Tantipathananandh Chayant Tantipathananandh with Tanya Berger-Wolf Constant-Factor Approximation...
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Transcript of Tantipathananandh Chayant Tantipathananandh with Tanya Berger-Wolf Constant-Factor Approximation...
Chayant TantipathananandhTantipathananandhwith Tanya Berger-Wolf
Constant-Factor Approximation Algorithms for Identifying Dynamic Communities
Constant-Factor Approximation Algorithms for Identifying Dynamic Communities
Dynamic NetworksDynamic Networks
Aggregated networkAggregated network
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4
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11
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•Interactions occur in the form of disjoint groups•Groups are not communities
…t=2
t=1
3322 114455
55 44 3311 22
55 22 33 44 11
55 22 33 44
55 22 44 11
t=1
t=2
55 44
1122
33
CommunitiesCommunities• What is community?
“Cohesive subgroups are subsets of actors among whom there are relatively strong, direct, intense, frequent, or positive ties.” [Wasserman & Faust 1994]
• Dynamic Community Identification– GraphScope [Sun et al 2005]– Metagroups [Berger-Wolf & Saia 2006]– Dynamic Communities [TBK 2007]– Clique Percolation [Palla et al 2007]– FacetNet [Lin et al 2009]– Bayesian approach [Yang et al 2009]
Ship of Theseus Ship of Theseus
Jeannot's knife “has had its blade changed fifteen times and its handle fifteen times, but is still the same knife.” [French story]
Jeannot's knife “has had its blade changed fifteen times and its handle fifteen times, but is still the same knife.” [French story]
from Wikipedia
“The ship … was preserved by the Athenians …, 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.” [Plutarch, Theseus]
“The ship … was preserved by the Athenians …, 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.” [Plutarch, Theseus]
Ship of Theseus Ship of Theseus
…
Individual parts never change identitiesCost for changing
identity
Ship of Theseus Ship of Theseus
…
Identity changes to match the group
Costs for visiting and being absent
Community = ColorCommunity = Color
Valid coloring: In each time step, different groups have different colors.
InterpretationInterpretation
Individual color: Who belong to community c at time t?
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Social Costs: ConservatismSocial Costs: Conservatism
Switching cost α
α
α
α
Absence cost β1 Visiting cost β2
α
α
α
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22
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22
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22
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22
22
22
Social Costs: LoyaltySocial Costs: Loyalty
β1
β1
β1
Absence cost β1 Visiting cost β2Switching cost α
β1
β1
β122 33
33
1111
3322
33
β1
Social Costs: LoyaltySocial Costs: Loyalty
β2
β2
Switching cost α Absence cost β1 Visiting cost β2
22
33
β2 22
β2 33
Problem ComplexityProblem Complexity
• Minimizing total cost is hardNP-complete and APX-hard [with Berger-Wolf and Kempe 2007]
• Constant-Factor Approximation [details in paper]
• Easy special caseIf no missing individuals and 2α ≤ β2 , thensimply weighted bipartite matching[details in paper]
Approximation via bipartite matchingApproximation via bipartite matching
– assume all individuals are observed at all time steps
Greedy ApproximationGreedy Approximation
time
No visiting or absence and minimizing switching
No visiting or absence and minimizing switching
Greedy ApproximationGreedy Approximation
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3
3
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3
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≈ maximizing path coverage ≈ maximizing path coverage
No visiting or absence andminimizing switching
No visiting or absence andminimizing switching
2
Improvement by dynamic programming
Improvement by dynamic programming
Greedy alg guaranteesmax{2, 2α/β1, 4α/β2}
in α, β1, β2, independent of input size
Greedy alg guaranteesmax{2, 2α/β1, 4α/β2}
in α, β1, β2, independent of input size
time
Southern Women Data Set [DGG 1941]Southern Women Data Set [DGG 1941]
• 18 individuals, 14 time steps• Collected in Natchez, MS, 1935
aggregated network
Optimal CommunitiesOptimal Communities
all costs equalwhite circles = unknown
Core Core
time
individuals
ethnography
ConclusionsConclusions
• Identity of objects that change over time (Ship of Theseus Paradox)
• Formulate an optimization problem• Greedy approximation– Fast– Near-optimal
• Future Work– Algorithm with guarantee not depending on α, β1, β2
– Network snapshots instead of disjoint groups
Arun Maiya
Saad Sheikh
Thank YouThank You
NSF grant, KDD student travel award
Habiba
David KempeJared Saia
Mayank Lahiri
Dan Rubenstein
Tanya Berger-Wolf
Rajmonda SuloRobert GrossmanSiva Sundaresan
Ilya Fischoff
Anushka Anand
Chayant
Ravi Kumar, Jasmine Novak, Prabhakar Raghavan, Andrew Tomkins IBM Almaden Research Center
On the Bursty Evolution of BlogspaceOn the Bursty Evolution of Blogspace
BlogspaceBlogspace
• Blogspace• Collection of blogs with their links
• Motivation– Sociological• Different with traditional web page
– Technical • From static snapshot to dynamic graphs
BackgroundBackground
• Web communities (Ravi Kumar,1999)• groups of individuals who share a common interest• characterized by dense directed bipartite subgraphs.
• Bursty communities of blogs• Exhibit striking temporal characteristics• Extract the community within a time interval
Time graphTime graph
• time graph G = (V,E)• v in V has an associated duaration D(v) • e in E is a triple (u, v, t)• t is a time in interval D(u) ∩ D(v).
• prefix of G at time t Gt = (Vt,Et) • Vt= {v in V | D(v) ∩ [0, t] ≠ Ø }
• Et = {(u, v, t) in E| t’ ≤ t}
ApproachApproach
• Two step approach– Community extraction• Extract dense subgraphs( potential communities)
– Bust analysis• analyze each dense subgraph to identfy and rank
bursts in these communities.
Community extractionCommunity extraction
• Finding the densest subgraph: NP-hard• Two steps:– Pruning• Remove vertices of degree no more than one• Vertices of degree two are K3
g• Output and remove communities (pass a threshold)• Repeat the 3 steps above
– Expanding• Determines the vertex containing the most links• Add it to the community If the links is larger than tk.
Burst analysisBurst analysis
• Kleinberg’s method (SIGKDD 2002)• model the generation of events by an automaton
– one of two states, “low” and “high.” high state is hypothesized as generating bursts of events.
• a cost is associated with any state transition to discourage short bursts.• find a low cost state sequence that is likely to generate
the stream.• solves the problem of enumerating all the bursts by
order of weight( dynamic programming)
Tuning the algorithmsTuning the algorithms
• Expansion in community extraction• Edges must grow to triangles; • communities of size up to six will only grow vertices
that link to all but one vertex; • Communities of size up to nine will only grow vertices
that link to all but two vertices; • communities up to size 20 will grow only vertices that
link to 70% of the community; • larger communities will grow only vertices that link to
at least 60% of the community