Re: Pair probabilities
- From: Ian Parker <ianparker2@xxxxxxxxx>
- Date: Mon, 01 Oct 2007 11:30:35 -0700
On 26 Sep, 21:03, Gareth Russell <russ...@xxxxxxxx> wrote:
Hi,
An ecologist here straying abroad. My apologies if this is trivial, but
it's outside my knowledge.
In classic probability example language...
I have a set of objects of c different types (say colors). The number
of objects of each type, n_1, n_2,... n_c is known.
I also have an arbitrary, symmetric n by n binary matrix that describes
'acceptable' pairing of types.
Pairs of objects are drawn at random from the set, and if they are an
acceptable pair, they are retained, and if they are not acceptable,
they are returned to the set.
This continues until all objects have been paired, or there remain in
the set only types that cannot be paired with each other.
What I would like to know is the probability distribution of pairings
(another symmetric matrix), plus the probability that an object of type
t does not end up in a pair at all.
You might guess that for me, the types are species that do or do not
interact. But this problem could equally represent reaction rates in a
well-mixed solution of many kinds of chemicals. I'm sure that it must
have been studied already. I would appreciate any pointers.
The way you have presented the problem it is fairly trivial. What you
will get at the end is pairings of one species, then pairings of
another species etc. Each species being orthogonal. Each species mates
as if the others wern't there.
If you postualate a more complicated problem where there is no clear
species demarcation. A mouse can't go with an elephant but a horse can
go with a donkey (a mule) you are in a radically different stuation.
One where there is no clear solution.
It may seem presumptous of me to challenge your assumptions but
personally I think the right model might be a much more thermodynamic
one. For example you have a pairing. The match has a given
suitability, given a more suitable pairing partners are swapped.
Reaction rates are described unambiguously by the Second Law of
Thermodynamics. You have what might be termed "redox" reactions where
a less reactive element is displaced by a more reactive. Chemistry is
thus not the problem you describe.
Do populations make their selections thermodynamically? Does a dance
floor have a "temperature"? Perhaps your researches might answer that
question.
If you look at the travelling salesperson you will find that an
acceptably short route (it may or may not be NP complete according to
complexity) is found by permuting cities. Matching may be done in a
similar way.
- Ian Parker
.
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