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 Sophia Antipolis, France 3Inria, France

Universidad Adolfo Ib` a˜ nez 2/21 Context 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 Context 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 Context 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 in Social Networks 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 Local dynamics at stake 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 Main problems 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 Main problems 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 Related work Network formation games Structural balance theory [Heider, 1946] → signed graphs ( friends or enemies )

Universidad Adolfo Ib` a˜ nez 6/21 Related work 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 Related work 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 Studying transient networks [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 Example of deviations Green edges ⇐ ⇒ positive interactions initially: no edges All missing edges ⇐ ⇒ negative interactions then one by one (if beneficial): (1) leave a community Figures ⇐ ⇒ size of community (2) join/create a community 0 0 0 0 0 0 0 0 0 0 0 0

Universidad Adolfo Ib` a˜ nez 8/21 Example of deviations Green edges ⇐ ⇒ positive interactions initially: no edges All missing edges ⇐ ⇒ negative interactions then one by one (if beneficial): (1) leave a community Figures ⇐ ⇒ size of community (2) join/create a community

Universidad Adolfo Ib` a˜ nez 8/21 Example of deviations Green edges ⇐ ⇒ positive interactions initially: no edges All missing edges ⇐ ⇒ negative interactions then one by one (if beneficial): (1) leave a community Figures ⇐ ⇒ size of community (2) join/create a community

Universidad Adolfo Ib` a˜ nez 8/21 Example of deviations Green edges ⇐ ⇒ positive interactions initially: no edges All missing edges ⇐ ⇒ negative interactions then one by one (if beneficial): (1) leave a community Figures ⇐ ⇒ size of community (2) join/create a community

Universidad Adolfo Ib` a˜ nez 8/21 Example of deviations Green edges ⇐ ⇒ positive interactions initially: no edges All missing edges ⇐ ⇒ negative interactions then k by k (if beneficial): (1) leave a community Figures ⇐ ⇒ size of community (2) join/create a community What about coalitions ?

Universidad Adolfo Ib` a˜ nez 8/21 Example of deviations Green edges ⇐ ⇒ positive interactions initially: no edges All missing edges ⇐ ⇒ negative interactions then k by k (if beneficial): (1) leave a community Figures ⇐ ⇒ size of community (2) join/create a community What about coalitions ?

Universidad Adolfo Ib` a˜ nez 9/21 Local process and individual optimization 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 Local process and individual optimization 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 • The partition representing communities is k -stable iff, there is no k -deviation. • A graph is called k -stable when there exists a k -stable partition.

Universidad Adolfo Ib` a˜ nez 9/21 Local process and individual optimization 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 • The partition representing communities is k -stable iff, there is no k -deviation. • A graph is called k -stable when there exists a k -stable partition. Existence ? Time of convergence ?

Universidad Adolfo Ib` a˜ nez 10/21 Prior work Theorem For all fixed k, the game dynamic converges to a k-stable partition. The maximum number of k-deviations before converging: k Literature O ( n 2 ) 1 O ( n 2 ) 2 O ( n 3 ) 3 ≥ 4 O (2 n ) Results were found by Jon M. Kleinberg and Katrina Ligett, using potential functions.

Universidad Adolfo Ib` a˜ nez 11/21 Our results The maximum number of k-deviations before converging k Literature Contributions O ( n 2 ) ∼ 2 3 n 3 / 2 1 O ( n 2 ) ∼ 2 3 n 3 / 2 2 O ( n 3 ) Ω( n 2 ) 3 Ω( n c ln( n ) ), O ( e √ n ) ≥ 4 O (2 n ) Resolving of a conjecture from Jon M. Kleinberg and Katrina Ligett.

Universidad Adolfo Ib` a˜ nez 12/21 Next step: extending the model 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 Modeling the Social Network with an edge-weighted graph → The (positive, or zero, or negative) weight of an edge represents what both users receive when they are in the same community. - ∞ u w 4 3 v

Universidad Adolfo Ib` a˜ nez 14/21 Communities partition users 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 Local process revisited 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 Local process revisited 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 • The partition representing communities is k -stable iff, there is no k -deviation. • A graph is called k -stable when there exists a k -stable partition.

Universidad Adolfo Ib` a˜ nez 15/21 Local process revisited 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 • The partition representing communities is k -stable iff, there is no k -deviation. • A graph is called k -stable when there exists a k -stable partition. Existence ? Time of convergence ?

Universidad Adolfo Ib` a˜ nez 16/21 Our contribution: counter-examples to stability 1-stable partition but no 2-stable partition exists.

Universidad Adolfo Ib` a˜ nez 16/21 Our contribution: counter-examples to stability 1-stable partition but no 2-stable partition exists. → Importance of the weights ?

Universidad Adolfo Ib` a˜ nez 17/21 Results → W = fixed set of weights → k ( W ) = max. k s.t. all graphs with weights in W are k -stable.