the results from message passing and matrix multiplication are different

[zhf-0]

I will illustrate my question by the following examples. Supposing the adjacent matrix is

[  a_00     0       0
     0     a_11   a_12
     0     a_21   a_22 ]

so, the edge_index and edge_weight are

edge_index   = [    0        1       1        2         2
                    0        1       2        1         2   ]
edge_weigth = [   a_00      a_11    a_12    a_21      a_22  ]

If I use GCNConv and the node features are x_0 x_1 x_2, the tensor x_j is (x_j is source)

x_j = [            x_0      x_1     x_1     x_2        x_2   ]

According the code GCNConv

def message(self, x_j: Tensor, edge_weight: OptTensor) -> Tensor:
    return x_j if edge_weight is None else edge_weight.view(-1, 1) * x_j

The result is

result = [ a_00 * x_0   a_11 * x_1  a_12 * x_1  a_21 *  x_2  a_22 *  x_2 ]

and updated features are

0  =>  a_00 * x_0
1  =>  a_11 * x_1 + a_21 *  x_2
2  =>  a_12 * x_1 + a_22 *  x_2 

But, if I caculate the updated features through matrix multiplication

[  a_00     0       0        [ x_0           [ a_00 * x_0
     0     a_11   a_12         x_1     =>      a_11 * x_1 + a_12 * x_2    
     0     a_21   a_22 ]       x_2 ]           a_21 * x_1 + a_22 * x_2 ]

Only if the adjacent matrix is transposed, the updated features are the same. Why is this?

[rusty1s]

Yes, this is intended. The default message passing flow is from source nodes to target nodes (i.e. from edge_index[0] to edge_index[1]). You can change it by setting flow="target_to_source". For dense GCN, this corresponds to adj.t() @ x. Since GCN is only properly defined for undirected graphs, this usually does not matter though.

[zhf-0]

Thank you very much!