I am studying the well-known paper of Prof. Newman title "Finding community structure in networks using the eigenvectors of matrices" which can be found in this address:
https://arxiv.org/pdf/physics/0605087.pdf
I am OK with the paper till it reaches to equation (21), that is,
$\sum_{i j} P_{i j} = f(k_{i}) \sum_{j=1}^{n} f(k_{j}) = k_{i}$ (21)
The above mentioned equation sounds. Then explains:
"for all i and hence $f(k_{i}) = Ck_{i}$ for some constant $C$. And Eq. (19) says that"
$2m = \sum_{i j} P_{i j} = C^{2} \sum_{i j} k_{i} k_{j} = (2mC)^2$ (22)
"and hence $C = \frac{1}{\sqrt{}{2m}}$" and
$P_{i j} = \frac{k_{i} k_{j}}{2m}$ equation (23)
Now my question are the following.
1) if we assume that the network is undirected and the adjacency matrix is symmetric So there should not be any $C$.
2) The main question: how he comes up with the conclusion in (23), i.e. how he realize the arguments of two sums in the LHS and RHS of (22) are equal. Because clearly there can exist many contradictions that it is not true at all.
P.S. Since it's a well-known paper I think I am missing something maybe he does not derive (23) from (22).
Any help is appreciated.
Thanks to the help of my supervisor I figure out where was the problem. In the paragraph above equation (21), the authors make an assumption --which is a conclusion of his previous discussion. To be more precise, $P_{ij}$ is the product of $f(k_{i})f(k_{j})$ of separate functions, and then my two questions can be replied with some algebraic manipulations.