets
when giving the probability measure of a set of one element, so we use P(G) instead
of P({G}).
Note that, if p = 1/2, then G(n, p) gives the uniform probability measure on Ω;
this means that every element of Ω is equally likely.
The model for G(n, p) can be expressed as follows: Take n vertices, {v1 , . . . , vn }.
For each pair of vertices {vi , vj }, independently, add an edge {vi , vj } with probability
p.
Random regular graphs. Let Gn,d be the set of all d-regular graphs with vertex set
{v1 , . . . , vn }. Let Gn,d be the probability space where Ω = Gn,d , F = 2Ω , and P is the
uniform probability measure. This is well-defined, but it is hard to actually generate
regular graphs according to this distribution. There are two models: the first is to
enumerate all d-regular graphs on n vertices, and then choose one of them uniformly
at random. The problem is that this enumeration is a notoriously hard problem; no
closed-form formula for the number of d-regular graphs of certain size exists.
A second method is the configuration model. To generate a d-regular graph on
n vertices (dn even), make d copies vi,1 , . . . vi,d of each vertex vi . Now uniformly at
random (uar) choose a matching between these dn vertices. Then, merge all copies of
each vertex into a single vertex. This will generate a d-regular multi-graph; the graph
may have loops or multiple edges. If we restrict the probability space to the case
where the resulting graph is simple, then this gives the uniform distribution on Gn,d .
Precisely, let Pu be the uniform distribution on Gn,d , and let Pc be the probability
measure produced by the configuration model. Then
Pu (G) = Pc (G| G simple).
So, in principle it is possible to generate regular graphs uniformly by repeating
the configuration model until a simple graph results. However, counting arguments
can show that the probability that a simple graph is produced is extremely small,
and becomes smaller as n gets large.
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There are two models to generate random regular graphs which give a distribution
that is almost uniform. The degree restricted model is the following: start with
isolated vertices {v1 , . . . , vn }. At each step, uar choose two distinct, non-adjacent
vertices, both of degree less than d, and add an edge between them. Repeat this step
until no such vertices can be found.
This process always produces a simple graph, but it does not always produce a
d-regular graph. Thus Ω = Gn,d ∪ D, where D is the collection of graphs that can
be the outcome of the process, but that are not d-regular. We can think of them
as “dead ends”. It can be shown (not trivial) that P(D) goes to zero as n goes
to infinity. However, even when conditioned on the event G 6∈ D, the probability
measure obtained by this model is not the uniform distribution.
Another model is the d-star model: start with isolated vertices {v1 , . . . , vn }. At
each step, uar choose a vertex v of degree less than d, and then uar choose a set S of
size d − deg(v), containing vertices distinct from v, and of degree less than d. Repeat
this step until no such vertices can be found.
Again, the sample space will contain “dead ends”, and the probability measure
is not uniform. However, both models are easy to implement, so they may be more
useful if the goal is to generate d-regular graphs for the purpose of simulation etc.
In general, it is not easy to generate random combinatorial objects, if you wish all
such objects to be equally likely. Consider these examples. How would you generate
them? Is the probability measure you obtain the uniform distribution?
• How to generate a permutation of n objects uniformly at random?
• A bipartite graph is a graph with vertex set V = X ∪ Y , where X ∩ Y = ∅,
so that all its edges have one endpoint in X and one in Y . A matching is a
collection of edges that do not share an endpoint. A matching is maximal if
you cannot add any edge of the graph to the matching without violating the
condition. Given a bipartite graph, how can we generate a maximal matching
uniformly at random?
• The degree sequence of a graph is the sequence representing the degrees of all
the vertices, in non-increasing order. Similarly, the degree distribution of a
graph is the information about how many vertices of a given degree the graph
has. Given a degree sequence, or degree distribution, how can we generate a
graph with this degree sequence uniformly at random?
• A partial order is a relation on a given set that is reflexive, antisymmetric, and
transitive. How can we generate a partial order on a given set uniformly at
random?
• A latin square is a square of size n × n so that each row and each column
contains each of the numbers 1, 2, . . . , n exactly once. How can we generate a
latin square uniformly at random?
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