I am working on a graph clustering problem and i've seen that applying two consecutive normalizations on the adjacency matrix gives much better performance than when applying a single one.
I first apply the symmetric normalization:
$$T = D^{-1/2} A D^{-1/2}$$
Then apply row normalization on the resulting matrix:
$$S = D_T^{-1}T$$
I tried working out the overall formula of S w.r.t to A but it didn't yield anything particularly significant (i think). Here's the formula i ended up with:
$$s_{ij}=\frac{a_{ij}}{\sqrt{d_j}\sum_k\frac{a_{ik}}{\sqrt{d_k}}}=\frac{a_{ij}}{a_{ij}+\sqrt{d_j}\sum_{k\neq j}\frac{a_{ik}}{\sqrt{d_k}}}$$
I know that when applied separately they have a connection with the graph cut problem but does it hold any meaning to apply the row normalization on the symmetric normalized adjacency matrix in the context of clustering?