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