This document proposes a method for community detection that accounts for both positive and negative links. It extends the modularity measure by separating the positive and negative parts of the network and defining different "allowed" and "forbidden" link types for each. This results in two Hamiltonians that can be optimized independently or together. The approach generalizes modularity to networks with negative links while reducing to existing methods in certain cases. An empirical example demonstrates its ability to detect communities under different parameter settings.

1. Community Detection with Negative Links
Vincent Traag1 Jeroen Bruggeman2
1Catholic University of Louvain, Belgium
2University of Amsterdam, Netherlands
June 9, 2009
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 1 / 15

2. Outline
1 Introduction
2 Social Balance Theory
3 Modularity
4 Including negative links
5 Empirical example
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 2 / 15

3. Introduction
• Community detection is succesfully applied in a number of fields.
• Whether a link is positive or negative usually ignored.
• It is highly relevant for
• Hyperlinks on webpages (“good” sites, instead of “important” sites)
• References in blogs (opinion clustering, not thematical)
• Trust relationships (e.g. P2P systems)
• International relationships (conflict and cooperation)
• . . .
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 3 / 15

4. Social Balance Theory
C1
C2
AB
C
D
• Triads (sets of three nodes) are balanced if
their relationships are “symmetric”.
• Triad i, j, k is balanced if AijAikAjk = 1.
• If network is balanced, is can be split in two
clusters. (Harary, 1953)
• A network is said to be k-balanced if it can be
split into k clusters.
• For unbalanced networks, how can the nodes
be clustered?
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 4 / 15

5. Social Balance Theory
C1
C2 C3
AB
D E
C
• Triads (sets of three nodes) are balanced if
their relationships are “symmetric”.
• Triad i, j, k is balanced if AijAikAjk = 1.
• If network is balanced, is can be split in two
clusters. (Harary, 1953)
• A network is said to be k-balanced if it can be
split into k clusters.
• For unbalanced networks, how can the nodes
be clustered?
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 4 / 15

6. Frustration
• Try to come close to the ideal ’balanced’ network.
• Minimize links that violate the conditions of k-balance:
• Negative links within clusters,
• Positive links between clusters.
Definition
Frustration
F =
ij
αA−
ij δ(σi , σj ) + (1 − α)A+
ij (1 − δ(σi , σj )).
• If α = 1
2 this is equivalent to minimizing
F =
ij
(A+
ij − A−
ij )δ(σi , σj ) =
ij
Aij δ(σi , σj ).
Approach by Doreian and Mrvar, Social Networks, Vol. 18, (1996).
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 5 / 15

7. Problems with frustration
• If there are no negative links,
there is only one cluster.
• Even minimally postive
connected group is in one
cluster.
• Absent links do not join or
seperate a cluster.
• Defines unclearly a community.
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 6 / 15

8. Modularity
Modularity has been succesfully applied in community detection.
Definition (Modularity)
Q =
1
m
ij
(Aij − pij )δ(σi , σj )
=
1
m c
ac − ec.
Newman & Girvan, Phys Rev E 69, (2004).
Maximizing modularity yields a ”good” community assignment.
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 7 / 15

9. Potts approach
• Potts approach by Reichardt and Bornholdt (2006): reward “allowed”
links, penalise “forbidden” links.
Allowed • Links within communities
(reward aij = 1 − γpij).
Forbidden • Absent links within communities
(penalty bij = γpij).
• Formulated as an “energy/cost” function (Hamiltonian):
H =
ij
−aijAijδ(σi , σj ) + bij (1 − Aij)δ(σi , σj )
• Reformulated equals modularity (if γ = 1)
Q = −
1
m
H =
1
m
ij
(Aij − γpij )δ(σi , σj )
• Results in a tuneable (γ) version of modularity.
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 8 / 15

10. Problem with negative links
ak = 1 b k = 1
c k = −1
Negative links poses problem for modularity.
Expected values pij not well defined.
A =


+ + −
+ + −
− − +


Q =
1
m
ij
Aij −
ki kj
m
δ(σi , σj )
= 0
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11. Allowing negative links
• Solution is to separate the positive and negative part.
• Then change “allowed” and “forbidden” links:
Allowed • Positive links within communities
(reward aij = γp+
ij ).
• Absent negative links within communities
(reward dij = λp−
ij ).
Forbidden • Absent positive links within communities
(penalty bij = 1 − γp+
ij ).
• Negative links within communities
(penalty cij = 1 − λp−
ij ).
• Results in two separate Hamiltonians
H+ = − ij (A+
ij − γp+
ij )δ(σi , σj ) and
H− = ij (A−
ij − λp−
ij )δ(σi , σj ).
Vincent Traag (UC Louvain) Community Detection with Negative Links June 9, 2009 10 / 15

12. Hamiltonian
• When both Hamiltonians are weighted equally this equals minimizing
H = H+
+ H−
=
ij
(Aij − (γp+
ij − λp−
ij ))δ(σi , σj )
• This is similar to modularity, but with different expected values.
• If there are no negative links, (and γ = 1) this equals modularity.
• Equivalent to choosing the appropriate null-model.
• If γ = λ = 0, or if graph is complete and balanced this is equal to
minimizing frustration.
• Implemented in the simulated annealing scheme used by Reichardt &
Bornholdt (2006).
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13. Empirical example
γ = 1, λ = 1
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14. Empirical example
γ = 0.3, λ = 1
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15. Empirical example
γ = 1, λ = 2
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16. Conclusions
• Proposed a solution for finding communities with negative links.
• Is in agreement with techniques for community detection with
positive links only.
• Results similar for ”social balance” clustering if network is (almost
complete) and balanced.
• Yields good community assignments.
• Can be readily implemented in existing modularity optimization
techniques.
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