Consider an undirected graph $G=(V,E)$ where $V$ is the set of vertices and $E$ is the set of edges. We use a Graph Neural Network $($GNN$)$ for node classification in this graph. The GNN updates the feature matrix $H^{(t)}$ of nodes at iteration $t$ using a message$-$passing mechanism, where each row of $H^{(t)}$ represents the feature vector of a node. Suppose $h_v^{(t)}$ denotes the feature vector of node $v$ at iteration $t.$

The message$-$passing operation at iteration $t$ is defined as follows:

$h_v^{(t+1)}=(W{h}_v^{(t)}+{\displaystyle \underset{\text{u}\text{s}\text{u}\text{b}\text{N}(v)}{\overset{?}{}}}{W}^{\text{'}}{h}_u^{(t)})$

where:

$W$ and $W^{\text{'}}$ are learnable weight matrices,

$N(v)$ is the set of neighbors of node $v,$

$$ is a non$-$linear activation function.

Prove that the above GNN is invariant to any permutation of nodes. In other words, show that for any permutation matrix $P,$ the following equation holds after $t$ iterations:

$P{H}_P^{(t+1)}=H_P^{(t+1)}$