Universidad Adolfo Ib anez~ 1/21 - Inria
Transcript of Universidad Adolfo Ib anez~ 1/21 - Inria
Universidad Adolfo Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 6/21
Related work
Network formation games
Structural balance theory [Heider, 1946]
→ signed graphs (friends or enemies)
Universidad Adolfo Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 8/21
Example of deviations
Green edges ⇐⇒ positive interactions
All missing edges ⇐⇒ negative interactions
Figures ⇐⇒ size of community
What about coalitions ?
initially: no edges
then one by one (if beneficial):
(1) leave a community
(2) join/create a community
0 0 0
000
00
0 0
0
0
Universidad Adolfo Ibanez 8/21
Example of deviations
Green edges ⇐⇒ positive interactions
All missing edges ⇐⇒ negative interactions
Figures ⇐⇒ size of community
What about coalitions ?
initially: no edges
then one by one (if beneficial):
(1) leave a community
(2) join/create a community
Universidad Adolfo Ibanez 8/21
Example of deviations
Green edges ⇐⇒ positive interactions
All missing edges ⇐⇒ negative interactions
Figures ⇐⇒ size of community
What about coalitions ?
initially: no edges
then one by one (if beneficial):
(1) leave a community
(2) join/create a community
Universidad Adolfo Ibanez 8/21
Example of deviations
Green edges ⇐⇒ positive interactions
All missing edges ⇐⇒ negative interactions
Figures ⇐⇒ size of community
What about coalitions ?
initially: no edges
then one by one (if beneficial):
(1) leave a community
(2) join/create a community
Universidad Adolfo Ibanez 8/21
Example of deviations
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 Ibanez 8/21
Example of deviations
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 Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 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
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 Ibanez 11/21
Our results
The maximum number of k-deviations before converging
k Literature Contributions
1 O(n2) ∼ 23n3/2
2 O(n2) ∼ 23n3/2
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 Ibanez 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 Ibanez 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
v
4 3
-∞
Universidad Adolfo Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 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 Ibanez 16/21
Our contribution: counter-examples to stability
1-stable partition but no 2-stable partition exists.
→ Importance of the weights ?
Universidad Adolfo Ibanez 16/21
Our contribution: counter-examples to stability
1-stable partition but no 2-stable partition exists.
→ Importance of the weights ?
Universidad Adolfo Ibanez 17/21
Results
→ 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 Ibanez 17/21
Results
→ 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 Ibanez 17/21
Results
→ 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 Ibanez 18/21
A consequence on the complexity
Theorem
∀k, counter-example to k-stability ⇐⇒ deciding k-stability is NP-complete
K3 K3
K3 K3
K3
G1
x0
G0G0\x0
G2
Universidad Adolfo Ibanez 18/21
A consequence on the complexity
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
G0G0\x0
G2
Universidad Adolfo Ibanez 19/21
Optimality
Question: are k-stable partitions “useful” ?
fixed sets of weights.
Chosen metrics = global utility = Σ individual utilities
Universidad Adolfo Ibanez 20/21
Our results
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 Ibanez 20/21
Our results
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 Ibanez 20/21
Our results
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 Ibanez 20/21
Our results
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 Ibanez 21/21
Perspectives
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
Universidad Adolfo Ibanez 21/21
Perspectives
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
Universidad Adolfo Ibanez 21/21
Perspectives
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