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, MagdonIsmail 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
a
b
a
a
c
c
a
b
b
a
Strength or probability
of interaction
over a period of time
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b
c
c
a
1/3
1/3
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b
c
b
1/3
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b
Individuals
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
f
a
e
c
d
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 Post
Jan-Dec 2001
2000
Aug-Sep
Jan-Apr
May-Jul
2001
Pentagon
Pennsylvania
WTC North
WTC South
file
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9
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t=3
t=4
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t=1
t=2
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1
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 sim ( git1 , g jt2 ) 
2 git1  g jt2
git1  g jt2
•Metagroup is a path of length ≥ α with
edges of weight ≥ β
2/3
2/3
2/3
2/3
2/3
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10
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Time
Time step
step =
=4
1 seconds
second
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t=1
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11
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β=.8 t=3
t=2 β=.5
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.
Group Connectivity
Given groups g1,…,gl, are they in the
same metagroup?
…
g1
g2
gl-1
gl
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={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 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
K-path Vertex Shattering Set
NP-hard: 2-path shattering set
k=2
= independent set
?
k=T
Polynomial: T-path shattering set
(T is the longest path length) –
min vertex cut in a DAG
Critical Individual Set
The smallest set of individuals whose
absence leaves no metagroups (for
given α and β)
a
b
a b cd
ab cd
a
a
b
b
ab cd
c
c
c
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d
…
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