[PPT] - Scalable Preference Aggregation in Social Networks IFCAM Workshop PowerPoint Presentation

SLIDE 1

Scalable Preference Aggregation in Social Networks

IFCAM Workshop on Social Networks Indian Institute of Science, Bangalore

Y. Narahari

Joint work with Swapnil Dhamal

Game Theory Lab Department 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

SLIDE 2

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

SLIDE 3

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

SLIDE 4

Homophily in Social Networks

What constitutes a social network? Individuals and friendships

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24

SLIDE 5

Homophily in Social Networks

What constitutes a social network? Individuals and friendships What causes friendships? Similarity of individuals

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24

SLIDE 6

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

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24

SLIDE 7

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

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 1 / 24

SLIDE 8

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

SLIDE 9

Preference Aggregation

Agents or Voters have certain preferences over a set of Alternatives

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

SLIDE 10

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

SLIDE 11

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

SLIDE 12

Preference Aggregation

Aggregation Rule f outputs an aggregate preference for each preference profile

X Y Z Y X Z Y X Z

Plurality i j p

Preference of voter i Preference Profile Aggregation Rule Y Z X

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 2 / 24

SLIDE 13

Preference Aggregation

Aggregate Preference f(P) summarizes the preferences of the voters

X Y Z Y X Z Y X Z

Plurality i j p

Aggregate Preference Preference of voter i Preference Profile Aggregation Rule Y Z X

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 2 / 24

SLIDE 14

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 1

3

Distance between (X, Y , Z) and (Y , Z, X) is 2

3

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

SLIDE 15

Motivation for the Work

Many situations where we need to obtain a satisfactory aggregate preference given the individual preferences: meetings, committees, voting, poll surveys, product ranking, search engine aggregation, collaborative filtering, etc.

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 4 / 24

SLIDE 16

Motivation for the Work

Many situations where we need to obtain a satisfactory aggregate 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 preferences from all the voters due to a variety of factors: time, lack of interest of the voters, etc.

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 4 / 24

SLIDE 17

Motivation for the Work

Many situations where we need to obtain a satisfactory aggregate 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 preferences from all the voters due to a variety of factors: time, lack of interest of the voters, etc. Most interesting aggregation rules are computationally intensive Estimate the aggregate preference of the population by selecting a subset of voters, taking into account the social network

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 4 / 24

SLIDE 18

Current Art and Research Gaps (1)

Social networks do influence voting in elections 1 2

1Sheingold, C. A. 1973. Social networks and voting: the resurrection of a research

agenda. American Sociological Review 712-720.

2Burstein, P. 1976. Social networks and voting: Some Israeli data. Social Forces

54(4):833–847.

3Conitzer, V. 2012. Should social network structure be taken into account in

elections? Mathematical Social Sciences 64(1):100-102.

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 5 / 24

SLIDE 19

Current Art and Research Gaps (1)

Social networks do influence voting in elections 1 2 Network structure can be ignored in many contexts 3

1Sheingold, C. A. 1973. Social networks and voting: the resurrection of a research

agenda. American Sociological Review 712-720.

2Burstein, P. 1976. Social networks and voting: Some Israeli data. Social Forces

54(4):833–847.

3Conitzer, V. 2012. Should social network structure be taken into account in

elections? Mathematical Social Sciences 64(1):100-102.

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 5 / 24

SLIDE 20

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 research

agenda. American Sociological Review 712-720.

2Burstein, P. 1976. Social networks and voting: Some Israeli data. Social Forces

54(4):833–847.

3Conitzer, V. 2012. Should social network structure be taken into account in

elections? Mathematical Social Sciences 64(1):100-102.

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 5 / 24

SLIDE 21

Current Art and Research Gaps (2)

Node selection in voting using attributes of nodes and alterna- tives without taking social network into account 4

4Soufiani, H. A.; Parkes, D. C.; and Xia, L. 2013. Preference elicitation for general

random utility models. In The Twenty-Ninth Conference on Uncertainty In Artificial Intelligence, 596-605.

5N.R. Suri and Y. Narahari. IEEE - TASE. 2012 6Easley, 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

SLIDE 22

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

4Soufiani, H. A.; Parkes, D. C.; and Xia, L. 2013. Preference elicitation for general

random utility models. In The Twenty-Ninth Conference on Uncertainty In Artificial Intelligence, 596-605.

5N.R. Suri and Y. Narahari. IEEE - TASE. 2012 6Easley, 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

SLIDE 23

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 into account

4Soufiani, H. A.; Parkes, D. C.; and Xia, L. 2013. Preference elicitation for general

random utility models. In The Twenty-Ninth Conference on Uncertainty In Artificial Intelligence, 596-605.

5N.R. Suri and Y. Narahari. IEEE - TASE. 2012 6Easley, 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

SLIDE 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

SLIDE 25

Survey Questions

Personal issues Favorite place to meet Favorite recent movie Favorite food cuisine

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 7 / 24

SLIDE 26

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

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SLIDE 27

Survey Network

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 8 / 24

SLIDE 28

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 for social issues

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 9 / 24

SLIDE 29

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 for social issues The social network had higher influence on rankings related to personal issues than social issues

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 9 / 24

SLIDE 30

Distribution of Distance

Histogram for different questions for a given pair fit by truncated Gaussian distribution having range [0, 1] Considered a discrete version of the truncated Gaussian distribution, D

1 2 mean 1 normalized 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

SLIDE 31

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

SLIDE 32

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 an aggregate preference that is close enough to the aggregate preference of N using f .

7Swapnil Dhamal and Y. Narahari. Scalable Preference Aggregation in

Social Networks. Proceedings of the First AAAI Conference on Human Computation and Crowdsourcing (HCOMP), November 2013.

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 11 / 24

SLIDE 33

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 an aggregate preference that is close enough to the aggregate preference of N using f .

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 11 / 24

SLIDE 34

Starting to Solve the Problem

i M N Distance between set M ⊆ N and node i ∈ N d(M, i) = min

j∈M d(j, i)

Representative of node i in set M Φ(M, i) ∈ arg min

j∈M

d(j, i)

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 12 / 24

SLIDE 35

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

SLIDE 36

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

SLIDE 37

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

SLIDE 38

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 an aggregate preference that is close enough to the aggregate preference of N using f .

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 13 / 24

SLIDE 39

What is ‘Close Enough’?

Actual Obtained f(P) f(R)

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24

SLIDE 40

What is ‘Close Enough’?

Actual Obtained f(P) f(R)

For any y ∈ f (R), distance = min

x∈f (P) δ(x, y)

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24

SLIDE 41

What is ‘Close Enough’?

Actual Obtained f(P) f(R)

?

For any y ∈ f (R), distance = min

x∈f (P) δ(x, y)

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24

SLIDE 42

What is ‘Close Enough’?

Actual Obtained f(P) f(R) u.a.r.

For any y ∈ f (R), distance = min

x∈f (P) δ(x, y)

y ∈ f (R) u.a.r., f (P) ∆ f (R) = Ey∼Uf (R)

min

x∈f (P) δ(x, y)

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24

SLIDE 43

What is ‘Close Enough’?

Actual Obtained f(P) f(R) u.a.r.

For any y ∈ f (R), distance = min

x∈f (P) δ(x, y)

y ∈ f (R) u.a.r., f (P) ∆ f (R) = Ey∼Uf (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

SLIDE 44

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 an aggregate preference that is close enough to the aggregate preference of N using f .

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 14 / 24

SLIDE 45

Issues in Solving this Problem

Find a set M of size k that maximizes h(M) = 1 − E[f (P) ∆ f (R)] Given M, computing h(M) hard for many aggregation rules h(·) not monotone and neither submodular nor supermodular even for simple aggregation rules apart from dictatorship Aggregation rule may be needed to be changed frequently (to tackle strategic users) An approach agnostic to the aggregation rule ρ(M) = min

i∈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

SLIDE 46

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 ≤ ǫ = ⇒ f (P) ∆ f (P′) ≤ ǫ

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 16 / 24

SLIDE 47

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

SLIDE 48

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 ≤ ǫ = ⇒ E[f (P) ∆ f (P′)] ≤ ǫ

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 17 / 24

SLIDE 49

Empirical Satisfaction of Expected Weak Insensitivity under Distribution 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

SLIDE 50

Objective Functions in the New Problem

Maximize minimum expected similarity: ρ(M) = min

i∈N c(M, i)

Maximize average expected similarity: ψ(M) = avg

i∈N

c(M, i)

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 19 / 24

SLIDE 51

What Next?

Again ... Solving the new problem is also NP-hard.

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 20 / 24

SLIDE 52

What Next?

Again ... Solving the new problem is also NP-hard. However ... Objective functions are non-negative, monotone, and submodular.

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 20 / 24

SLIDE 53

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 − 1

e ) approximate optimal solution. a

aNemhauser, G. L.; Wolsey, L. A.; and Fisher, M. L. 1978. An analysis of

approximations for maximizing submodular set functions-I. Mathematical Programming 14(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

SLIDE 54

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

SLIDE 55

Experimental Results

Method How to select nodes? How to name 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) = min

i∈N c(M, i)

ψ(M) =

i∈N

c(M, i) Q : Profile containing only preferences of nodes in M Q′ : Profile containing weighted preferences of nodes in M

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 21 / 24

SLIDE 56

Experimental Results

Average case Worst case Personal issues

1 6 11 16 21 26 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 26 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

Social issues

1 6 11 16 21 26 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 26 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

SLIDE 57

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

SLIDE 58

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

SLIDE 59

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

T U H O K N Y A H A T N U O K Y N K T U H Y A O T H A N K Y O U A O T N H Y U K U O Y K N A H T T N A K H Y U O

f

T U H O K N Y A H A T N U O K Y N K T U H Y A O T H A N K Y O U A O T N H Y U K U O Y K N A H T T N A K H Y U O

SLIDE 60

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)

?

Y. Narahari

(IISc) Scalable Preference Aggregation in Social Networks January 16, 2014 24 / 24

SLIDE 61

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