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Basic Network Creation GamesBasic Network Creation Games

MohammadTaghi HajiAghayi MohammadTaghi HajiAghayi AT&T Labs & U. of MarylandAT&T Labs & U. of Maryland

Joint work with Joint work with

Noga Alon, Erik Demaine, and Tom Noga Alon, Erik Demaine, and Tom LeightonLeighton

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Motivations for Network Creation Games

Distributed way to create a network Undirected graph. Nodes Selfish agents Agents’ cost:

Creation cost (network design)Usage cost (network routing)

Combine two costs by defining the cost of an edge to be α and agents minimize their sum

Several studies so far: FLMPS03, CP05, AEEMR06,DHMZ07,HM07,A08,DHMZ09,etc

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Basic Network Creation Games The simplest and the heart of all such games while

avoiding α Motivations: Cash-oblivious model No cost transformation but every edge cost is the

same Thus each edge only perform edge swap, replacing

an existing edge with another incident edge We focus on structures of equilibria

Diameter Price of anarchy (PoA)

Agents try to minimize sum or maximum distance to all other vertices

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Formal Definitions

Sum equilibrium: for every edge vw and every node w’, swapping edge vw with vw’ does not decrease the total sum distances from v to all others

Max equilibrium: for every edge vw and every node w’, swapping edge vw with vw’ does not decrease the max distances from v to all others

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Our Results

For sum equilibrium: Upper bound 2 O(√ log n) for diameter (and thus PoA) and

lower bound 3 for general graphs Give an evidence (and a conjecture) for a polylog upper

bound for diameter (and PoA) Tight bound 2 for trees

For max equilibrium: Lower bound √n for diameter in general graphs even for

insertion-stable equilibria Tight bound 3 for trees

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Sum Equilibrium on Trees: Diameter 2

Swap for v is a net win unless sb+sw<=sa

Swap for w is a net win unless sv+sa<=sb

Summing up: sb+sw+sv+sa<=sa+sb

Contradiction since sw+sv>=2 by definition

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Max Equilibrium on Trees: Diameter 3

None of three swaps around a with edge av is helpful for the local diameter of a

Proof of upper bound is different from Sum

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A Diameter-3 Sum Equilibrium Graph

The proof is involved and uses some lemmas and case analysis

We do not know any example of diameter 4 or higher!

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A Diameter- √n Max Equilibrium Graph

The proof is cute and more involved The graph is indeed insertion-stable as well

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Sum Equilibrium: Diameter 2 O(√ log n)

First we need the following lemma whose proof is a little bit involved:

Lm: In any sum equilibrium, the addition of any edge uv decreases the sum of distance from u by at most 5n lg n

Bk(u) denote the number of vertices within distance at most k from u and Bk= minu Bk(u)

We prove either B4k>n/2 or B4k >=(k/20lg n)Bk

Assume there is a u with B4k(u)<= n/2. Certainly B3k(u)<= n/2.

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Diameter 2O(√ logn) (Cont.))()( 133 uNuND kk

D

3ku

t2

2C

t1kdistC 2:1

l

iiCD

1

iAti

iC

w

v3k

≤3k+1 >2k

• Distance of any v outside of B3k(u) from one vertex of T is at most dist(u,v)- k • By Pigeonhole principle, there are at least n/(2|T|) vertices Ai whose distances from the same vertex t in T is at most dist(u,v)- k• Adding an edge from u to t improves the sum of distances from u by at least (k-1) n/(2|T|) <= 5nlg n (by the Lm) which implies |T|>= k/(20lg n)•The balls of radius k centered at the vertices of T are all pairwise disjoint, all lie within distance 4k from u and each of them has at least Bk vertices. Thus B4k >=(k/20lg n)Bk

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Evidence for polylog diameter An n-vertex graph is ε-distance uniform if there is a

value r such that from every vertex v the number of vertices w at distance exactly r from v >= (1- ε)n

We can connect high-diameter sum equilibria graphs to high-diameter distance uniformity. More formally:

Thm: Any sum equil. graph G with at least 24 vertices and diameter d> 2lg n induces an ε-distance–uniform graph G’ with n vertices and diameter (εd/lg n )

Conj: Distance-uniform graphs have diameter O(lg n) If Conj above is correct, Thm gives O(lg2n) diameter We know Conj is true e.g., for Cayley graphs

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Main Open Problem

Can we prove a polylog upper bound (even O(log n) or smaller) for diameter of sum equilibria esp. by proving the conjecture?

Consider convergences of (basic) network creation games

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