Basic Network Creation Games

1. Basic Network Creation Games MohammadTaghiHajiAghayi AT&T Labs & U. of Maryland Joint work with NogaAlon, Erik Demaine, and Tom Leighton

2. 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 αand agents minimize their sum • Several studies so far: FLMPS03, CP05, AEEMR06,DHMZ07,HM07,A08,DHMZ09,etc

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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!

9. A Diameter- √n Max Equilibrium Graph • The proof is cute and more involved • The graph is indeed insertion-stable as well

10. 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.

11. t2 u t1 v ti 3k D 3k w Diameter 2O(√ logn) (Cont.) ≤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

12. 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

13. 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