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

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 ?)

Context

Privacy

Context

Privacy

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

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.

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

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

Related work

Network formation games

Structural balance theory [Heider, 1946]

→ signed graphs (friends or enemies)

Related work

Theorem

After a graph has evolved to avoid ”forbidden” triangles, the users are

partitioned in one (or a few) rival communities.

Related work

Theorem

After a graph has evolved to avoid ”forbidden” triangles, the users are

partitioned in one (or a few) rival communities.

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.

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

initially: no edges

then k by k (if beneficial):

initially: no edges

then k by k (if beneficial):

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 ?

Definition

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.

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√

Resolving of a conjecture from Jon M. Kleinberg and Katrina Ligett.

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

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.

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.

Local process revisited

Definition

Our contribution: counter-examples to stability

1-stable partition but no 2-stable partition exists.

→ Importance of the weights ?

Our contribution: counter-examples to stability

1-stable partition but no 2-stable partition exists.

→ Importance of the weights ?

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) ∞

Results

Theorem

∀W, k(W) ≥ 1.

W k(W)

−∞, a, b, 0 < a < b 1

−∞, 0, 1 2

Results

Theorem

∀W, k(W) ≥ 1.

W k(W)

−∞, a, b, 0 < a < b 1

−∞, 0, 1 2

A consequence on the complexity

Theorem

∀k, counter-example to k-stability ⇐⇒ deciding k-stability is NP-complete

G0G0\x0

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

G0G0\x0

Optimality

Question: are k-stable partitions “useful” ?

fixed sets of weights.

Chosen metrics = global utility = Σ individual utilities

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.

Our results

Definition

Theorem

p(n, 1) =∞ !

Theorem

Our results

Definition

Theorem

p(n, 1) =∞ !

Theorem

Our results

Definition

Theorem

p(n, 1) =∞ !

Theorem

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

Perspectives

Extending the model

Perspectives

Extending the model

Universidad Adolfo Ib anez~ 1/21 - Inria

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