Convergence of Coloring Games with Collusions Augustin Chaintreau 1 - - PowerPoint PPT Presentation
Universidad Adolfo Ib` a nez 1/21 Convergence of Coloring Games with Collusions Augustin Chaintreau 1 Guillaume Ducoffe 2 Dorian Mazauric 3 1Columbia University in the City of New York 2Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, 06900
Universidad Adolfo Ib` a˜ nez 1/21
Augustin Chaintreau 1 Guillaume Ducoffe 2 Dorian Mazauric 3
1Columbia University in the City of New York
3Inria, France
Universidad Adolfo Ib` a˜ nez 2/21
Object of study: evolution over time of the social networks → creation/removal of social ties (= edges in the social graph)
Universidad Adolfo Ib` a˜ nez 2/21
Object of study: evolution over time of the social networks → creation/removal of social ties (= edges in the social graph) Information-sharing in social networks → cornerstone of social network formation → an edge between two nodes ⇐ ⇒ information-sharing between two users → in this work: only one information flow considered
Universidad Adolfo Ib` a˜ nez 2/21
Object of study: evolution over time of the social networks → creation/removal of social ties (= edges in the social graph) Information-sharing in social networks → cornerstone of social network formation → an edge between two nodes ⇐ ⇒ information-sharing between two users → in this work: only one information flow considered Privacy → keep private content produced/received → who receives the content? (how is the graph constructed ?)
Universidad Adolfo Ib` a˜ nez 3/21
Communities = groups of connected users. → every user is in only one community (= ⇒ partition) → users share information ⇐ ⇒ they are in the same community → maximal clique We focus on: the evolution over time of communities → formation of communities → dynamic
Universidad Adolfo Ib` a˜ nez 4/21
Any user can change her community at any time. → remove incident edges + create new incident edges → selfish users: maximizing individual utility under private preferences. → Local process.
Universidad Adolfo Ib` a˜ nez 5/21
To understand (and to anticipate) the dynamics that shape communities. → Can the dynamics stop ? (stable partitions) → How long to converge ?
Universidad Adolfo Ib` a˜ nez 5/21
To understand (and to anticipate) the dynamics that shape communities. → Can the dynamics stop ? (stable partitions) → How long to converge ? To study the impact of selfishness on the local process. → Measurement on global utility → Incentive to better choices for the users
Universidad Adolfo Ib` a˜ nez 6/21
Network formation games Structural balance theory [Heider, 1946] → signed graphs (friends or enemies)
Universidad Adolfo Ib` a˜ nez 6/21
Network formation games Structural balance theory [Heider, 1946] Theorem After a graph has evolved to avoid ”forbidden” triangles, the users are partitioned in one (or a few) rival communities.
Universidad Adolfo Ib` a˜ nez 6/21
Network formation games Structural balance theory [Heider, 1946] Theorem After a graph has evolved to avoid ”forbidden” triangles, the users are partitioned in one (or a few) rival communities.
Universidad Adolfo Ib` a˜ nez 7/21
[Kleinberg and Ligett] → Signed edges = weights -∞, 1 → No assumption on the graph → Individual goals: choosing a community:
with no enemies; the largest possible.
Universidad Adolfo Ib` a˜ nez 8/21
Green edges ⇐ ⇒ positive interactions All missing edges ⇐ ⇒ negative interactions Figures ⇐ ⇒ size of community initially: no edges then one by one (if beneficial): (1) leave a community (2) join/create a community
Universidad Adolfo Ib` a˜ nez 8/21
Green edges ⇐ ⇒ positive interactions All missing edges ⇐ ⇒ negative interactions Figures ⇐ ⇒ size of community initially: no edges then one by one (if beneficial): (1) leave a community (2) join/create a community
Universidad Adolfo Ib` a˜ nez 8/21
Green edges ⇐ ⇒ positive interactions All missing edges ⇐ ⇒ negative interactions Figures ⇐ ⇒ size of community initially: no edges then one by one (if beneficial): (1) leave a community (2) join/create a community
Universidad Adolfo Ib` a˜ nez 8/21
Green edges ⇐ ⇒ positive interactions All missing edges ⇐ ⇒ negative interactions Figures ⇐ ⇒ size of community initially: no edges then one by one (if beneficial): (1) leave a community (2) join/create a community
Universidad Adolfo Ib` a˜ nez 8/21
Green edges ⇐ ⇒ positive interactions All missing edges ⇐ ⇒ negative interactions Figures ⇐ ⇒ size of community What about coalitions ? initially: no edges then k by k (if beneficial): (1) leave a community (2) join/create a community
Universidad Adolfo Ib` a˜ nez 8/21
Green edges ⇐ ⇒ positive interactions All missing edges ⇐ ⇒ negative interactions Figures ⇐ ⇒ size of community What about coalitions ? initially: no edges then k by k (if beneficial): (1) leave a community (2) join/create a community
Universidad Adolfo Ib` a˜ nez 9/21
Definition k-deviation ⇐ ⇒ any subset of ≤ k users joining the same community —or creating a new one— so that all the users in the subset increase their utility.
Universidad Adolfo Ib` a˜ nez 9/21
Definition k-deviation ⇐ ⇒ any subset of ≤ k users joining the same community —or creating a new one— so that all the users in the subset increase their utility. Definition
Universidad Adolfo Ib` a˜ nez 9/21
Definition k-deviation ⇐ ⇒ any subset of ≤ k users joining the same community —or creating a new one— so that all the users in the subset increase their utility. Definition
Existence ? Time of convergence ?
Universidad Adolfo Ib` a˜ nez 10/21
Theorem For all fixed k, the game dynamic converges to a k-stable partition. The maximum number of k-deviations before converging: k Literature 1 O(n2) 2 O(n2) 3 O(n3) ≥ 4 O(2n)
Results were found by Jon M. Kleinberg and Katrina Ligett, using potential functions.
Universidad Adolfo Ib` a˜ nez 11/21
The maximum number of k-deviations before converging k Literature Contributions 1 O(n2) ∼ 2
3n3/2
2 O(n2) ∼ 2
3n3/2
3 O(n3) Ω(n2) ≥ 4 O(2n) Ω(nc ln(n)), O(e
√n)
Resolving of a conjecture from Jon M. Kleinberg and Katrina Ligett.
Universidad Adolfo Ib` a˜ nez 12/21
some drawbacks of Kleinberg and Ligett’s model: − → no neutral interaction (= complete signed graphs) − → realistic only for small-size networks − → weight uniformity: no best friend, no worst enemy
Universidad Adolfo Ib` a˜ nez 13/21
→The (positive, or zero, or negative) weight of an edge represents what both users receive when they are in the same community.
Universidad Adolfo Ib` a˜ nez 14/21
The utility of user u equals the sum of the weights of the edges between herself and the other users in her community.
Universidad Adolfo Ib` a˜ nez 15/21
Definition k-deviation ⇐ ⇒ any subset of ≤ k users joining the same community —or creating a new one— so that all the users in the subset increase their utility.
Universidad Adolfo Ib` a˜ nez 15/21
Definition k-deviation ⇐ ⇒ any subset of ≤ k users joining the same community —or creating a new one— so that all the users in the subset increase their utility. Definition
Universidad Adolfo Ib` a˜ nez 15/21
Definition k-deviation ⇐ ⇒ any subset of ≤ k users joining the same community —or creating a new one— so that all the users in the subset increase their utility. Definition
Existence ? Time of convergence ?
Universidad Adolfo Ib` a˜ nez 16/21
1-stable partition but no 2-stable partition exists.
Universidad Adolfo Ib` a˜ nez 16/21
1-stable partition but no 2-stable partition exists. → Importance of the weights ?
Universidad Adolfo Ib` a˜ nez 17/21
→ W = fixed set of weights → k(W) = max. k s.t. all graphs with weights in W are k-stable.
Universidad Adolfo Ib` a˜ nez 17/21
→ W = fixed set of weights → k(W) = max. k s.t. all graphs with weights in W are k-stable. Theorem ∀W, k(W) ≥ 1.
Universidad Adolfo Ib` a˜ nez 17/21
→ W = fixed set of weights → k(W) = max. k s.t. all graphs with weights in W are k-stable. Theorem ∀W, k(W) ≥ 1. Beyond 1-stability: characterization of k-stable graphs W k(W) {−∞, a, b}, 0 < a < b 1 {−∞, 0, 1} 2 {−∞, 1} (uniform case) ∞
Universidad Adolfo Ib` a˜ nez 18/21
Theorem ∀k, counter-example to k-stability ⇐ ⇒ deciding k-stability is NP-complete
Universidad Adolfo Ib` a˜ nez 18/21
Sketch: embedding of the counter-example into a supergraph. to “break” the counter-example: need of a large clique in the network − → reduction to Maximum Clique Problem
K3 K3 K3 K3 K3
G1
x0
G0 G0\x0
G2
Universidad Adolfo Ib` a˜ nez 19/21
Question: are k-stable partitions “useful” ? fixed sets of weights. Chosen metrics = global utility = Σ individual utilities
Universidad Adolfo Ib` a˜ nez 20/21
Definition p(n, k) ⇐ ⇒ price of anarchy ⇐ ⇒ best utility for any partition/ worst utility for a k-stable partition
Universidad Adolfo Ib` a˜ nez 20/21
Definition p(n, k) ⇐ ⇒ price of anarchy ⇐ ⇒ best utility for any partition/ worst utility for a k-stable partition Theorem p(n, 1) = ∞ !
Universidad Adolfo Ib` a˜ nez 20/21
Definition p(n, k) ⇐ ⇒ price of anarchy ⇐ ⇒ best utility for any partition/ worst utility for a k-stable partition Theorem p(n, 1) = ∞ ! Theorem For any k ≥ 2, Ω(n/k) ≤ p(n, k) ≤ O(n).
Universidad Adolfo Ib` a˜ nez 20/21
Definition p(n, k) ⇐ ⇒ price of anarchy ⇐ ⇒ best utility for any partition/ worst utility for a k-stable partition Theorem p(n, 1) = ∞ ! Theorem For any k ≥ 2, Ω(n/k) ≤ p(n, k) ≤ O(n). → The price of anarchy improves as the number of stable partitions decreases.
Universidad Adolfo Ib` a˜ nez 21/21
Extending the model → Asymmetrical weights (Twitter) → Transitive weights: modelization using hypergraphs → Overlapping within communities
Universidad Adolfo Ib` a˜ nez 21/21
Extending the model → Asymmetrical weights (Twitter) → Transitive weights: modelization using hypergraphs → Overlapping within communities (Distributed) Algorithmic → Existence of a k-stable partition → Local algorithm for computing a k-stable partition → Price of stability and price of anarchy → Incentive process
Universidad Adolfo Ib` a˜ nez 21/21
Extending the model → Asymmetrical weights (Twitter) → Transitive weights: modelization using hypergraphs → Overlapping within communities (Distributed) Algorithmic → Existence of a k-stable partition → Local algorithm for computing a k-stable partition → Price of stability and price of anarchy → Incentive process To understand and to improve the existing social networks