带有邻接矩阵的乘法和点积（numpy）

Question

I am using the following chunk of code with networkx, when I discovered the following oddity. In the first case, I used the ufunc multiply(*) on a sparse matrix that unexpectedly correctly giving me a degree sequence. However, when the same is done with an ordinary matrix, it is giving me a 10 x 10 matrix, and as expected np.dot(...) is giving me the correct result.

import numpy as np
import networks as nx

ba = nx.barabasi_albert_graph(n=10, m=2)

A = nx.adjacency_matrix(ba)
# <10x10 sparse matrix of type '<class 'numpy.int64'>'
# with 32 stored elements in Compressed Sparse Row format>

A * np.ones(10)

# output: array([ 5.,  3.,  4.,  5.,  4.,  3.,  2.,  2.,  2.,  2.])

nx.degree(ba)

# output {0: 5, 1: 3, 2: 4, 3: 5, 4: 4, 5: 3, 6: 2, 7: 2, 8: 2, 9: 2}

B = np.ones(100).reshape(10, 10)

B * np.ones(10)

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

np.dot(B, np.ones(10))
# array([ 10.,  10.,  10.,  10.,  10.,  10.,  10.,  10.,  10.,  10.])

I was expecting that I should be doing np.dot(A, np.ones(10)) but that returns an array of 10, 10 x 10 matrices

array([ <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>,
   <10x10 sparse matrix of type '<class 'numpy.float64'>'
with 32 stored elements in Compressed Sparse Row format>], dtype=object)

What is the nuance here?

Answer 1

For regular numpy arrays, * multiply is element by element (with broadcasting ). np.dot is the matrix product, the sum-of-products. For the np.matrix subclass * is the matrix product, the dot . sparse.matrix is not a subclass, but it is modeled on that. * is the matrix product.

In [694]: A = sparse.random(10,10,.2, format='csr')
In [695]: A
Out[695]: 
<10x10 sparse matrix of type '<class 'numpy.float64'>'
    with 20 stored elements in Compressed Sparse Row format>
In [696]: A *np.ones(10)
Out[696]: 
array([ 0.6349177 ,  0.        ,  1.25781168,  1.12021258,  2.43477065,
        1.10407149,  1.95096264,  0.6253589 ,  0.44242708,  0.50353061])

The sparse matrix has the dot method, which behaves the same:

In [698]: A.dot(np.ones(10))
Out[698]: 
array([ 0.6349177 ,  0.        ,  1.25781168,  1.12021258,  2.43477065,
        1.10407149,  1.95096264,  0.6253589 ,  0.44242708,  0.50353061])

The dense version:

In [699]: np.dot(A.A,np.ones(10))
Out[699]: 
array([ 0.6349177 ,  0.        ,  1.25781168,  1.12021258,  2.43477065,
        1.10407149,  1.95096264,  0.6253589 ,  0.44242708,  0.50353061])

I thought np.dot was supposed to handle sparse matrices right, that is differ to their own method. But np.dot(A,np.ones(10)) does not do that right, producing the object array of 2 sparse matrices. I can dig into why, but for now, avoid it.

In general, use sparse functions and methods with sparse matrices. Don't assume numpy functions will have them correctly.

---

np.dot works fine when both arrays are sparse,

In [702]: np.dot(A,A)
Out[702]: 
<10x10 sparse matrix of type '<class 'numpy.float64'>'
    with 32 stored elements in Compressed Sparse Row format>
In [703]: np.dot(A,A.T)
Out[703]: 
<10x10 sparse matrix of type '<class 'numpy.float64'>'
    with 31 stored elements in Compressed Sparse Row format>

In [705]: np.dot(A, sparse.csr_matrix(np.ones(10)).T)
Out[705]: 
<10x1 sparse matrix of type '<class 'numpy.float64'>'
    with 9 stored elements in Compressed Sparse Row format>
In [706]: _.A
Out[706]: 
array([[ 0.6349177 ],
       [ 0.        ],
       [ 1.25781168],
       [ 1.12021258],
       [ 2.43477065],
       [ 1.10407149],
       [ 1.95096264],
       [ 0.6253589 ],
       [ 0.44242708],
       [ 0.50353061]])

For what it's worth the sparse sum is performed with this kind of matrix product:

In [708]: A.sum(axis=1)
Out[708]: 
matrix([[ 0.6349177 ],
        [ 0.        ],
        [ 1.25781168],
        [ 1.12021258],
        [ 2.43477065],
        [ 1.10407149],
        [ 1.95096264],
        [ 0.6253589 ],
        [ 0.44242708],
        [ 0.50353061]])