Matrix Calculation

A
    a  b  c
a  0  1  1
b  0  0  1
c  0  0  0

B

    a'  b'  c'

a'  0  1  1

b'  0  0  1

c'  0  0  0

Transpose row-> col

AT
    a  b  c

a  0  0  0

b  1  0  0

c  1  1  0

BT

    a'  b'  c'

a'  0  0  0

b'  1  0  0

c'  1  1  0

Vlondel &Van Dooren Measure

C=A Kronecker matrix product B

  0B | 1B | 1B

---------------

  0B | 0B | 1B

---------------

  0B | 0B | 0B

0 0 0 | 0 1 1 | 0 1 1

0 0 0 | 0 0 1 | 0 0 1

0 0 0 | 0 0 0 | 0 0 0

-------------------

0 0 0 | 0 0 0 | 0 1 1

0 0 0 | 0 0 0 | 0 0 1

0 0 0 | 0 0 0 | 0 0 0

---------------------

0 0 0 | 0 0 0 | 0 0 0

0 0 0 | 0 0 0 | 0 0 0

0 0 0 | 0 0 0 | 0 0 0

D= (AT Kronecker matrix product BT)

  0BT | 0BT | 0BT

---------------

  1BT | 0BT | 0BT

---------------

  1BT | 1BT | 0BT

0 0 0 | 0 0 0 | 0 0 0

0 0 0 | 0 0 0 | 0 0 0

0 0 0 | 0 0 0 | 0 0 0

-------------------

0 0 0 | 0 0 0 | 0 0 0

1 0 0 | 0 0 0 | 0 0 0

1 1 0 | 0 0 0 | 0 0 0

---------------------

0 0 0 | 0 0 0 | 0 0 0

1 0 0 | 1 0 0 | 0 0 0

1 1 0 | 1 1 0 | 0 0 0

C+D=

0 0 0 | 0 1 1 | 0 1 1

0 0 0 | 0 0 1 | 0 0 1

0 0 0 | 0 0 0 | 0 0 0

-------------------

0 0 0 | 0 0 0 | 0 1 1

1 0 0 | 0 0 0 | 0 0 1

1 1 0 | 0 0 0 | 0 0 0

---------------------

0 0 0 | 0 0 0 | 0 0 0

1 0 0 | 1 0 0 | 0 0 0

1 1 0 | 1 1 0 | 0 0 0

Semitics of the each element in C. For example: e(0,0)  = (a and a) *(a and a) =1 indicates a must connect to a and a' must connect to a' . We can use elements in original graph to denote e(0,0) in C, i.e., e(A00, B00)

C=A Kronecker matrix product B

  0B | 1B | 1B

---------------

  0B | 0B |1B

---------------

  0B | 0B | 0B.

can be rewritten to:

 A(0, 0) B | A(0, 1)B  |A(0, 2)B

--------------------------------

 A(1, 0) B | A(1, 1)B  |A(1, 2)B

--------------------------------

 A(2, 0) B | A(2, 1)B  |A(2, 2)B

Furthermore, each block can be expended to 3X3 matrix. For example, the block one A(0, 0)B, or A(a, a)B , is

 A(0, 0) B(0, 0)  | A(0, 0) B(0, 1)  |A(0, 0) B(0, 2)

--------------------------------------------------

 A(0, 0) B(1, 0)  | A(0, 0) B(1, 1)  |A(0, 0) B(1, 2)

-----------------------------------------------------

 A(0, 0) B(2, 0)  | A(0, 0) B(2, 1)  |A(0, 0) B(2, 2) 

For forward propagation based on the figure (see the matrix C below). There are 9 edges including dashed lines. It corresponds to the 9 1s' in the matrix.

0 0 0 | 0 1 1 | 0 1 1

0 0 0 | 0 0 1 | 0 0 1

0 0 0 | 0 0 0 | 0 0 0

-------------------

0 0 0 | 0 0 0 | 0 1 1

0 0 0 | 0 0 0 | 0 0 1

0 0 0 | 0 0 0 | 0 0 0

---------------------

0 0 0 | 0 0 0 | 0 0 0

0 0 0 | 0 0 0 | 0 0 0

0 0 0 | 0 0 0 | 0 0 0

Induced propagation graph is C+D, where D is the backward propagation. As transport is backward of forward propagation.

-----------------------------------------------------------------------------------------------------------
Melnik propagation graph: 

Sk+1=[w o P]Sk, note the formula is not normalized. P is adjacency matrix, w is weight as connections in the joint graph may have different weight.

So we have to use other formulas based on above:

1. basic iteration formula (Norm)
2. || Sk-Sk01|| <= 0.05
3. Initial score =0.001