Scalable Preference Aggregation in Social Networksifcam/new_avenue/Slides/Seminars/Narahar… · Should social network structure be taken into account in elections? Mathematical Social

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


Overview


2 A Sample Survey



5 Conclusions


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.






























Preference Aggregation

Agents or Voters have certain preferences over a set ofAlternatives

X Y Z

Y X Z

i

j

p

Y Z X



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



Preference Profile P is a vector of preferences of voters

X Y Z

Y X Z

i

j

p


Preference Profile

Y Z X



Aggregation Rule f outputs an aggregate preference for eachpreference profile

X Y Z

Y X Z

Y X ZPlurality

i

j

p


Preference Profile

Aggregation RuleY Z X



Aggregate Preference f(P) summarizes the preferences of thevoters

X Y Z

Y X Z

Y X ZPlurality

i

j

p

Aggregate Preference


Preference Profile

Aggregation RuleY Z X


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


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














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.



















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.


















Overview


2 A Sample Survey



5 Conclusions


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


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


Survey Network


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


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


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(·, ·)


Overview


2 A Sample Survey



5 Conclusions


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.


Problem Statement

Problem Statement



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)


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



v j

t s

i

f(P)

P

f

i

j

s

t

v

f(Q)

Q

f

j

v



v j

t s

i

f(P)

P

f

i

j

s

t

v

f(Q')

Q'

f

j

j

j

j

v


Problem Statement

Problem Statement



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)]



Actual

Obtained

f(P)

f(R)


δ(x , y)


[min

x∈f (P)δ(x , y)




Actual

Obtained

f(P)

f(R)

?


δ(x , y)


[min

x∈f (P)δ(x , y)




Actual

Obtained

f(P)

f(R)

u.a.r.


δ(x , y)


[min

x∈f (P)δ(x , y)

]

Our objective is to minimize E[f (P) ∆ f (R)]



Actual

Obtained

f(P)

f(R)

u.a.r.


δ(x , y)


[min

x∈f (P)δ(x , y)



Problem Statement

Problem Statement



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)


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


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


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


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.


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)


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)


What Next?

Again ...


However ...


That means ...





What Next?

Again ...


However ...


That means ...





Overview


2 A Sample Survey



5 Conclusions


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


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


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


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


E[

f(P

) ∆

f(

R)]

Greedy−min

Greedy−avg

Random−poll

Random−rep


Overview


2 A Sample Survey



5 Conclusions


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


Cartoons: 3.bp.blogspot.com, altruhelp.files.wordpress.com, standwitharizona.com.


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)


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)


Documents

Scalable Preference Aggregation in Social Networksifcam/new_avenue/Slides/Seminars/Narahar… · Should social network structure be taken into account in elections? Mathematical Social