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Exponential Ranking: Taking into
account negative links.
V.A. Traag1, Y.E. Nesterov2, P. Van Dooren1
1ICTEAM
Universit´e C...
Negative links?
Negative links underrated
• Negative links (negative weight) often disregarded
• Hostility instead of frie...
Iterative formulation
Iterative steps
1 Assign each node a reputation ki
2 Let nodes vote for reputation of others
3 Assig...
Iterative formulation
Iterative steps
1 Assign each node a reputation ki
2 Let nodes vote for reputation of others
3 Assig...
Iterative formulation
Iterative steps
1 Assign each node a reputation ki
2 Let nodes vote for reputation of others
3 Assig...
Iterative formulation
Iterative steps
1 Assign each node a reputation ki
2 Let nodes vote for reputation of others
3 Assig...
Iterative formulation
Iterative steps
1 Assign each node a reputation ki
2 Let nodes vote for reputation of others
3 Assig...
Iterative formulation
Iterative steps
1 Assign each node a reputation ki
2 Let nodes vote for reputation of others
3 Assig...
Example
c
a
b
d e
Example cycles for µ = 0
Reputations
1 2 3 4 5 6 7
a 1.00 0.40 0.67 0.50 0.67 0.50 0.67
b 1.00 0.40 0.33...
Example
c
a
b
d e
Example convergence for µ = 1
Reputations
1 2 3 4 5 6 7
a 1.00 0.40 0.43 0.43 0.43 0.43 0.43
b 1.00 0.40...
Preliminary tests
Generate test network
1 Generate random network (n = 1100)
ER graphs Each link with probability p = 0.01...
Preliminary tests
Generate test network
1 Generate random network (n = 1100)
2 Divide network in Good and Bad agents (rati...
Results
Faithful results
Erd¨os-Reny´ı
0 0.1 0.2 0.3 0.4 0.5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
q
CP
Exponenti...
Results
Semi-deceptive results
Erd¨os-Reny´ı
0 0.1 0.2 0.3 0.4 0.5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
q
CP
Exp...
Results
Deceptive results
Erd¨os-Reny´ı
0 0.1 0.2 0.3 0.4 0.5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
q
CP
Exponent...
Conclusions
Method & Convergence
• New ranking method taking into account negative links
• Converges relatively quickly to...
Debate example
• Debate in opinion pages of Dutch newspapers 1990–2005
• Authors refer to each other to express (dis)agree...
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Exponential Ranking: Taking into account negative links.

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Presentation at Benelux Meeting, March 15, 2011

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Exponential Ranking: Taking into account negative links.

  1. 1. Exponential Ranking: Taking into account negative links. V.A. Traag1, Y.E. Nesterov2, P. Van Dooren1 1ICTEAM Universit´e Catholique de Louvain 2CORE Universit´e Catholique de Louvain 15 March 2011
  2. 2. Negative links? Negative links underrated • Negative links (negative weight) often disregarded • Hostility instead of friendliness • Vote against, instead of vote in favor • Distrust instead of trust • Important for understanding networks Empirical networks • International Relations (Conflict vs. Alliances) • Citation Networks (Disapproving vs. Approving) • Social networks (Dislike vs. Like) • Trust networks (Distrust vs. Trust)
  3. 3. Iterative formulation Iterative steps 1 Assign each node a reputation ki 2 Let nodes vote for reputation of others 3 Assign new reputation based on weighted votes 4 Repeat (1)-(3) until reputations converge
  4. 4. Iterative formulation Iterative steps 1 Assign each node a reputation ki 2 Let nodes vote for reputation of others 3 Assign new reputation based on weighted votes 4 Repeat (1)-(3) until reputations converge Starting reputation • Start with some reputation for each node (say ki = 1) • Unique fixed point, so starting reputation has no effect
  5. 5. Iterative formulation Iterative steps 1 Assign each node a reputation ki 2 Let nodes vote for reputation of others 3 Assign new reputation based on weighted votes 4 Repeat (1)-(3) until reputations converge New reputation • Select node with highest ‘real’ reputation as judge • ‘Real’ reputation = observed reputation + random error • Standard deviation of random error proportional to µ
  6. 6. Iterative formulation Iterative steps 1 Assign each node a reputation ki 2 Let nodes vote for reputation of others 3 Assign new reputation based on weighted votes 4 Repeat (1)-(3) until reputations converge Trust probability • The probability to be chosen as judge is pi = exp ki /µP j exp kj /µ • Votes of judge i are Aij • Expected new reputation is ki = j pj Aji
  7. 7. Iterative formulation Iterative steps 1 Assign each node a reputation ki 2 Let nodes vote for reputation of others 3 Assign new reputation based on weighted votes 4 Repeat (1)-(3) until reputations converge Dual iterative formulations • In terms of trust probabilities: p(t + 1) = exp ATp(t)/µ exp ATp(t)/µ 1 • In terms of reputation: k(t + 1) = AT exp k(t)/µ exp k(t)/µ 1
  8. 8. Iterative formulation Iterative steps 1 Assign each node a reputation ki 2 Let nodes vote for reputation of others 3 Assign new reputation based on weighted votes 4 Repeat (1)-(3) until reputations converge Variance determining convergence • Sufficiently large µ, convergence to unique point • For smaller µ, convergence is not guaranteed • In the limit of µ → 0, cycles will emerge
  9. 9. Example c a b d e Example cycles for µ = 0 Reputations 1 2 3 4 5 6 7 a 1.00 0.40 0.67 0.50 0.67 0.50 0.67 b 1.00 0.40 0.33 0.50 0.33 0.50 0.33 c 1.00 0.40 0.67 0.50 0.67 0.50 0.67 d 1.00 0.20 - - - - - e 1.00 0.20 - - - - -
  10. 10. Example c a b d e Example convergence for µ = 1 Reputations 1 2 3 4 5 6 7 a 1.00 0.40 0.43 0.43 0.43 0.43 0.43 b 1.00 0.40 0.39 0.39 0.39 0.39 0.39 c 1.00 0.40 0.43 0.43 0.43 0.43 0.43 d 1.00 0.20 0.17 0.17 0.17 0.17 0.17 e 1.00 0.20 0.17 0.17 0.17 0.17 0.17
  11. 11. Preliminary tests Generate test network 1 Generate random network (n = 1100) ER graphs Each link with probability p = 0.01 SF graphs Network generated through BA model with m = 3 2 Divide network in Good and Bad agents (ratio 10 : 1) 3 Assign sign to each link between Good and Bad agents G B G + − B + − Faithful G B G + − B + + Semi-deceptive G B G + − B − + Deceptive 4 Perturb: flip sign of link with probability 0 < q < 1/2 Prediction and measure
  12. 12. Preliminary tests Generate test network 1 Generate random network (n = 1100) 2 Divide network in Good and Bad agents (ratio 10 : 1) 3 Assign sign to each link between Good and Bad agents 4 Perturb: flip sign of link with probability 0 < q < 1/2 Prediction and measure 1 Predict Good/Bad agents (reputation k ≥ 0 or k < 0) Exponential Ranking Method suggested here PageRank+ Apply PageRank on positive links + 1 step of (dis)trust (pos. and neg.) Degree Weighted degree 2 Succes: Fraction of correctly predicted Bad agents (100 runs)
  13. 13. Results Faithful results Erd¨os-Reny´ı 0 0.1 0.2 0.3 0.4 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 q CP Exponential Ranking Degree PageRank+ Scale-Free 0 0.1 0.2 0.3 0.4 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 q CP Exponential Ranking Degree PageRank+
  14. 14. Results Semi-deceptive results Erd¨os-Reny´ı 0 0.1 0.2 0.3 0.4 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 q CP Exponential Ranking Degree PageRank+ Scale-Free 0 0.1 0.2 0.3 0.4 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 q CP Exponential Ranking Degree PageRank+
  15. 15. Results Deceptive results Erd¨os-Reny´ı 0 0.1 0.2 0.3 0.4 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 q CP Exponential Ranking Degree PageRank+ Scale-Free 0 0.1 0.2 0.3 0.4 0.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 q CP Exponential Ranking Degree PageRank+
  16. 16. Conclusions Method & Convergence • New ranking method taking into account negative links • Converges relatively quickly to unique point Performance & Application • Seems to perform well for trust systems, detecting ‘bad’ nodes • Further testing is required • Might have applications as research tool in various networks Questions?
  17. 17. Debate example • Debate in opinion pages of Dutch newspapers 1990–2005 • Authors refer to each other to express (dis)agreement 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.5 1 1.5 2 2.5 x 10 −3 PageRank ExponentialRank Data from Justus Uitermark, Erasmus University Rotterdam

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