Analysis of Dynamic Social Networks Tanya Berger-Wolf

Department of Computer Science University of Illinois at Chicago © 2006 Board of Trustees of the University of Illinois Authored by Tanya Berger-Wolf

Zebras

Dan Rubenstein, Siva Sandaresan, Ilya Fischhoff (Princeton) Movie credit: “Champions of the Wild”, © Omni-Film Productions.

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

Context

disease modeling

Eubank et.al.‘04, Keeling’99, Kretzschmar&Morris’96

cultural and information transmission

Baumes 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 surveillance

Airoldi&Malin’04,Baumes et.al.’04, Kolata’05, Malin’04, Magdon-Ismail et.al.’03

business management

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

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Social Networks: Static

vs

Dynamic

t=1 t=2 t=3 t=4 t=5 1 2 3 4 5 6 1 2 3 4 5 6 Strength or probability of interaction over a period of time 1/5 1/5 1/5

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

1/5 1/5 1 2 3 4 5 6 Individuals

Input – Individual Information

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

file © Christopher Sadler 4 9

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

© Christopher Sadler 3 © Christopher Sadler 1 1

4 8 1 3 1 1 4 9 3 4 8 t=1 t=2

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

t=3 4 1 t=4

Questions

Communities over time – persistent groups A group persist in time (is a metagroup ) if some (big) fraction β exists some (big) fraction α of time of it 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 gatherings Gatherings that facilitate most (least) spreading Critical times Timesteps when interaction pattern changes Persistent demographic configurations Repeated groups with the same demographic pattern © 2006 Board of Trustees of the University of Illinois. Authored by Tanya Berger-Wolf

4 9 3 1 1 1 8/9 1/2 4 8 1 3 1 1 3/4 4 8 t=1 t=2

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

t=3 1 8/9 1/2 4 1 t=4

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.

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

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

Southern Women Metagroups

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13 .61-.7

.71-.8

.81-.9

.91-1 © 2006 Board of Trustees of the University of Illinois. Authored by Tanya Berger-Wolf

Linton Freeman Metanalysis, 2003

DGG 1941 Intuition Homans Phillips and Conviser Breiger Breiger, Boorman & Arabie 1975 1974 Matrix Algebra Computational Bonacich 1951 Intuition 1972 Information Theory 1978 Boolean Algebra Doreian Bonacich Freeman Everett & Borgatti 1979 Algebraic Topology 1991 Correspondence Analysis 1992 G-Transitivity 1993 Regular Coloring Freeman Freeman & White Borgatti & Everett Skvoretz & Faust Roberts 1993 Genetic Algorithm I & II 1993 Galois Lattices I & II 1997 Bipartite Analyses I, II & III 1999 p* Model 2000 Normalized SVD Osbourn 2000 VERI Procedure Newman 2001 Weighted Proximities © 2006 Board of Trustees of the University of Illinois. Authored by Tanya Berger-Wolf

Southern Women Communities

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

Southern Women: Core vs Periphery

BW&S

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Group Connectivity

Given groups g 1 ,…,g l , are they in the same metagroup?

…

g 1 g 2 g l-1 g l

Most persistent/largest/loudest/.. metagroup that contains these groups A metagroup that contains largest number of these groups – dynamic programming © 2006 Board of Trustees of the University of Illinois. Authored by Tanya Berger-Wolf

Individual Connectivity

Given individuals S={s 1 ,…,s l }, 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 © 2006 Board of Trustees of the University of Illinois. Authored by Tanya Berger-Wolf

Critical Group Set

The smallest set of groups whose absence leaves no metagroups (for given α and β) Formally: remove fewest vertices in a DAG so there are no paths of length > k-1 K-path Vertex Shattering Set © 2006 Board of Trustees of the University of Illinois. Authored by Tanya Berger-Wolf

K-path Vertex Shattering Set

k=2

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

k=T

Polynomial: T-path shattering set (T is the longest path length) – min vertex cut in a DAG © 2006 Board of Trustees of the University of Illinois. Authored by Tanya Berger-Wolf

Critical Individual Set

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

a b cd c d a b a b cd

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

c d a b a b cd c d a b

…

Other questions:

Close Group: individuals that appear together more than others.

Loyal Individuals: appear most frequently in any metagroup.

Individual Membership: of groups in which a given individual occurs.

Extra/Introvert: metagroup which maximizes the cardinality of the set 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:

k

largest values of α, β for which there exists at least 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.

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

Conclusions

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 of work!

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

Credits: Jared Saia (UNM) Dan Rubenstein (Princeton) Siva Sundaresan Ilya Fischoff Simon Levin (Princeton) S. Muthu Muthukrishnan (Rutgers) David Kempe (USC) Habiba Habiba (UIC) Mayank Lahiri (UIC) Chayant Tantipasanandth (UIC)

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

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