Can anyone back up me solve this problem?Taking a random graph G=(V,E) of n vertices such that V is any set of size n. V={1,2,...,n}. For any two distinct vertices: v1,v2 belong to G the advance (v1,v2) belongs to E with probability p and does not be to it with probability q=1-p where 0<p<1 is arbitrary fixed. Show that the probability of G being connected converges to 1 as n->oo. Thanks.
(1)because we undergo to ensure that none of the vertices in
is connected to a vertex which is not in
and we do away with the conditions that
has to be connected. Now assume that the connected component of
with minimal request has request
Then since any two connected components are disjoints it follows that there can be no more than one connected component whicn means that
is disconnected if and only if there exist a
be the number of connected components with request
It follows from above that the probability that
From this measure move it is clear that
which means that the probability that
as desired. I wish for not too many mistakes. Pierre.
The argument given above goes a good deal advance than was asked by the problem (down to
). Actually. I think you can get all the way drink to
by making a few small adjustments to it (apply: prove that for
the graph is almost surely NOT connected)When
is fixed you can alter the argument by making the following observation:To show
is connected it suffices to show that every pair of vertices has a common neighbor.
As far as I bequeath the sign result from Erdös and Rényi states that for any pertinent function
is connected (which means that the probabilty that it is connected converges to
) and almost every graph with
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