For an undirected graph without labels on the edges, a function that we calculate in each layer of a neural network graph must satisfy certain special properties in order to be able to use the same function $($with weight sharing$)$ in different nodes of the graph. Suppose for a specific node $i$ in the graph, hi$(l-1$ represents the state calculated in the previous layer for this node, while the messages from the previous layer of the ni neighbors of node i are denoted by $ml-1$i$,$j where $j$ ranges from $1$ to ni$.$ We will use subscripts and superscripts to indicate learnable weights. If a superscript is absent$,$ the weights are shared across layers. Assume that all dimensions work correctly. Explain which of these are valid functions for calculating the next message hi$(l)$ for this node. For each invalid choice, briefly state the reason.

Note: Validity means that they must satisfy the Invariance and Equivariance properties required for using them as a GNN on an undirected graph.

$($a$)$ hi$(l$ w$1$hi$(l-1$ nij $=1ml-1i,j$

$($c$)$ hi $(l)=$max$(wl1hi(l-1),w2ml-1i,1,w2ml-1i,2,$dots,$w2ml-1i,ni)$ where max operates element$-$wise on vectors.