Talk:Configuration model

Error
The statement "the expected number of self-loops and multi-links goes to zero in the N → ∞ limit" in the Algorithm section does not appear to be true. Other sections/paragraphs refer to the density of self-loops and multi-links going to zero which seems to be the correct statement.


 * orange money 197.215.32.46 (talk) 18:18, 16 February 2022 (UTC)


 * orange money 197.215.32.46 (talk) 18:18, 16 February 2022 (UTC)

Error/abuse in probability distribution
One can read in the section "Properties" and sub-section "Edge probability" that it is written " Since node $$i$$ has $$k_i$$ stubs, the probability of $$i$$ being connected to $$j$$ is $$\frac{k_ik_j}{2m-1}$$ ($$\frac{k_ik_j}{2m}$$ for sufficiently large $$m$$). The probability of self-edges cannot be described by this formula, but as the density of self-edges goes to zero as $$N\rightarrow\infty$$, it usually gives a good estimate. "

But, if I am right, in the particular case of this example (i.e. when the "probability of self-edges is not described"), the probability of $$i$$ being connected to $$j$$ is: $$ 1 - \frac{ \binom{2m-k_j-k_i}{k_i} }{   \binom{2m-k_i}{k_i} } $$. This formula is simply 1 minus the proportion of configurations leading to the situation "$$i$$ not connected to $$j$$ nor to $$i$$". At the numerator it is the number of way to connect the $$k_i$$ stubs of $$i$$ to the $$2m-k_j-k_i$$ allowed stubs ($$k_j$$ stubs are not allowed because one must avoid stubs of $$j$$ and $$k_i$$ stubs are not allowed because $$i$$ must not be connected to itself). At the denominator it is the number of configurations where $$i$$ is not connected to itself.

Anyway... The formula displayed above should be the true one when $$i$$ is not connected to itself instead of the one of the page (i.e. $$\frac{k_j k_i}{2m-1}$$). That is what astonishs me. Can any one give me his point of view. I am not daring correcting the wikipage because I have no access to the reference given.

Moreover, the approximation of this formula is indeed $$\frac{k_j k_i}{2m}$$ as $$m$$ tends to $$\infty$$ if one uses the asymptotic developpement of $$\binom mk$$ (if $$m \rightarrow \infty$$, $$k \in \mathbb{N} $$ then $$\binom mk \underset{m\rightarrow \infty}{\sim} \frac{m^k}{k!}$$ as described in the french wikipedia page). It looks like the "true probability" given in the wikipage is already an approximation...

I am looking forward to having your point of view. Thank you. --AOMckey (talk) 11:39, 6 July 2020 (UTC)
 * You are right, the value written for the edge probability is not correct. See also https://math.stackexchange.com/questions/786862/probability-there-is-no-vertex-at-distance-larger-than-d-away-from-source-in-r?rq=1. The section should be edited. Pier.d94 (talk) 12:28, 14 April 2021 (UTC)