network creation games - heinz nixdorf institut · pdf filenetwork creation games a game...

Network Creation GamesA Game Theoretic Approach to Network Evolution

November 14th, 2013

Andreas Cord-Landwehr

Source: Fotolia, Jürgen Priewe

Network Creation Games Andreas Cord-Landwehr 1

HEINZ NIXDORF INSTITUTEUniversity of PaderbornAlgorithms and Complexity

Motivation: We want to understand the creation of Internet-like networks by selfishagents, without central design.

Concrete Question: How costly is the lack of coordination?

Model Ideaeach network peer perceived as agentpeers work strategically to improve their own utilitiesutility given by connection maintenance costs plus communication costs

Understand the Evolution of the Internet. . . or a little bit of it

Definition (Network Creation Game [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03])

players: 0, 1, . . . , n− 1 =: [n]

strategy space of a player i: Si := 2[n]\i

network graph G[s] that results from the joint strategiess := (s0, . . . , sn−1) ∈ S0 × . . .× Sn−1 is given by

1 nodes [n]

2 edgesi, j ∈ [n]× [n] : j ∈ si ∨ i ∈ sj

=⋃n−1

i=0 (i × si)cost of a player i under the joint strategy s, α > 0:

ci(s) = α · |si |+n−1∑i=0

dG[s](i, j)

(Pure) Nash equilibrium: given by joint strategy s ∈ S0 × . . .× Sn−1 s.t. for eachplayer i and for all s′ ∈ S0 × . . .× Sn−1 with s and s′ to differ only in the i-thcomponent it holds ci(s) ≤ ci(s′).

Network Creation Games by Fabrikant et al.The Model

Definition (Network Creation Game [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03])

players: 0, 1, . . . , n− 1 =: [n]

strategy space of a player i: Si := 2[n]\i

network graph G[s] that results from the joint strategiess := (s0, . . . , sn−1) ∈ S0 × . . .× Sn−1 is given by

1 nodes [n]

2 edgesi, j ∈ [n]× [n] : j ∈ si ∨ i ∈ sj

=⋃n−1

i=0 (i × si)cost of a player i under the joint strategy s, α > 0:

ci(s) = α · |si |+n−1∑i=0

dG[s](i, j)

(Pure) Nash equilibrium: given by joint strategy s ∈ S0 × . . .× Sn−1 s.t. for eachplayer i and for all s′ ∈ S0 × . . .× Sn−1 with s and s′ to differ only in the i-thcomponent it holds ci(s) ≤ ci(s′).

Network Creation Games by Fabrikant et al.The Model

NCG with α = 1.5

Network Creation Games by Fabrikant et al.Example

NCG with α = 1.5

Best Reponses are NP-hardGiven s ∈ S0 × . . .× Sn−1 and i ∈ [n] it is NP-hard to compute the best response forplayer i.

Sketch. We perform reduction from Dominating Set (NP-hard). Assume we hadan algorithm BRi(G[s]) that computes the best response edge selection for node igiven strategy profile graph.

Dominating Set (DS): Given undirected, unweighted graph G = (V ,E). Task is tocompute set D ⊆ V of minimal size s.t. ∀ v ∈ V \ D : ∃ u ∈ D ∧ (v, u) ∈ E

Consider instance G = (V ,E) for DS, then:1 add node n + 1 to G and define α := 1.5 (any value in (1, 2) works)2 D← BRn+1(G)

3 return DResult is dominating set (proof this!) ⇒ BR is NP-hard.

Network Creation Games by Fabrikant et al.Best Response Computation

Best Reponses are NP-hardGiven s ∈ S0 × . . .× Sn−1 and i ∈ [n] it is NP-hard to compute the best response forplayer i.

Sketch. We perform reduction from Dominating Set (NP-hard). Assume we hadan algorithm BRi(G[s]) that computes the best response edge selection for node igiven strategy profile graph.

Dominating Set (DS): Given undirected, unweighted graph G = (V ,E). Task is tocompute set D ⊆ V of minimal size s.t. ∀ v ∈ V \ D : ∃ u ∈ D ∧ (v, u) ∈ E

Consider instance G = (V ,E) for DS, then:1 add node n + 1 to G and define α := 1.5 (any value in (1, 2) works)2 D← BRn+1(G)

3 return DResult is dominating set (proof this!) ⇒ BR is NP-hard.

Network Creation Games by Fabrikant et al.Best Response Computation

Social costs: c(G) =∑

i∈V ci = α · |E|+∑

i,j dG(i, j)

LemmaFor any strategy profile s, we have the social cost lower bound:

c(G[s]) ≥ 2n(n− 1) + (α− 2)|E|

Proof. Note that 2|E| many distances are exactly 1 and hence n(n− 1)− 2|E|distances are at least of length 2. This can be used for the following estimation:

c(G[s]) = α|E|+∑i,j

dG[s](i, j)

≥ α|E|+ 2|E|+ 2[n(n− 1)− 2|E|]= 2n(n− 1) + (α− 2)|E|

Network Creation Games by Fabrikant et al.Social Cost Lower Bound

i∈V ci = α · |E|+∑

i,j dG(i, j)

c(G[s]) ≥ 2n(n− 1) + (α− 2)|E|

c(G[s]) = α|E|+∑i,j

dG[s](i, j)

≥ α|E|+ 2|E|+ 2[n(n− 1)− 2|E|]= 2n(n− 1) + (α− 2)|E|

i∈V ci = α · |E|+∑

i,j dG(i, j)

c(G[s]) ≥ 2n(n− 1) + (α− 2)|E|

c(G[s]) = α|E|+∑i,j

dG[s](i, j)

≥ α|E|+ 2|E|+ 2[n(n− 1)− 2|E|]= 2n(n− 1) + (α− 2)|E|

TheoremFor α < 1 the price of anarchy is 1.

Claim 1: optimal solution is complete graphLooking at social costs

c(G) = α · |E|+∑i,j

dG(i, j)

sum minimized when |E| is maximized.

Claim 2: every Nash equilibrium has diameter of 1Assume there exists an NE that is not a complete graph:

then there is a “non-existing edge” that can be bought by a player i to improve itsprivate costi.e.: if there is a non-existing edge i, j we have d(i, j) ≥ 2by adding j to si we get c(i)− c′(i) > 0 and hence a contradiction, since α < 1

Hence, the only NE is the complete graph.

NCG Price of Anarchy BoundsSimple Bounds (1/3)

TheoremFor α < 1 the price of anarchy is 1.

Claim 1: optimal solution is complete graphLooking at social costs

c(G) = α · |E|+∑i,j

dG(i, j)

sum minimized when |E| is maximized.

Claim 2: every Nash equilibrium has diameter of 1Assume there exists an NE that is not a complete graph:

then there is a “non-existing edge” that can be bought by a player i to improve itsprivate costi.e.: if there is a non-existing edge i, j we have d(i, j) ≥ 2by adding j to si we get c(i)− c′(i) > 0 and hence a contradiction, since α < 1

Hence, the only NE is the complete graph.

TheoremFor 1 ≤ α < 2 the price of anarchy is 4/3.

Claim 1: optimal solution is complete graphConsider lower bound c(G[s]) ≥ 2n(n− 1) + (α− 2)|E| that is reached for completegraph.Claim 2: every Nash equilibrium has diameter at most 2Assume ∃ NE with nodes u, v s.t. d(u, v) > 2⇒ su ∪ (u, v) is IR for u⇒ contradiction

Compute worst case ratio:

PoA =C(Star)C(Kn)

=(n− 1)α + (n− 1)(n− 2) · 22 + 2(n− 1)

α · n(n−1)2 + n(n− 1)

=(n− 1)(α− 2 + 2n)

(n− 1)n(α/2 + 1)

nα+22

+α− 2

n(α/2 + 2)<

4α + 2 ≤ 4/3

PoA =C(Star)C(Kn)

=(n− 1)α + (n− 1)(n− 2) · 22 + 2(n− 1)

α · n(n−1)2 + n(n− 1)

=(n− 1)(α− 2 + 2n)

(n− 1)n(α/2 + 1)

nα+22

+α− 2

n(α/2 + 2)<

4α + 2 ≤ 4/3

PoA =C(Star)C(Kn)

=(n− 1)α + (n− 1)(n− 2) · 22 + 2(n− 1)

α · n(n−1)2 + n(n− 1)

=(n− 1)(α− 2 + 2n)

(n− 1)n(α/2 + 1)

nα+22

+α− 2

n(α/2 + 2)<

4α + 2 ≤ 4/3

TheoremFor α > n2 all Nash equilibria are trees and the price of anarchy is constant.

Claim 1: for α > n2 all Nash equilibria are trees

assume there exists a NE that contains a circle v0, . . . , vk , vk+1, . . . , v0 andconsider strategy of v0by removing (v0, v1), G still connected, max-distance improvement is n− 1cost change for v0 is at most (n− 1)(n− 1)− n2 < 0removing edge was IR

Claim 2: for trees the PoA is constant

worst case cost of any tree:c(Tree) = α · |E|+

∑i,j dG(i, j) ≤ α(n− 1) + (n− 1)n2 ≤ 2α(n− 1)

optimum cost:c(Star) = α(n− 1) + (n− 1)(n− 2) + 2(n− 1)

TheoremFor α > n2 all Nash equilibria are trees and the price of anarchy is constant.

Claim 1: for α > n2 all Nash equilibria are trees

assume there exists a NE that contains a circle v0, . . . , vk , vk+1, . . . , v0 andconsider strategy of v0by removing (v0, v1), G still connected, max-distance improvement is n− 1cost change for v0 is at most (n− 1)(n− 1)− n2 < 0removing edge was IR

Claim 2: for trees the PoA is constant

worst case cost of any tree:c(Tree) = α · |E|+

∑i,j dG(i, j) ≤ α(n− 1) + (n− 1)n2 ≤ 2α(n− 1)

optimum cost:c(Star) = α(n− 1) + (n− 1)(n− 2) + 2(n− 1)

TheoremFor α < n2 the price of anarchy is O(

√α).

Proof. (consider only n2 > α ≥ 2)Claim 1: PoA ∈ O(α|E|+n2

αn+n2 ).1 star is social cost optimum: our cost lower bound is reached when |E| is

minimized⇒ the social optimum has cost Θ(αn + n2)

2 social cost of every NE at most α|E|+ n2 · 2√α

assume ∃ u, v ∈ V with dG(u, v) ≥ 2√α and consider u creating edge to v

u’s costs change by:

≤ α−b√αc∑

i=12i = α− 2

√α(√α+ 1)

this is contradiction and hence diameter < 2√α

Using this, we can compute the price of anarchy as:α|E|+

∑i,j d(i, j)

(n− 1)(α + 2n− 2)≤ α|E|+ n2 · 2

αn + n2 = O(α|E|+ n2α

αn + n2

NCG Price of Anarchy BoundsA bound for α < n2 (1/4)

√α).

≤ α−b√αc∑

i=12i = α− 2

√α(√α+ 1)

∑i,j d(i, j)

(n− 1)(α + 2n− 2)≤ α|E|+ n2 · 2

αn + n2 = O(α|E|+ n2α

αn + n2

√α).

≤ α−b√αc∑

i=12i = α− 2

√α(√α+ 1)

∑i,j d(i, j)

(n− 1)(α + 2n− 2)≤ α|E|+ n2 · 2

αn + n2 = O(α|E|+ n2α

αn + n2

Claim 2: |E| = O(n2/√α)

(Idea: we associate α-many non-edges with each existing edge.)

Consider arbitrary v ∈ V with own edges e1, . . . , el and define sets Ti :

Ti := u ∈ V |ei ∈ shortest canonical path from v to u

shortest canonical path: arbitrary fixed chosen shortest path for every pair (ensuresthat Tis are disjoint)

The Plan: Consider G′ := (V ,E \ ei) and analyze cases:1 Ti and v are connected in G′

2 Ti and v are not connected in G′

Note: for any path v to u in G′ its length is either∞ or less than 2 · diam(G) < 4√α.

Claim 2: |E| = O(n2/√α)

(Idea: we associate α-many non-edges with each existing edge.)

Consider arbitrary v ∈ V with own edges e1, . . . , el and define sets Ti :

Ti := u ∈ V |ei ∈ shortest canonical path from v to u

shortest canonical path: arbitrary fixed chosen shortest path for every pair (ensuresthat Tis are disjoint)

The Plan: Consider G′ := (V ,E \ ei) and analyze cases:1 Ti and v are connected in G′

2 Ti and v are not connected in G′

Note: for any path v to u in G′ its length is either∞ or less than 2 · diam(G) < 4√α.

Remember G′ = (V ,E \ ei)

Case Ti and v are connected in G′: for all u ∈ Ti we have:dG′(v, u)− dG(v, u) < 4

since ei must improve i ’s distance cost by ≥ α it holds:

α ≤∑u∈Ti

(dG′(v, u)− dG(v, u)) < |Ti |4√α

this gives |Ti | ∈ Ω(√α), i.e., there are Ω(

√α) many vertices u

s.t. (v, u) does not existCase Ti and v are not connected in G′: G′ consists of two components

denote endpoint of ei by wfor v there exist (|Ti | − 1)-many nodes w ′ s.t. (v,w ′) 6∈ G′for w ∈ Ti there are (|V \ Ti | − 1)-many nodes that are notconnected to whence, |Ti | − 1+ |V \Ti | − 1 = n− 2 = Ω(

√α) edges do not exist

For both cases, non-edges counted at most twice (for ordered pairs (v,w) and (w, v))⇒ |E| is an O(1/

√α) fraction of the number of vertex pairs

α ≤∑u∈Ti

(dG′(v, u)− dG(v, u)) < |Ti |4√α

α ≤∑u∈Ti

(dG′(v, u)− dG(v, u)) < |Ti |4√α

Summary:

PoA ∈ O(α|E|+n2√α

αn+n2 )

|E| = O(n2/√α)

This gives PoA ∈ O(√α)

Network Creation Games [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker: PODC ’03]

Model: Each node v buys a set of edges Sv (every edge costs α), s.t. v minimizessum of buying cost plus connection costs.Average Distance Model: c(v) = α · |sv |+

∑u∈V d(v, u)

PoA bound

0 1 2 3√n/2√

n/2 O(n1−ε) 65n 12n lg n ∞

1 ≤ 43 ≤ 4 ≤ 6 Θ(1) 2O(

√log n) < 5 ≤ 1.5

Maximal Distance Model: c(v) = α · |sv |+ maxu∈V d(v, u)

PoA bound

0 1n−2 O(n−1/2) 129 n ∞

1 Θ(1) 2√

log n < 4 ≤ 2

Problems with this model:Equilibria quality/price of anarchy are tightly connected to value of α.Computation of Best-Response is NP-hard.

NCG Price of Anarchy BoundsState of the Art PoA Bounds

∑u∈V d(v, u)

PoA bound

0 1 2 3√n/2√

n/2 O(n1−ε) 65n 12n lg n ∞

1 ≤ 43 ≤ 4 ≤ 6 Θ(1) 2O(

√log n) < 5 ≤ 1.5

PoA bound

0 1n−2 O(n−1/2) 129 n ∞

1 Θ(1) 2√

log n < 4 ≤ 2

∑u∈V d(v, u)

PoA bound

0 1 2 3√n/2√

n/2 O(n1−ε) 65n 12n lg n ∞

1 ≤ 43 ≤ 4 ≤ 6 Θ(1) 2O(

√log n) < 5 ≤ 1.5

PoA bound

0 1n−2 O(n−1/2) 129 n ∞

1 Θ(1) 2√

log n < 4 ≤ 2

∑u∈V d(v, u)

PoA bound

0 1 2 3√n/2√

n/2 O(n1−ε) 65n 12n lg n ∞

1 ≤ 43 ≤ 4 ≤ 6 Θ(1) 2O(

√log n) < 5 ≤ 1.5

PoA bound

0 1n−2 O(n−1/2) 129 n ∞

1 Θ(1) 2√

log n < 4 ≤ 2

Further Models1 Basic Network Creation Games

un-owned edgesnodes can swap connected edgesno parameter α in cost functionBR computation is in P, similar structural results as in NCGs

2 Greedy Network Creation Gamesnodes are only allowed to perform one-edge-change operationsequilibria are 3-approximations to NCG equilibriaBR computation is in P

Other Questions1 convergence dynamics2 price of stability3 effects of local knowledge4 . . .

NCG Price of Anarchy BoundsOther Models & Other Questions

Thank you for your attention!Thank you for your attention!

Andreas Cord-Landwehr

Heinz Nixdorf Institute& Department of Computer ScienceUniversity of Paderborn

Address: Fürstenallee 1133102 PaderbornGermany

Phone: +49 5251 60-6428Fax: +49 5251 60-6482E-mail: [email protected]: http://wwwhni.upb.de/en/alg/

Source: Fotolia, Jürgen Priewe

network creation games - heinz nixdorf institut · pdf filenetwork creation games a game...

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