Exponential Ranking: Taking into

account negative links.

V.A. Traag1, Y.E. Nesterov2, P. Van Dooren1

1Department of Applied MathematicsUniversite Catholique de Louvain

2COREUniversite Catholique de Louvain

27 October 2010

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)

Ranking with negative links

Analysis of empirical networks

• Centrality (popularity) of nodes

• Study roles of nodes

• Aids analysis of negative links

Trust

• Which node is trustworthy?

• Including distrust could improve trust mechanisms

• How to deal with cyclic distrust?

• What is the enemy of my enemy?

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

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

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

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 pjAji

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) = expATp(t)/µ‖ expATp(t)/µ‖1

• In terms of reputation: k(t + 1) = AT exp k(t)/µ‖ exp k(t)/µ‖1

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

Example

c

a

b

d e

Example cycles for µ = 0

Reputations

1 2 3 4 5 6 7a 1.00 0.40 0.67 0.50 0.67 0.50 0.67b 1.00 0.40 0.33 0.50 0.33 0.50 0.33c 1.00 0.40 0.67 0.50 0.67 0.50 0.67d 1.00 0.20 - - - - -e 1.00 0.20 - - - - -

Example

c

a

b

d e

Example convergence for µ = 1

Reputations

1 2 3 4 5 6 7a 1.00 0.20 0.21 0.22 0.22 0.22 0.22b 1.00 0.20 0.21 0.21 0.21 0.21 0.21c 1.00 0.20 0.21 0.22 0.22 0.22 0.22d 1.00 0.20 0.17 0.17 0.17 0.17 0.17e 1.00 0.20 0.17 0.17 0.17 0.17 0.17

Preliminary tests

Generate test network

1 Generate random network (n = 1100)

ER graphs Each link with probability p = 0.01SF 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

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 herePageRank+ 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)

Results

Faithful results

Erdos-Renyı

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

q

CP

Exponential RankingDegreePageRank+

Scale-Free

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

q

CP

Exponential RankingDegreePageRank+

Results

Semi-deceptive results

Erdos-Renyı

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

q

CP

Exponential RankingDegreePageRank+

Scale-Free

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

q

CP

Exponential RankingDegreePageRank+

Results

Deceptive results

Erdos-Renyı

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

q

CP

Exponential RankingDegreePageRank+

Scale-Free

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

q

CP

Exponential RankingDegreePageRank+

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 0.040

0.5

1

1.5

2

2.5x 10

−3

PageRank

Exp

onen

tial R

ank

Data from Justus Uitermark, Erasmus University Rotterdam

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?