multi-layer graph perceptron

A(n $\ell$-layer) multi-layer graph perceptron (MGLP) is simply several graph perceptrons in sequence. This forms a function with input $x_0=x$, and output $y=x_L$.

Where at each layer, $$x_{\ell} = \sigma(u_{\ell}) = \sigma\left( \sum_{k=0}^{K-1} h_{\ell,k} S^k x_{\ell-1} \right), ; 1 \leq \ell \leq L$$
We call $x_{\ell }$ the $\ell$-layer embedding.

For conciseness, we define $H=\{h_{k,\ell }|0\le k\le K-1,1\le \ell \le L\}$ and represent our MLGP as $$y = \Phi(x, S; H)$$