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 ${\frac {k_{i}k_{j}}{2m-1}}$ (${\frac {k_{i}k_{j}}{2m}}$ 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 ${\binom {m}{k}}$ (if $m\rightarrow \infty$, $k\in \mathbb {N}$ then ${\binom {m}{k}}{\underset {m\rightarrow \infty }{\sim }}{\frac {m^{k}}{k!}}$ 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)