Community Detection with Negative Links

Vincent Traag1 Jeroen Bruggeman2

1Catholic University of Louvain, Belgium

2University of Amsterdam, Netherlands

June 9, 2009

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Outline

1 Introduction

2 Social Balance Theory

3 Modularity

4 Including negative links

5 Empirical example

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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)• . . .

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Social Balance Theory

C1

C2

AB

C

D

• Triads (sets of three nodes) are balanced iftheir relationships are “symmetric”.

• Triad i , j , k is balanced if AijAikAjk = 1.

• If network is balanced, is can be split in twoclusters. (Harary, 1953)

• A network is said to be k-balanced if it can besplit into k clusters.

• For unbalanced networks, how can the nodesbe clustered?

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Social Balance Theory

C1

C2 C3

AB

D E

C

• Triads (sets of three nodes) are balanced iftheir relationships are “symmetric”.

• Triad i , j , k is balanced if AijAikAjk = 1.

• If network is balanced, is can be split in twoclusters. (Harary, 1953)

• A network is said to be k-balanced if it can besplit into k clusters.

• For unbalanced networks, how can the nodesbe clustered?

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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 = 12 this is equivalent to minimizing

F =∑

ij

(A+ij − A−

ij )δ(σi , σj) =∑

ij

Aijδ(σi , σj).

Approach by Doreian and Mrvar, Social Networks, Vol. 18, (1996).

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Problems with frustration

• If there are no negative links,there is only one cluster.

• Even minimally postiveconnected group is in onecluster.

• Absent links do not join orseperate a cluster.

• Defines unclearly a community.

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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.

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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

mH =

1

m

∑

ij

(Aij − γpij)δ(σi , σj)

• Results in a tuneable (γ) version of modularity.

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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 −kikj

m

)

δ(σi , σj )

= 0

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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).

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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 tominimizing frustration.

• Implemented in the simulated annealing scheme used by Reichardt &Bornholdt (2006).

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Empirical example

γ = 1, λ = 1

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Empirical example

γ = 0.3, λ = 1

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Empirical example

γ = 1, λ = 2

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Conclusions

• Proposed a solution for finding communities with negative links.

• Is in agreement with techniques for community detection withpositive links only.

• Results similar for ”social balance” clustering if network is (almostcomplete) and balanced.

• Yields good community assignments.

• Can be readily implemented in existing modularity optimizationtechniques.

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