ConstantFactor Approximation Algorithms for Identifying Dynamic Communities Chayant

Constant-Factor Approximation Algorithms for Identifying Dynamic Communities Chayant Tantipathananandh with Tanya Berger-Wolf

Social Networks These are snapshots and networks change over time

Dynamic Networks 2 3 t=2 1 2 3 5 4 5 2 3 4 1 … 5 2 5 5 4 2 3 5 2 1 3 1 • Interactions occur in the form of disjoint groups • Groups are not communities 5 2 t=2 4 4 2 1 1 3 4 Aggregated network t=1 2 3 1 … 1 t=1

Communities • 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 – – – Graph. Scope [Sun et al 2005] Metagroups [Berger-Wolf & Saia 2006] Dynamic Communities [TBK 2007] Clique Percolation [Palla et al 2007] Facet. Net [Lin et al 2009] Bayesian approach [Yang et al 2009]

Ship of Theseus 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] Jeannot's knife “has had its blade changed fifteen times and its handle fifteen times, but is still the same knife. ” [French story]

Ship of Theseus Individual Cost for parts changing never change identity identities …

Ship of Theseus Costs Identity for changes visiting and to match beingthe absent group …

Approach

Community = Color Valid coloring: In each time step, different groups have different colors.

Interpretation Group color: How does community c interact at time t?

Interpretation 2 2 2 1 1 2 2 Individual color: Who belong to community c at time t? 1 1 1

Social Costs: Conservatism 2 α α 2 2 2 α 2 Switching cost α Absence cost β 1 2 2 α 2 Visiting cost β 2

Social Costs: Loyalty 3 2 3 3 β 1 β 1 1 1 Switching cost α Absence cost β 1 2 β 1 3 β 1 Visiting cost β 2

Social Costs: Loyalty β 2 3 β 2 2 Switching cost α Absence cost β 1 3 2 Visiting cost β 2

Problem Complexity • Minimizing total cost is hard NP-complete and APX-hard [with Berger-Wolf and Kempe 2007] • Constant-Factor Approximation [details in paper] • Easy special case If no missing individuals and 2α ≤ β 2 , then simply weighted bipartite matching [details in paper]

Greedy Approximation No visiting or absence and minimizing switching time

Greedy Approximation 3 4 2 ≈ maximizing path coverage 3 Greedy alg guarantees 7 2 No visiting or absence and minimizing switching max{2, 2α/β 1, 4α/β 2} in α, β 1, β 2, independent of input size 3 4 time 3 Improvement by dynamic programming

Southern Women Data Set [DGG 1941] • 18 individuals, 14 time steps • Collected in Natchez, MS, 1935 aggregated network

Ethnography [DGG 1941] Core note: columns not ordered by time

Optimal Communities individuals time ethnography Core all costs equal white circles = unknown

Approximate Optimal time Core ethnography Core

Approximation Power 0, 80 OPT≥ Greedy Grevys Greedy+DP 0, 70 300, 00 OPT≥ Greedy+DP Greedy Plains ≤Guarantee 1, 60 OPT≥ ≤Guarantee 1, 40 250, 00 0, 60 ≤Guarantee 1, 20 0, 40 0, 30 200, 00 total cost (k) 0, 50 total cost (k) Greedy Onagers Greedy+DP 150, 00 100, 00 0, 20 1, 00 0, 80 0, 60 0, 40 50, 00 0, 10 0, 00 0, 20 0, 00 1/3 1/2 1/1 2/1 switch/visit 28 inds, 44 times 3/1 0, 00 1/3 1/2 1/1 2/1 switch/visit 3/1 29 inds, 82 times 1/3 1/2 1/1 2/1 switch/visit 3/1 313 inds, 758 times

Approximation Power 7, 00 OPT≥ 14, 00 OPT≥ 80, 00 OPT≥ 6, 00 12, 00 70, 00 5, 00 10, 00 Greedy+DP Greedy ≤Guarantee Haggle Infocom (264) 4, 00 3, 00 8, 00 6, 00 50, 00 40, 00 30, 00 2, 00 4, 00 1, 00 2, 00 10, 00 1/3 1/2 1/1 2/1 switch/visit 3/1 41 inds, 418 times Greedy+DP Greedy ≤Guarantee Reality Mining 60, 00 total cost (k) Greedy+DP Greedy ≤Guarantee Haggle Infocom (41) 20, 00 1/3 1/2 1/1 2/1 switch/visit 3/1 264 inds, 425 times 1/3 1/2 1/1 2/1 switch/visit 3/1 96 inds, 1577 times

Conclusions • 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, β 1, β 2 – Network snapshots instead of disjoint groups

Thank You NSF grant, KDD student travel award David Kempe Jared Saia Chayant Mayank Lahiri Arun Maiya Ilya Fischoff Tanya Berger-Wolf Habiba Saad Sheikh Dan Rubenstein Siva Sundaresan Robert Grossman Anushka Anand Rajmonda Sulo