I have $N$ values defined on a graph with $N$ nodes called $x_i$ for node $i$. I define the pairwise distance between these values as follow:
$D(x) = \frac{1}{2}\sum_{i,j}a_{ij}||x_i - x_j||_2 $
Now, I will multiply the normalized adjacency matrix (with self-loops) and define new values for each node as follow:
$y_i = \sum\limits_{j\in Neighbors(i)} a_{ij}x_j$
where $a_{ij}$ is $\frac{1}{\sqrt{d_id_j}}$ if there is an edge between node $i$ and node $j$ and zero otherwise (remember we have also self-edges)
Is there any way to prove that D(y) is less than D(x)? I mean:
$\sum_{i,j}a_{ij}||y_i - y_j||_2 \leq \sum_{i,j}a_{ij}||x_i - x_j||_2 $
It is really intuitive for me. But I cannot prove it mathematically. I would appreciate any hint or suggestion.