Scalable Preference Aggregationin Social Networks
IFCAM Workshop on Social NetworksIndian Institute of Science, Bangalore
Y. NarahariJoint work with Swapnil Dhamal
Game Theory LabDepartment of Computer Science and Automation
Indian Institute of Science, Bangalore
January 16, 2014
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 0 / 24
Overview
1 Introduction and Motivation
2 A Sample Survey
3 Problem Formulation
4 Experimental Results
5 Conclusions
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 0 / 24
Overview
1 Introduction and Motivation
2 A Sample Survey
3 Problem Formulation
4 Experimental Results
5 Conclusions
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 0 / 24
Homophily in Social Networks
What constitutes a social network?Individuals and friendships
What causes friendships?Similarity of individuals
What do friendships cause?Individuals become more similar
What is homophily?A bias in friendships towards similar individuals
Homophily plays a key role in social networks.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24
Homophily in Social Networks
What constitutes a social network?Individuals and friendships
What causes friendships?Similarity of individuals
What do friendships cause?Individuals become more similar
What is homophily?A bias in friendships towards similar individuals
Homophily plays a key role in social networks.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24
Homophily in Social Networks
What constitutes a social network?Individuals and friendships
What causes friendships?Similarity of individuals
What do friendships cause?Individuals become more similar
What is homophily?A bias in friendships towards similar individuals
Homophily plays a key role in social networks.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24
Homophily in Social Networks
What constitutes a social network?Individuals and friendships
What causes friendships?Similarity of individuals
What do friendships cause?Individuals become more similar
What is homophily?A bias in friendships towards similar individuals
Homophily plays a key role in social networks.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24
Homophily in Social Networks
What constitutes a social network?Individuals and friendships
What causes friendships?Similarity of individuals
What do friendships cause?Individuals become more similar
What is homophily?A bias in friendships towards similar individuals
Homophily plays a key role in social networks.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24
Preference Aggregation
Agents or Voters have certain preferences over a set ofAlternatives
X Y Z
Y X Z
i
j
p
Y Z X
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 2 / 24
Preference Aggregation
Preference of a voter is a complete ranked list of alternatives
X Y Z
Y X Z
i
j
p
Preference of voter i
Y Z X
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 2 / 24
Preference Aggregation
Preference Profile P is a vector of preferences of voters
X Y Z
Y X Z
i
j
p
Preference of voter i
Preference Profile
Y Z X
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 2 / 24
Preference Aggregation
Aggregation Rule f outputs an aggregate preference for eachpreference profile
X Y Z
Y X Z
Y X ZPlurality
i
j
p
Preference of voter i
Preference Profile
Aggregation RuleY Z X
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 2 / 24
Preference Aggregation
Aggregate Preference f(P) summarizes the preferences of thevoters
X Y Z
Y X Z
Y X ZPlurality
i
j
p
Aggregate Preference
Preference of voter i
Preference Profile
Aggregation RuleY Z X
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 2 / 24
Normalized Kendall-Tau Distance
r = number of alternatives
Normalized Kendall-Tau Distance =Number of pair inversions(
r
2
)
Distance between (X ,Y ,Z ) and (X ,Z ,Y ) is 13
Distance between (X ,Y ,Z ) and (Y ,Z ,X ) is 23
Distance between (X ,Y ,Z ) and (Z ,Y ,X ) is 1
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 3 / 24
Motivation for the Work
Many situations where we need to obtain a satisfactoryaggregate preference given the individual preferences:meetings, committees, voting, poll surveys, product ranking,search engine aggregation, collaborative filtering, etc.
For large networks, it is infeasible to gather the preferencesfrom all the voters due to a variety of factors: time, lack ofinterest of the voters, etc.
Most interesting aggregation rules are computationallyintensive
Estimate the aggregate preference of the population by selecting asubset of voters, taking into account the social network
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 4 / 24
Motivation for the Work
Many situations where we need to obtain a satisfactoryaggregate preference given the individual preferences:meetings, committees, voting, poll surveys, product ranking,search engine aggregation, collaborative filtering, etc.
For large networks, it is infeasible to gather the preferencesfrom all the voters due to a variety of factors: time, lack ofinterest of the voters, etc.
Most interesting aggregation rules are computationallyintensive
Estimate the aggregate preference of the population by selecting asubset of voters, taking into account the social network
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 4 / 24
Motivation for the Work
Many situations where we need to obtain a satisfactoryaggregate preference given the individual preferences:meetings, committees, voting, poll surveys, product ranking,search engine aggregation, collaborative filtering, etc.
For large networks, it is infeasible to gather the preferencesfrom all the voters due to a variety of factors: time, lack ofinterest of the voters, etc.
Most interesting aggregation rules are computationallyintensive
Estimate the aggregate preference of the population by selecting asubset of voters, taking into account the social network
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 4 / 24
Current Art and Research Gaps (1)
Social networks do influence voting in elections 1 2
Network structure can be ignored in many contexts 3
Opinions are divided
1Sheingold, C. A. 1973. Social networks and voting: the resurrection of a researchagenda. American Sociological Review 712-720.
2Burstein, P. 1976. Social networks and voting: Some Israeli data. Social Forces54(4):833–847.
3Conitzer, V. 2012. Should social network structure be taken into account inelections? Mathematical Social Sciences 64(1):100-102.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 5 / 24
Current Art and Research Gaps (1)
Social networks do influence voting in elections 1 2
Network structure can be ignored in many contexts 3
Opinions are divided
1Sheingold, C. A. 1973. Social networks and voting: the resurrection of a researchagenda. American Sociological Review 712-720.
2Burstein, P. 1976. Social networks and voting: Some Israeli data. Social Forces54(4):833–847.
3Conitzer, V. 2012. Should social network structure be taken into account inelections? Mathematical Social Sciences 64(1):100-102.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 5 / 24
Current Art and Research Gaps (1)
Social networks do influence voting in elections 1 2
Network structure can be ignored in many contexts 3
Opinions are divided
1Sheingold, C. A. 1973. Social networks and voting: the resurrection of a researchagenda. American Sociological Review 712-720.
2Burstein, P. 1976. Social networks and voting: Some Israeli data. Social Forces54(4):833–847.
3Conitzer, V. 2012. Should social network structure be taken into account inelections? Mathematical Social Sciences 64(1):100-102.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 5 / 24
Current Art and Research Gaps (2)
Node selection in voting using attributes of nodes and alterna-tives without taking social network into account 4
Node selection in influence maximization, influence limitation,virus inoculation, etc. taking social network into account 5 6
Our interest: Node selection in voting taking social network intoaccount
4Soufiani, H. A.; Parkes, D. C.; and Xia, L. 2013. Preference elicitation for generalrandom utility models. In The Twenty-Ninth Conference on Uncertainty In ArtificialIntelligence, 596-605.
5N.R. Suri and Y. Narahari. IEEE - TASE. 20126Easley, D., and Kleinberg, J. 2010. Networks, Crowds, and Markets: Reasoning
About a Highly Connected World. Cambridge University Press.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 6 / 24
Current Art and Research Gaps (2)
Node selection in voting using attributes of nodes and alterna-tives without taking social network into account 4
Node selection in influence maximization, influence limitation,virus inoculation, etc. taking social network into account 5 6
Our interest: Node selection in voting taking social network intoaccount
4Soufiani, H. A.; Parkes, D. C.; and Xia, L. 2013. Preference elicitation for generalrandom utility models. In The Twenty-Ninth Conference on Uncertainty In ArtificialIntelligence, 596-605.
5N.R. Suri and Y. Narahari. IEEE - TASE. 20126Easley, D., and Kleinberg, J. 2010. Networks, Crowds, and Markets: Reasoning
About a Highly Connected World. Cambridge University Press.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 6 / 24
Current Art and Research Gaps (2)
Node selection in voting using attributes of nodes and alterna-tives without taking social network into account 4
Node selection in influence maximization, influence limitation,virus inoculation, etc. taking social network into account 5 6
Our interest: Node selection in voting taking social network intoaccount
4Soufiani, H. A.; Parkes, D. C.; and Xia, L. 2013. Preference elicitation for generalrandom utility models. In The Twenty-Ninth Conference on Uncertainty In ArtificialIntelligence, 596-605.
5N.R. Suri and Y. Narahari. IEEE - TASE. 20126Easley, D., and Kleinberg, J. 2010. Networks, Crowds, and Markets: Reasoning
About a Highly Connected World. Cambridge University Press.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 6 / 24
Overview
1 Introduction and Motivation
2 A Sample Survey
3 Problem Formulation
4 Experimental Results
5 Conclusions
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 6 / 24
Survey Questions
Personal issues
Favorite place to meet
Favorite recent movie
Favorite food cuisine
Social issues
Most deserving Test batsman for the vacant spot
Most deserving Prime Minister
Most likely Prime Minister
Most deplorable crime
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 7 / 24
Survey Questions
Personal issues
Favorite place to meet
Favorite recent movie
Favorite food cuisine
Social issues
Most deserving Test batsman for the vacant spot
Most deserving Prime Minister
Most likely Prime Minister
Most deplorable crime
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 7 / 24
Survey Network
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 8 / 24
Observations on the Survey Network
Somewhat similar rankings by connected nodes
Very similar rankings by connected nodes belonging to big clus-ters
First and last alternatives mostly consistent for connected nodes
Somewhat similar rankings even by (un)connected nodes forsocial issues
The social network had higher influence on rankings related topersonal issues than social issues
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 9 / 24
Observations on the Survey Network
Somewhat similar rankings by connected nodes
Very similar rankings by connected nodes belonging to big clus-ters
First and last alternatives mostly consistent for connected nodes
Somewhat similar rankings even by (un)connected nodes forsocial issues
The social network had higher influence on rankings related topersonal issues than social issues
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 9 / 24
Distribution of Distance
Histogram for different questions for a given pair fit bytruncated Gaussian distribution having range [0, 1]
Considered a discrete version of the truncated Gaussiandistribution, D
1
2
0
mean
1normalized Kendall-Tau distance
count
Distance between i and j followed distribution D with mean d(i , j)
c(·, ·) = 1− d(·, ·)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 10 / 24
Overview
1 Introduction and Motivation
2 A Sample Survey
3 Problem Formulation
4 Experimental Results
5 Conclusions
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 10 / 24
Problem Statement 7
Problem Statement
Given a network with a set of nodes N and an aggregation rule f ,select a subset of nodes M ⊆ N of cardinality k , and deduce anaggregate preference that is close enough to the aggregatepreference of N using f .
7Swapnil Dhamal and Y. Narahari. Scalable Preference Aggregation inSocial Networks. Proceedings of the First AAAI Conference on HumanComputation and Crowdsourcing (HCOMP), November 2013.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 11 / 24
Problem Statement
Problem Statement
Given a network with a set of nodes N and an aggregation rule f ,select a subset of nodes M ⊆ N of cardinality k , and deduce anaggregate preference that is close enough to the aggregatepreference of N using f .
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 11 / 24
Starting to Solve the Problem
i
M
N
Distance between set M ⊆ N and node i ∈ N
d(M, i) = minj∈M
d(j , i)
Representative of node i in set M
Φ(M, i) ∈ arg minj∈M
d(j , i)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 12 / 24
How to Aggregate Preferences of Selected Nodes?
v j
t s
i
f(P)
P
f
i
j
s
t
v
?
f(R)
R
f
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 13 / 24
How to Aggregate Preferences of Selected Nodes?
v j
t s
i
f(P)
P
f
i
j
s
t
v
f(Q)
Q
f
j
v
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 13 / 24
How to Aggregate Preferences of Selected Nodes?
v j
t s
i
f(P)
P
f
i
j
s
t
v
f(Q')
Q'
f
j
j
j
j
v
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 13 / 24
Problem Statement
Problem Statement
Given a network with a set of nodes N and an aggregation rule f ,select a subset of nodes M ⊆ N of cardinality k , and deduce anaggregate preference that is close enough to the aggregatepreference of N using f .
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 13 / 24
What is ‘Close Enough’?
Actual
Obtained
f(P)
f(R)
For any y ∈ f (R), distance = minx∈f (P)
δ(x , y)
y ∈ f (R) u.a.r., f (P) ∆ f (R) = Ey∼U f (R)
[min
x∈f (P)δ(x , y)
]Our objective is to minimize E[f (P) ∆ f (R)]
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24
What is ‘Close Enough’?
Actual
Obtained
f(P)
f(R)
For any y ∈ f (R), distance = minx∈f (P)
δ(x , y)
y ∈ f (R) u.a.r., f (P) ∆ f (R) = Ey∼U f (R)
[min
x∈f (P)δ(x , y)
]Our objective is to minimize E[f (P) ∆ f (R)]
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24
What is ‘Close Enough’?
Actual
Obtained
f(P)
f(R)
?
For any y ∈ f (R), distance = minx∈f (P)
δ(x , y)
y ∈ f (R) u.a.r., f (P) ∆ f (R) = Ey∼U f (R)
[min
x∈f (P)δ(x , y)
]Our objective is to minimize E[f (P) ∆ f (R)]
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24
What is ‘Close Enough’?
Actual
Obtained
f(P)
f(R)
u.a.r.
For any y ∈ f (R), distance = minx∈f (P)
δ(x , y)
y ∈ f (R) u.a.r., f (P) ∆ f (R) = Ey∼U f (R)
[min
x∈f (P)δ(x , y)
]
Our objective is to minimize E[f (P) ∆ f (R)]
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24
What is ‘Close Enough’?
Actual
Obtained
f(P)
f(R)
u.a.r.
For any y ∈ f (R), distance = minx∈f (P)
δ(x , y)
y ∈ f (R) u.a.r., f (P) ∆ f (R) = Ey∼U f (R)
[min
x∈f (P)δ(x , y)
]Our objective is to minimize E[f (P) ∆ f (R)]
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24
Problem Statement
Problem Statement
Given a network with a set of nodes N and an aggregation rule f ,select a subset of nodes M ⊆ N of cardinality k , and deduce anaggregate preference that is close enough to the aggregatepreference of N using f .
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24
Issues in Solving this Problem
Find a set M of size k that maximizesh(M) = 1− E[f (P) ∆ f (R)]
Given M, computing h(M) hard for many aggregation rules
h(·) not monotone and neither submodular nor supermodulareven for simple aggregation rules apart from dictatorship
Aggregation rule may be needed to be changed frequently (totackle strategic users)
An approach agnostic to the aggregation rule
ρ(M) = mini∈N
c(M, i) ψ(M) =∑i∈N
c(M, i)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 15 / 24
Weak Insensitivity Property
≤ E
≤ E
f(P) f(P')
P P'
f f
i
j
s
t
v
i
j
s
t
v
f(P) Δ f(P')
≤ E
≤ E
≤ E
≤ E
Deviations for all i ≤ ε=⇒ f (P) ∆ f (P ′) ≤ ε
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 16 / 24
Weak Insensitivity Property
≤ E
≤ E
f(P) f(P')
P P'
f f
i
j
s
t
v
i
j
s
t
v
f(P) Δ f(P')
≤ E
≤ E
≤ E
≤ E
Only Dictatorship seems to satisfy this property
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 16 / 24
Expected Weak Insensitivity Property
≤ E
≤ E
f(P) f(P')
P P'
f f
i
j
s
t
v
i
j
s
t
v
[f(P) Δ f(P')]
≤ E
≤ E
≤ E
≤ E
E
mean
mean
mean
mean
mean
Deviations for all i from distribution with mean ≤ ε=⇒ E[f (P) ∆ f (P ′)] ≤ ε
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 17 / 24
Empirical Satisfaction of Expected Weak Insensitivity underDistribution D, Kendall-Tau Distance, and the Defined ∆
YES NO
Plurality
Dictatorship
Minmax
Bucklin
Smith set
Veto
Borda
Kemeny
Schulze
Copeland
Survey of Voting Rules 8 9
8Brandt, F.; Conitzer, V.; and Endriss, U. Computational social choice.www.cs.duke.edu/∼conitzer/comsocchapter.pdf.
9Wikipedia. 2013. Voting system – wikipedia, the free encyclopedia.wikipedia.org/w/index.php?title=Voting system.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 18 / 24
Objective Functions in the New Problem
Maximize minimum expected similarity:
ρ(M) = mini∈N
c(M, i)
Maximize average expected similarity:
ψ(M) = avgi∈N
c(M, i)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 19 / 24
What Next?
Again ...
Solving the new problem is also NP-hard.
However ...
Objective functions are non-negative, monotone, and submodular.
That means ...
Greedy hill-climbing gives (1− 1e ) approximate optimal solution. a
aNemhauser, G. L.; Wolsey, L. A.; and Fisher, M. L. 1978. An analysis ofapproximations for maximizing submodular set functions-I. Mathematical Programming14(1):265–294.
Until |M| = k , select j ∈ N\M that maximizes h(M ∪ {j})− h(M)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 20 / 24
What Next?
Again ...
Solving the new problem is also NP-hard.
However ...
Objective functions are non-negative, monotone, and submodular.
That means ...
Greedy hill-climbing gives (1− 1e ) approximate optimal solution. a
aNemhauser, G. L.; Wolsey, L. A.; and Fisher, M. L. 1978. An analysis ofapproximations for maximizing submodular set functions-I. Mathematical Programming14(1):265–294.
Until |M| = k , select j ∈ N\M that maximizes h(M ∪ {j})− h(M)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 20 / 24
What Next?
Again ...
Solving the new problem is also NP-hard.
However ...
Objective functions are non-negative, monotone, and submodular.
That means ...
Greedy hill-climbing gives (1− 1e ) approximate optimal solution. a
aNemhauser, G. L.; Wolsey, L. A.; and Fisher, M. L. 1978. An analysis ofapproximations for maximizing submodular set functions-I. Mathematical Programming14(1):265–294.
Until |M| = k , select j ∈ N\M that maximizes h(M ∪ {j})− h(M)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 20 / 24
Overview
1 Introduction and Motivation
2 A Sample Survey
3 Problem Formulation
4 Experimental Results
5 Conclusions
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 20 / 24
Experimental Results
Method How to select nodes? How toname aggregate?
Greedy-min Greedy hill-climbing maximize ρ(·) f (Q ′)
Greedy-avg Greedy hill-climbing maximize ψ(·) f (Q ′)
Random-poll Random f (Q)
Random-rep Random f (Q ′)
ρ(M) = mini∈N
c(M, i) ψ(M) =∑i∈N
c(M, i)
Q : Profile containing only preferences of nodes in MQ ′ : Profile containing weighted preferences of nodes in M
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 21 / 24
Experimental Results
Average case Worst caseP
erso
nal
issu
es
1 6 11 16 21 260
0.1
0.2
0.3
0.4
Number of selected nodes (k)
E[
f(P
) ∆
f(
R)]
Greedy−min
Greedy−avg
Random−poll
Random−rep
1 6 11 16 21 260
0.2
0.4
0.6
0.8
Number of selected nodes (k)
E[
f(P
) ∆
f(
R)]
Greedy−min
Greedy−avg
Random−poll
Random−rep
So
cial
issu
es
1 6 11 16 21 260
0.05
0.1
0.15
0.2
Number of selected nodes (k)
E[
f(P
) ∆
f(
R)]
Greedy−min
Greedy−avg
Random−poll
Random−rep
1 6 11 16 21 260
0.2
0.4
0.6
Number of selected nodes (k)
E[
f(P
) ∆
f(
R)]
Greedy−min
Greedy−avg
Random−poll
Random−rep
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 22 / 24
Overview
1 Introduction and Motivation
2 A Sample Survey
3 Problem Formulation
4 Experimental Results
5 Conclusions
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 22 / 24
Future Work
Explore other forms of modified preference profile R given P
Conduct a survey on a larger scale
Study the problem when agents are strategic
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 23 / 24
Cartoons: 3.bp.blogspot.com, altruhelp.files.wordpress.com, standwitharizona.com.
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 23 / 24
Modeling Homophily for Unconnected Nodes
Initializations
d(i , j) known for connected pairs {i , j} [0 for i = j ]
d(i , j) = 1 for all unconnected pairs
p
j
i
d(p,j)
d(p,i)
?
All pairs shortest path with update rule
if d(p, i) +©r d(p, j) < d(i , j) then d(i , j) = d(p, i) +©r d(p, j)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 24 / 24
Modeling Homophily for Unconnected Nodes
Initializations
d(i , j) known for connected pairs {i , j} [0 for i = j ]
d(i , j) = 1 for all unconnected pairs
p
j
i
d(p,j)
d(p,i)
d(i,j)
All pairs shortest path with update rule
if d(p, i) +©r d(p, j) < d(i , j) then d(i , j) = d(p, i) +©r d(p, j)
Y. Narahari (IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 24 / 24