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- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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