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Presentation at Université catholique de Louvain at Feb 13, 2009.
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Outline Social Influence Community Detection
Social Influence & Community Detection
V.A. Traag
February 13, 2009
Outline Social Influence Community Detection
Outline
1 Social InfluenceIntroductionBA-modelSocial influence modelEmpirical resultsFurther research
2 Community DetectionIntroductionModularity & Potts modelNegative linksEmpirical exampleFurther research
Outline Social Influence Community Detection
Outline
1 Social InfluenceIntroductionBA-modelSocial influence modelEmpirical resultsFurther research
2 Community DetectionIntroductionModularity & Potts modelNegative linksEmpirical exampleFurther research
Outline Social Influence Community Detection
Introduction
• What items (e.g. movies, books) become popular?
• Based on an extension of the BA-model.(Social influence balancing parameter)
• Idea emerged from web based experiment of Salganik et al.(Science, 2006)
Outline Social Influence Community Detection
Experiment from Salganik et al.
More social influence 1...
More social influence 8
Social influence 1...
Social influence 8
No social influence 1...
No social influence 8
User arrival
Outline Social Influence Community Detection
Experiment from Salganik et al.
More social influence 1...
More social influence 8
Social influence 1...
Social influence 8
No social influence 1...
No social influence 8
User arrival
Mor
ein
equal
ity
and
unce
rtai
nty
Outline Social Influence Community Detection
BA-model
• Rich-get-richer effect.
• Web sites (items) attract links (votes) proportional to thenumber of links (votes).
k̇i = mki
∑
j kj
• Yields stationary degree distribution.
Pr(X = k) = 2m2k−3
Outline Social Influence Community Detection
Social influence
• Additional good-get-richer effect.
• Introduce quality φ ≥ 0 with mean quality µ and variance σ.
• Balance quality and popularity through parameter 0 ≤ λ ≤ 1.
• New differential equation
k̇i = m
[
(1 − λ)φi
∑
j φj
+ λki
∑
j kj
]
.
Outline Social Influence Community Detection
Theoretical results
Result is
ki (t) =
[
(
t
ti
)λ
− 1
]
(1 − λ)mφi
µλ.
from which we can see that:
• Votes increase with time
• Older items obtain more votes
• Better items obtain more votes (might catch up with older,but worse, items)
• Higher social influence, changes growth pattern: less quicklyat introduction, but keeps growing more.
Outline Social Influence Community Detection
Theoretical results
• For invariant quality, the “uncertainty” distribution is
Pr(X = k|φ) =µ((1 − λ)mφ)
1λ
(kλµ + (1 − λ)mφ)(1+1λ).
• Mean popularity and variance
E (X |φ) =mφ
µand Var(X |φ) =
E (X |φ)2
1 − 2λ.
• Expected number of votes rise with quality
• Uncertainty rises with quality and with social influence
• In congruence with experiment from Salganik et al.
Outline Social Influence Community Detection
Theoretical results
• Quality distribution is ρ(φ) with mean µ and variance σ.
• The “popularity” distribution can be deduced as
Pr(X = k) =
∫ φmax
φmin
ρ(φ) Pr(X = k|φ)dφ.
• In general, mean popularity and variance is
E (X ) = m and Var(X ) =m2(2σ(1 − λ) + µ2)
µ2(1 − 2λ).
Outline Social Influence Community Detection
Empirical results
• Quality usually a problem, how to estimate it?
• Workaround: assume a quality distribution (e.g. Dirac,Exponential).
• Compare empirical popularity distribution (#views, #sales) totheoretical distribution.
• Estimate social influence parameter λ using MLE.
Outline Social Influence Community Detection
10-4
10-3
10-2
10-1
100
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103
HollywoodYouTube
Fit (Hollywood)Fit (YouTube)
k
Pr(
x>
k)
YouTube1 λ ≈ 0.878
Hollywood1 λ ≈ 0.663
1Assuming an exponential distribution
Outline Social Influence Community Detection
Other results
• Other research from Pennock et al. shows additional results.
• Hyperlink distribution per category of websites.
• Relatively high for companies (0.950) and newspapers (0.948).
• Relatively low for universities (0.612) and scientists (0.602).
• Might be used as a rough estimate of the amount of socialinfluence.
Outline Social Influence Community Detection
Social Influence
• Introduce parameter social influence parameter λ on network.
• Balance between own preferences and preferences of others.
• Spreading (cascading) of preferences.
• Updating of exclusive preferences might result in communitydetection algorithm.
• Popularity of items = size of communities?
One separate topic: estimate social influence in citationdistributions over the last few years. Has it increased?
Outline Social Influence Community Detection
Social Balance Theory
E1
E2
AB
C
D • Triads (a triple set of nodes) are balancedif their relationships are “symmetric”.
• Triad i , j , k is balanced if AijAikAjk = 1.
• If network is balanced, is can be split intwo communities. (Harary, 1953)
• Social balance can be extended tok-balanced: a k-cycle does not containexactly one negative edge.
• For unbalanced (or k-balanced) networks,how can communities be assigned suchthat nodes form cohesive groups?
Outline Social Influence Community Detection
Modularity
Definition
Modularity Q = 1m
∑
ij(Aij − pij)δ(σi , σj).
Newman & Girvan.
• Modularity can also be expressed as
Q =1
m
∑
c
ac − ec .
• Optimising modularity yields a good community assignment.
Outline Social Influence Community Detection
Potts approach
• Potts approach by Reichardt and Bornholdt: reward “allowed”links, penalise “forbidden” links.
Allowed • Links within communities(reward aij = γpij).
Forbidden • Absent links within communities(penalty bij = 1 − γpij).
• Formulated as a “energy/cost” function (Hamiltonian):
H =∑
ij
−aijAijδ(σi , σj ) + bij(1 − Aij)δ(σi , σj )
• Reformulated equals modularity (if γ = 1)
−1
mQ = H = −
∑
ij
(Aij − γpij)δ(σi , σj)
• Results in a tuneable (γ) version of modularity.
Outline Social Influence Community Detection
Problem with negative links
ik = 1 j k = 1
k k = −1
Negative links poses problem formodularity. Probabilities pij not welldefined.
A =
+ + −+ + −− − +
Q =1
m
∑
ij
(
Aij −kikj
m
)
δ(σi , σj)
= 0
Outline Social Influence Community Detection
Negative links
• Solution is to 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 Hamiltonian
H+ = −∑
ij
(A+ij − γp+
ij )δ(σi , σj) and
H− = −∑
ij
(A−
ij − λp−
ij )δ(σi , σj).
Outline Social Influence Community Detection
Hamiltonian
• We weigh both Hamiltonians equally.
• This results in
−1
mQ = H+ −H−
= −∑
ij
(Aij − (γp+ij − λp−
ij ))δ(σi , σj)
• Changing the expected values in modularity, allowscommunity detection in networks with negative links.
Outline Social Influence Community Detection
Empirical example
γ = 1, λ = 1
Outline Social Influence Community Detection
Empirical example
γ = 0.3, λ = 1
Outline Social Influence Community Detection
Empirical example
γ = 1, λ = 2
Outline Social Influence Community Detection
Further research
• Apply community detection scheme to citation networks.
• Communities in unsigned networks are ’thematic’ clusters.
• Communities in signed networks are ’positional’ clusters.
• For example: Dutch opinion makers.