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Fast Direction-Aware Proximity for Graph Mining
KDD 2007, San JoseHanghang Tong, Yehuda Koren,
Christos Faloutsos
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Defining Direction-Aware Proximity (DAP): escape probability
• Define Random Walk (RW) on the graph• Esc_Prob(AB)– Prob (starting at A, reaches B before returning to A)
Esc_Prob = Pr (smile before cry)
A Bthe remaining graph
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Esc_Prob(1->5) =
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
P=
I - +
-1
1 5
3
2
6
4
0.5 0.5
0.5
0.50.5
0.5
0.5
1
0.5 1
P: Transition matrix (row norm.)
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Intuition of Formula
1 2 3
2
,
,
1. = + + ,
2. tells the probability that start from , take two
steps to arrive at
3. gives the stationary distribution.
4. tells the probability we started from and
i j
i j
Q I P I P P P
P i
j
Q
Q i
ended with .j
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
P*P=
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
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Esc_Prob(1->5) =
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
P=
I - +
-1
1 5
3
2
6
4
0.5 0.5
0.5
0.50.5
0.5
0.5
1
0.5 1
P: Transition matrix (row norm.)
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• Case 1, Medium Size Graph– Matrix inversion is feasible, but…– What if we want many proximities?– Q: How to get all (n ) proximities efficiently?– A: FastAllDAP!
• Case 2: Large Size Graph – Matrix inversion is infeasible– Q: How to get one proximity efficiently?– A: FastOneDAP!
Challenges
2
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FastAllDAP
• Q1: How to efficiently compute all possible proximities on a medium size graph?– a.k.a. how to efficiently solve multiple linear
systems simultaneously?• Goal: reduce # of matrix inversions!
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1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
FastAllDAP: Observation
1 5
3
2
6
4
0.5 0.5
0.5
0.50.5
0.5
0.5
1
0.5 1
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
Need two different matrix inversions!
P=
P=
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1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
FastAllDAP: Rescue
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
p p p p p p
Redundancy among different linear systems!
P=
P=
Overlap between two gray parts!
Prox(1 5)
Prox(1 6)
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FastAllDAP: Theorem
• Theorem:
• Proof: by SM Lemma
• Example:
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FastAllDAP: Algorithm
• Alg.– Compute Q– For i,j =1,…, n, compute
• Computational Save O(1) instead of O(n )!
• Example– w/ 1000 nodes, – 1m matrix inversion vs. 1 matrix!
2
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FastOneDAP
• Q1: How to efficiently compute one single proximity on a large size graph?– a.k.a. how to solve one linear system
efficiently?• Goal: avoid matrix inversion!
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FastOneDAP: Observation
1 5
3
2
6
4
0.5 0.5
0.5
0.50.5
0.5
0.5
1
0.5 1
Partial Info. (4 elements /2 cols ) of Q is enough!
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FastOneDAP: Observation
• Q: How to compute one column of Q?• A: Taylor expansion
Reminder:
i col of Qth
[0, …0, 1, 0, …, 0]T
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FastOneDAP: Observation
x x x
Sparse matrix-vector multiplications!
….
i col of Qth[0, …0, 1, 0, …, 0]
T
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FastOneDAP: Iterative Alg.
• Alg. to estimate i Col of Qth
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FastOneDAP: Property• Convergence Guaranteed !
• Computational Save– Example: • 100K nodes and 1M edges (50 Iterations)• 10,000,000x fast!
• Footnote: 1 col is enough! – (details in paper)
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Esc_Prob is good, but…
• Issue #1: – `Degree-1 node’ effect
• Issue #2:–Weakly connected pair
Need some practical modifications!
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Issue#1: `degree-1 node’ effect[Faloutsos+] [Koren+]
• no influence for degree-1 nodes (E, F)!– known as ‘pizza delivery guy’ problem in undirected graph
• Solutions: Universal Absorbing Boundary!
A BD1 1
A BD1 1/3
E F
1/31/311
Esc_Prob(a->b)=1
Esc_Prob(a->b)=1
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Universal Absorbing Boundary
U-A-B is a black-hole!
A BD1 1
U-A-B
Footnote: fly-out probability = 0.1
A BD0.9 0.9
U-A-B0.1
0.1
0.1
1
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Introducing Universal-Absorbing-Boundary
A BD0.9 0.9
U-A-B0.1
0.1
0.1
A BD0.9 0.3
E F
0.30.30.90.9
U-A-B
0.1
0.10.10.10.1
Prox(a->b)=0.91
Prox(a->b)=0.74
A BD1 1
A BD1 1/3
E F
1/31/311
Footnote: fly-out probability = 0.1
Esc_Prob(a->b)=1
Esc_Prob(a->b)=1
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Issue#2: Weakly connected pair
A B1 1 1
wi j
Prox(AB) = Prox (BA)=0
Solution: Partial symmetry!
a w
i j
(1-a) w
.
.
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Practical Modifications: Partial Symmetry
A B1 1 1
Prox(AB) = Prox (BA)=0
A B0.9 0.9 0.9
0.1 0.1 0.1
Prox(AB) =0.081 > Prox (BA)=0.009
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Efficiency: FastAllDAP
Size of Graph
Time (sec)
Straight-Solver
FastAllDAP
1,000xfaster!
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Efficiency: FastOneDAP
Size of Graph
Time (sec)
FastOneDAP
Straight-Solver
1,0000xfaster!
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Link Prediction: direction
• Q: Given the existence of the link, what is the direction of the link?
• A: Compare prox(ij) and prox(ji)>70%
Prox (ij) - Prox (ji)
density
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