A Computational Framework for Analysis of Dynamic Social Networks Tanya Berger-Wolf University of Illinois at Chicago Joint work with Jared Saia University of New Mexico

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

Ants Stephen Pratt (Princeton)

People – Hidden Groups Baumes 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 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 biology and behavioral ecology Croft et.al.’04, Cross et.al.’05, Lusseau&Newman’04

Social Networks: Static vs Dynamic c b a b a c a b c b a c b a c b a c b a c 1/3 Individuals Strength 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 cdef

Individual Information Input – Problem: Objects within a cluster are closer to each other than to objects in other clusters

Input – Pairwaise Information Baumes et al.(RPI) and Washington PostWashington Post Pentagon Pennsylvania WTC North WTC South Jan-Dec 2000 Jan-Apr 2001May-Jul 2001Aug-Sep 2001

file

t= t= t= 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."Ship of Theseus

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 g 1t …g mt Similarity measure Metagroup is a path of length ≥ α with edges of weight ≥ β

2/3 2/5 3/5 2/5 1/5 1/2 1/3

Time step = 1 second / Time step = 4 seconds

1/10 1/9 1/ t= t= t= t= / /4 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 Connectivity Given groups g 1,…,g l, are they in the same metagroup? g1g1 g l-1 g2g2 glgl … Most persistent/largest/loudest/.. metagroup that contains these groups A metagroup that contains largest number of these groups – dynamic programming

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

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

NP-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 Set The smallest set of individuals whose absence leaves no metagroups (for given α and β) c d c d c d … cd cd a a a aaabbb b b b cd

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 Saia Dan Rubenstein Siva Sundaresan Ilya Fischoff Simon Levin S. Muthu Muthukrishnan Martin Pal