Steffen Staab [email protected] 1 WeST Web Science & Technologies University of Koblenz ▪ Landau, Germany Network Theory and Dynamic Systems Link Prediction Steffen Staab Christoph Ringelstein Acknowledgment: Jérôme Kunegis
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Steffen [email protected]
1WeST
Web Science & TechnologiesUniversity of Koblenz ▪ Landau, Germany
Network Theory and Dynamic SystemsLink Prediction
Steffen StaabChristoph Ringelstein
Acknowledgment: Jérôme Kunegis
Steffen [email protected]
2WeST 2
The Problem: Link Prediction
Motivation: Recommend connections in a social network Predict links in an undirected, unweighted network
Known links (A) Links to be predicted (B)
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4WeST
Recommender Systems
Predict who I will add as friend next
Standard algorithm: find friends-of-friends
me
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5WeST
Friend of a Friend
1 2 4 5 6
3
Count the number of ways a person can be found as the friend of a friend.
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6WeST
Adjacency Matrix
0 1 0 0 0 01 0 1 1 0 00 1 0 1 0 00 1 1 0 1 00 0 0 1 0 10 0 0 0 1 0
Aij = 1 when i and j are connectedAij = 0 when i and j are not connected
A is square and symmetric
1 2 4 5 6
3
A =
1
23
45
6
1 2 3 4 5 6
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7WeST
Baseline: Friend of a Friend Model
Count the number of ways a person can be found as the friend of a friend
Matrix product AA = A2
1 2 4
30 1 0 0 0 01 0 1 1 0 00 1 0 1 0 00 1 1 0 1 00 0 0 1 0 10 0 0 0 1 0
1 0 1 1 0 00 3 1 1 1 01 1 2 1 1 01 1 1 3 0 10 1 1 0 2 00 0 0 1 0 1
=
2
A2 =
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8WeST
Matrix product AA = A2
Friend of a Friend Computation
1 2 4
3
=
2
A2 =
0 1 0 0 0 01 0 1 1 0 00 1 0 1 0 00 1 1 0 1 00 0 0 1 0 10 0 0 0 1 0
1 0 1 1 0 00 3 1 1 1 01 1 2 1 1 01 1 1 3 0 10 1 1 0 2 00 0 0 1 0 1
Steffen [email protected]
9WeST
Recommender Systems
Predict who I will add as friend next
Standard algorithm: find friends-of-friends → Problem: Rest of the network is ignored!
:-(
me
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10WeST
Friend of a Friend of a Friend
1 2 4 5 6
3
Count the number of ways a person can be found as the friend of a friend of a friend.
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11WeST
Friend of a Friend of a Friend
Compute the number of friends-of-friends-of-friends:
Idea: Link recommendation by highest scoreProblem: A3 is not sparse!
1 2 4 5 6
3
1 2 3 4 5 6123456
0 1 0 0 0 01 0 1 1 0 00 1 0 1 0 00 1 1 0 1 00 0 0 1 0 10 0 0 0 1 0
0 3 1 1 1 03 2 4 5 1 11 4 2 4 1 11 5 4 2 4 01 1 1 4 0 20 1 1 0 2 0
3
=A3 =
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Eigenvalue Decomposition
A = UΛU-1
whereU are the eigenvectorsΛ are the eigenvalues Λij = 0
when i ≠ j
Λ is also called the spectrum of AProblem: complexity of computing U-1.
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Problem: complexity of computing U-1.
Exploit U:• U-1 = UT because U contains normalized eigenvectors (U is orthogonal)
A = UΛU-1 = UΛUT
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A0 1 0 0 0 01 0 1 1 0 00 1 0 1 0 00 1 1 0 1 00 0 0 1 0 10 0 0 0 1 0
Λ UU
Eigenvalue Decomposition: Example
1 2 4 5 6
3
Eigenvector Eigenvalue
U contains eigenvectors, Λ contains eigenvalues
=
−1 .74 0 0 0 0 00 −1 .37 0 0 0 00 0 −0 .59 0 0 00 0 0 0 .27 0 00 0 0 0 1.10 00 0 0 0 0 2 .33
T
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AΛ UU
Eigenvector-Centrality
1 2 4 5 6
3
Eigenvector Eigenvalue
U contains eigenvectors, Λ contains eigenvalues
0.55 0.56 0.30 0.13
0.48
0.23
=
−1 .74 0 0 0 0 00 −1 .37 0 0 0 00 0 −0 .59 0 0 00 0 0 0 .27 0 00 0 0 0 1.10 00 0 0 0 0 2 .33
T
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18WeST
0 3 1 1 1 03 2 4 5 1 11 4 2 4 1 11 5 4 2 4 01 1 1 4 0 20 1 1 0 2 0
Computing A3
1 2 3 4 5 612345
6
A3 =
Use the eigenvalue decomposition A = UΛUT
A3 = UΛUT UΛUT UΛUT = UΛ3UT
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19WeST
Computing A3
Exploit U and Λ:• U-1 = UT because U contains
eigenvectors• UTU = I because U-1 = UT
• (Λk)ii = Λiik because Λ contains eigenvalues
A3 = UΛUT UΛUT UΛUT = UΛΛΛUT = UΛ3UT
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Link Prediction
Prediction for node i: (A3)i * = (U Λ3 UT)i * = Ui * Λ3 UT
A3 = ( ) i * =iU Λ3 UT
U i * Λ3 UT i=
Optional: reduction to most dominant eigenvectors
fast ;)
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21WeST
Recommender Systems
Predict who I will add as friend next
Standard algorithm: find friends-of-friends → Problem: all paths have the same weights!
:-(
me
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22WeST 22
Follow paths
Number of paths of length k given by Ak
Nodes connected by many paths Nodes connected by short paths
Weight powers of A: αA² + βA³ + γA⁴ … with α > β > γ … [ > 0 ]
Examples:Exponential graph kernel: eαA = Σi αi/i!
Ai
Path Counting
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23WeST
exp(A) = I + A + 1/2 A2 + 1/6 A3 + . . .1 .66 1.72 0 .93 0 .98 0 .28 0 .06 0 .011.72 3 .57 2.70 2.93 1.04 0 .29 0 .060 .93 2.70 2.86 2 .71 0 .99 0 .28 0 .060 .98 2.93 2.71 3 .63 1.95 0 .76 0 .220 .28 1 .04 0 .99 1.95 2 .35 1 .59 0 .640 .06 0 .29 0 .28 0 .76 1 .59 2 .23 1 .380 .01 0 .06 0.06 0 .22 0 .64 1 .38 1 .59
Matrix Exponential
1 2 4 5
3
1 2 3 4 5 61234566
0 1 0 0 0 0 01 0 1 1 0 0 00 1 0 1 0 0 00 1 1 0 1 0 00 0 0 1 0 1 00 0 0 0 1 0 10 0 0 0 0 1 0
exp =
76
7
7
0.98 0.76 0.22
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24WeST
Spectral Transformations
Friend of a friend
Friend of a friend of a friend
Matrix exponential
…are link prediction functions!
A2 = UΛ2UT
A3 = UΛ3UT
exp(A) = Uexp(Λ)UT
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25WeST
Summary - Why the Matrix Exponential
An = Number of paths of length n
aA2 + bA3 + cA4 + . . .= Number of paths, weighted by path
length
→ New edges more likely to appear when there are many paths already
→ When a > b > c > . . . > 0, short paths are weighted higher
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