A Computational Framework for Analysis

of DynamicSocial Networks

Tanya Berger-WolfUniversity of Illinois

at Chicago

Joint work withJared Saia

University of New Mexico

ZebrasDan Rubenstein, Siva Sandaresan, Ilya Fischhoff (Princeton)

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

Ants Stephen Pratt (Princeton)

People – Hidden GroupsBaumes et al. (RPI)

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 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 biology and behavioral ecologyCroft et.al.’04, Cross et.al.’05, Lusseau&Newman’04

Social Networks: Static vs Dynamic

c

ba ba

c

a b

c

ba

c

ba

c

ba

c

ba

c

1/3

1/31/3 IndividualsStrength or probability

of interaction over a period of time

Advantage of Dynamic Networks:•More accurate information•Time related questions:– How do processes spread through

population? – Who are the individuals that change the

dynamics of interaction (leaders, interaction facilitators, etc.)? How do they emerge?

– How do social structures change with outside circumstances?

– What is the average lifespan of a social structure and are there recurring structures?

Input – Individual Information

b

a

cd

e

f

Individual Information Input – Problem:

Objects within a cluster are closer to eachother than to objects in other clusters

Input – Pairwaise InformationBaumes et al.(RPI) and Washington Post

PentagonPennsylvaniaWTC NorthWTC South

Jan-Dec 2000Jan-Apr 2001May-Jul 2001Aug-Sep 2001

4

93

1

1

file

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

Theseus’s Paradox• During a twelve month period 95% of all

the atoms that make up your 50 trillion cells are replaced

• FAA regulations: airplane = left rudder number

• Ship of Theseus"The ship wherein Theseus and the youth of Athens returned [from Crete] had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same."

A group persist in time (is a metagroup) if some (big)

fraction β of it exists some (big) fraction α of time

•A time snapshot is a partition g1t…gmt

•Similarity measure

•Metagroup is a path of length ≥ α with edges of weight ≥ β

21

2121

2),(jtit

jtit

gg

ggjtit ggsim

2/3 2/3 2/3 2/3 2/3

2/5

3/5 3/5

2/5 1/51/2

1/3

1 2 3 4

5

6789

10

Time step = 1 second

1234

2345

3/4

3456

3/4

4567

3/4

5678

3/4

Time step = 4 seconds

1234

5678

0

91012

0

3456

0

78910

0

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:• 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.

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

g1 gl-1g2 gl

…

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

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

Individual ConnectivityGiven 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 SetThe 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 SetNP-hard: 2-path shattering

set = independent set

?Polynomial: T-path shattering

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

k=2

k=T

Critical Individual SetThe 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:•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.

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 algorithmic questions – lots of work!

Credits:Jared SaiaDan RubensteinSiva SundaresanIlya FischoffSimon LevinS. Muthu MuthukrishnanMartin Pal