Tuesday, November 8, 2011
Monday, April 25, 2011
label similarity
Interesting paper
labeled graph
sequence
A: a1-x->a2-y->a3
B: b1-x->b2-y->b3-z->b4
x, y, and z are labels:
a1 a2 a3
a1 0 x 0
a2 0 0 y
a3 0 0 0
A Kronecker matrix product B
comments: can the similarity be percentages? can we define a function? association rules?
col of <a2, b2> : indegree =1
row of <a2, b2>: outdegree =1
F+Ft
The result is the column vector of the similarity matrix S (row by row)
b1 b2 b3 b4
a1
a2
a3
The column vector is [a1b1, a1b2, a1b3, a1b4, a2b1, a2b2, a2b3,a2b4,a3b1,a3b2,a3b3,a3b4]
=[1,0,0,0,0,2,0,0,0,0,1,0 ]
labeled graph
sequence
A: a1-x->a2-y->a3
B: b1-x->b2-y->b3-z->b4
x, y, and z are labels:
a1 a2 a3
a1 0 x 0
a2 0 0 y
a3 0 0 0
b1 b2 b3 b4
b1 0 x 0 0
b2 0 0 y 0
b3 0 0 0 z
b4 0 0 0 0A Kronecker matrix product B
0B xB 0B
---------------
0B 0B yB
---------------
0B 0B 0B
0 x 0 0 x*0 x*x x*0 x*0 0 x 0 0
0* 0 0 y 0 x*0 x*0 x*y 0*0 0* 0 0 y 0
0 0 0 z x*0 x*0 x*0 x*z 0 0 0 z
0 0 0 0 x*0 x*0 x*0 x*0 0 0 0 0
-----------------------------------------------------------------------------------------
0 x 0 0 0 x 0 0 y0 yx y0 y0
0* 0 0 y 0 0* 0 0 y 0 y0 y0 yy y0
0 0 0 z 0 0 0 z y0 y0 y0 yz
0 0 0 0 0 0 0 0 y0 y0 y0 y0
----------------------------------------------------------------------------------------
0 x 0 0 0 x 0 0 0 x 0 0
0* 0 0 y 0 0* 0 0 y 0 0* 0 0 y 0
0 0 0 z 0 0 0 z 0 0 0 z
0 0 0 0 0 0 0 0 0 0 0 0
0 x 0 0 0 x 0 0 0 x 0 0
results
0 0 0 0 0 xx 0 0 0 0 0 0
0 0 0 0 0 0 xy 0 0 0 0 0
0 0 0 0 0 0 0 xz 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 yx 0 0
0 0 0 0 0 0 0 0 0 0 yy 0
0 0 0 0 0 0 0 0 0 0 0 yz
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
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
finally F=
0 0 0 0 0 x 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 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 y 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 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 0
similarity edge (a1a2) - (b1b2) =1=x
similarity edge (a2a3) - (b2b3)=1 =y
similarity edge (a2a3) - (b2b3)=1 =y
similarity of pair <ai, bj> = logic similarity of <ai, bj> and neighbor similarity between ai, and bi
logic similarity of <ai, bj>= edges into the <ai, bj> + edges out the <ai, bj>
0 0 0 0 0 x 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 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 y 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
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
How to get <a2, b2>=2 ?
col of <a2, b2> : indegree =1
row of <a2, b2>: outdegree =1
F+Ft
0 0 0 0 0 x 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 0 0 0 0 0 0
---------------------------------------------------------------------------------------
0 0 0 0 0 0 0 0 0 0 0 0
x 0 0 0 0 0 0 0 0 0 y 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
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 y 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
============================================
The result is the column vector of the similarity matrix S (row by row)
b1 b2 b3 b4
a1
a2
a3
The column vector is [a1b1, a1b2, a1b3, a1b4, a2b1, a2b2, a2b3,a2b4,a3b1,a3b2,a3b3,a3b4]
=[1,0,0,0,0,2,0,0,0,0,1,0 ]
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