I encountered super strange behaviour of higher_order, both in old W and new Graph implementation.

g = graph.Graph.from_arrays([0, 1, 2, 3, 3, 4, 4], [0, 3, 4, 1, 4, 2, 3], [0, 1, 1, 1, 1, 1, 1])
g.adjacency

focal  neighbor
0      0           0
1      3           1
2      4           1
3      1           1
       4           1
4      2           1
       3           1
Name: weight, dtype: int64

g.higher_order(k=2, lower_order=True).adjacency

focal  neighbor
0      0           0
1      4           1
2      3           1
3      2           1
4      1           1
Name: weight, dtype: int64

As you can see, this includes only neighbors of second order, not the first one as it should. The reason is that the underlying matrix power computation does not include them.

g.sparse.todense()

array([[0., 0., 0., 0., 0.],
       [0., 0., 0., 1., 0.],
       [0., 0., 0., 0., 1.],
       [0., 1., 0., 0., 1.],
       [0., 0., 1., 1., 0.]])

sparse.linalg.matrix_power(int_g.sparse, 2).todense()

array([[0., 0., 0., 0., 0.],
       [0., 1., 0., 0., 1.],
       [0., 0., 1., 1., 0.],
       [0., 0., 1., 2., 0.],
       [0., 1., 0., 0., 2.]])

The solution is to sum the original sparse with the new one. It seems to be an edge case resulting from extremely sparse matrix but could be real nevertheless.

I think that the solution is to sum the result of matrix power(s) with the original sparse to ensure a value is present. Here