© 2006 board of trustees of the university of illinois authored by tanya berger-wolf analysis of...

© 2006 Board of Trustees of the University of Illinois

Authored by Tanya Berger-Wolf

Analysis of DynamicSocial Networks Analysis of DynamicSocial Networks

Tanya Berger-WolfDepartment of Computer ScienceUniversity of Illinois at Chicago

Tanya Berger-WolfDepartment of Computer ScienceUniversity of Illinois at Chicago

© 2006 Board of Trustees of the University of Illinois. Authored by Tanya Berger-Wolf

ZebrasZebrasDan Rubenstein, Siva Sandaresan, Ilya Fischhoff (Princeton)

Movie credit: “Champions of the Wild”, © Omni-Film Productions.


ContextContextdisease modelingEubank et.al.‘04, Keeling’99, Kretzschmar&Morris’96 cultural and information transmissionBaumes et.al.’04, Broido&Claffy’01, Carley’96, Chen&Carley’05, Kempe et.al.’03, Tsvetovat et.al.’03,Tyler et.al.’03, Wellman’97

intelligence and surveillanceAiroldi&Malin’04,Baumes et.al.’04, Kolata’05, Malin’04, Magdon-Ismail et.al.’03

business managementBernstein et.al.’02, Carley&Prietula’01, Papadimitriou’97, Papadimitriou&Servan-Schreiber’99

conservation and population biology Croft et.al.’04, Cross et.al.’05, Lusseau&Newman’04

disease modelingEubank et.al.‘04, Keeling’99, Kretzschmar&Morris’96 cultural and information transmissionBaumes et.al.’04, Broido&Claffy’01, Carley’96, Chen&Carley’05, Kempe et.al.’03, Tsvetovat et.al.’03,Tyler et.al.’03, Wellman’97

intelligence and surveillanceAiroldi&Malin’04,Baumes et.al.’04, Kolata’05, Malin’04, Magdon-Ismail et.al.’03

business managementBernstein et.al.’02, Carley&Prietula’01, Papadimitriou’97, Papadimitriou&Servan-Schreiber’99

conservation and population biology Croft et.al.’04, Cross et.al.’05, Lusseau&Newman’04


Social Networks: Static vs DynamicSocial Networks: Static vs Dynamic1

2

3

4

5

6

t=1 t=2 t=3 t=4 t=5

1 2 3 4 5 6

1

2

3

4

5

6

1

2

3

4

5

6

1/5 1/5 1/5 1/5 1/5

IndividualsStrength or probability

of interaction over a period of time


Input – Individual InformationInput – Individual Information

2

1

34

5

6


file

4

9

3

1

1

© Christopher Sadler© Christopher Sadler

© Christopher Sadler


4

9

3

1

1

t=1

4

8

3

22

1

t=2

9

4

4

4

9

1

t=4

4

4

4

8

22

t=3


Communities over time – persistent groups

Critical individuals (starters and blockers)

Critical gatherings

Critical times

Persistent demographic configurations

Communities over time – persistent groups

Critical individuals (starters and blockers)

Critical gatherings

Critical times

Persistent demographic configurations

Communities over time – persistent groupsA group persist in time (is a metagroup) if some (big) fraction β of it exists some (big) fraction α of time

Critical individuals (starters and blockers)Critical starter = a spreading process started with it will affect most individuals. Critical blocker = a spreading process will affect fewest individuals with it absent from the population.

Critical gatheringsGatherings that facilitate most (least) spreading

Critical timesTimesteps when interaction pattern changes

Persistent demographic configurationsRepeated groups with the same demographic pattern

Communities over time – persistent groupsA group persist in time (is a metagroup) if some (big) fraction β of it exists some (big) fraction α of time

Critical individuals (starters and blockers)Critical starter = a spreading process started with it will affect most individuals. Critical blocker = a spreading process will affect fewest individuals with it absent from the population.

Critical gatheringsGatherings that facilitate most (least) spreading

Critical timesTimesteps when interaction pattern changes

Persistent demographic configurationsRepeated groups with the same demographic pattern

QuestionsQuestions


1/10

1/9

1/4

1

4

9

3

1

1

t=1

4

8

3

22

1

t=2

9

4

4

4

9

1

t=4

4

4

4

8

22

t=3

1 1 1

8/9 1 8/9

11 1

1

3/4

1/2

1/2 1/2

β=.5β=.8


Simple Stats:Simple Stats:

Metagroup = path length ≥ α

Total #metagroups = #paths length ≥ α

Maximal metagroup length = max path length

Most persistent metagroup = longest path in a DAG

Let x be a member of MG is it appears in it at least γ times.Largest metagroup = dynamic programming on membership set.

Metagroup = path length ≥ α

Total #metagroups = #paths length ≥ α

Maximal metagroup length = max path length

Most persistent metagroup = longest path in a DAG

Let x be a member of MG is it appears in it at least γ times.Largest metagroup = dynamic programming on membership set.

Skip 3Skip 3 Skip 5Skip 5


Example – Southern Women(Natches, TN 1933)Example – Southern Women(Natches, TN 1933)

A. Davis, B. B. Gardner, and M. R. Gardner. Deep South. The U. of Chicago Press, Chicago, IL, 1941


Southern Women MetagroupsSouthern Women Metagroups

11 5 2 7 12 9 3 6 10 1 14 8 4 13

.61-.7

.71-.8

.81-.9

.91-1

101112131415

1112131415

121314

121314

1234567

9

123456

1234

678

1

345


DGG 1941 Intuition

Homans 1951 Intuition

Phillips and Conviser 1972 Information Theory

Breiger 1974 Matrix Algebra

Breiger, Boorman & Arabie 1975 Computational

Bonacich 1978 Boolean Algebra

Doreian 1979 Algebraic Topology

Bonacich 1991 Correspondence Analysis

Freeman 1992 G-Transitivity

Everett & Borgatti 1993 Regular Coloring

Freeman 1993 Genetic Algorithm I & II

Freeman & White 1993 Galois Lattices I & II

Borgatti & Everett 1997 Bipartite Analyses I, II & III

Skvoretz & Faust 1999 p* Model

Roberts 2000 Normalized SVD

Osbourn 2000 VERI Procedure

Newman 2001 Weighted Proximities

Linton Freeman Metanalysis, 2003


Southern Women CommunitiesSouthern Women Communities


Southern Women: Core vs Periphery

BW&S 1 3 4 5 | 2 6 | 7 9 12 13 14 | 11 15 | 10


Group ConnectivityGroup Connectivity

Given groups g1,…,gl, are they in the same metagroup?Given groups g1,…,gl, are they in the same metagroup?

Most persistent/largest/loudest/.. metagroup that contains these groups

A metagroup that contains largest number of these groups – dynamic programming

…g1 g2 gl-1

gl


Individual ConnectivityIndividual Connectivity

Given individuals S={s1,…,sl}, are they in the same metagroup?

Metagroup that contains max number of individuals in S

Most persistent/largest/shiniest.. metagroup that contains all individuals in S

Given individuals S={s1,…,sl}, are they in the same metagroup?

Metagroup that contains max number of individuals in S

Most persistent/largest/shiniest.. metagroup that contains all individuals in S


Critical Group SetCritical Group Set

The smallest set of groups whose absence leaves no metagroups (for given α and β)

The smallest set of groups whose absence leaves no metagroups (for given α and β)

Formally: remove fewest vertices in a DAGso there are no paths of length > k-1

K-path Vertex Shattering Set


K-path Vertex Shattering SetK-path Vertex Shattering Set

NP-hard: 2-path shattering set = vertex cover

NP-hard: 2-path shattering set = vertex cover

k=2

k=T

?

Polynomial: T-path shattering set (T is the longest path length) – min vertex cut in a DAG

Polynomial: T-path shattering set (T is the longest path length) – min vertex cut in a DAG


Critical Individual SetCritical Individual Set

The smallest set of individuals whose absence leaves no metagroups (for given α and β)

The smallest set of individuals whose absence leaves no metagroups (for given α and β)

c

d

c

d

c

d

…cd cda a a

a a a

b b bb b bcd


Other questions:Other questions:Close Group: individuals that appear together more than others.Loyal Individuals: appear most frequently in any metagroup.Individual Membership: metagroup which maximizes the cardinality of the set of groups in which a given individual occurs.Extra/Introvert: member of the largest/smallest number of metagroups.Metagroup Representative: an individual who occurs more in a metagroup than any other individual and occurs in it more than in any other metagroup.Demographic Distinction: given a coloring of individuals, is there a property that distinguishes one color from the others? Critical Parameter Values: largest values of α, β for which there exists at least k metagroups. Largest γ for which each metagroup has at least k members.Sampling Rate: largest time step such that the answer does not change if the time step is decreased but changes if it is increased.Critical Time Moments: e.g., the time when the groups' membership changes most, i.e. minimal edge weight sum between time steps.Data Augmented Solution Reconciliation: given partial sets of observations and a partial solution, find is the combined solution to the entire input.

Close Group: individuals that appear together more than others.Loyal Individuals: appear most frequently in any metagroup.Individual Membership: metagroup which maximizes the cardinality of the set of groups in which a given individual occurs.Extra/Introvert: member of the largest/smallest number of metagroups.Metagroup Representative: an individual who occurs more in a metagroup than any other individual and occurs in it more than in any other metagroup.Demographic Distinction: given a coloring of individuals, is there a property that distinguishes one color from the others? Critical Parameter Values: largest values of α, β for which there exists at least k metagroups. Largest γ for which each metagroup has at least k members.Sampling Rate: largest time step such that the answer does not change if the time step is decreased but changes if it is increased.Critical Time Moments: e.g., the time when the groups' membership changes most, i.e. minimal edge weight sum between time steps.Data Augmented Solution Reconciliation: given partial sets of observations and a partial solution, find is the combined solution to the entire input.


ConclusionsConclusions

New data structure with explicit time component of social interactions

Generic – applicable in many contexts

Powerful – can ask meaningful questions (finding leaders of zebras)

But! (And?) many hard questions – lots of work!

New data structure with explicit time component of social interactions

Generic – applicable in many contexts

Powerful – can ask meaningful questions (finding leaders of zebras)

But! (And?) many hard questions – lots of work!


Credits:

Jared Saia (UNM)Dan Rubenstein (Princeton)

Siva SundaresanIlya Fischoff

Simon Levin (Princeton)S. Muthu Muthukrishnan (Rutgers)David Kempe (USC)

Habiba Habiba (UIC)Mayank Lahiri (UIC)

Chayant Tantipasanandth (UIC)

MicrosoftNSF

© 2006 Board of Trustees of the University of Illinois

Authored by Tanya Berger-Wolf

© 2006 Microsoft Corporation. All rights reserved.Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries.The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation.Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft,and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation.MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

© 2006 board of trustees of the university of illinois authored by tanya berger-wolf analysis of...

Documents