Network Creation Game* Presented by Miriam Allalouf

1. Network Creation Game* Presented by Miriam Allalouf On a Network Creation Game by A.Fabrikant, A. Luthra, E. Maneva, C. H. Papadimitriou, and S. Shenker, [FLMPS], PODC 2003 *Part of the Slides are taken from Alex Fabrikant PPT presentation On Nash Equilibria for a Network Creation Game by Albers, S. Eilts, E. Even-Dar, Y. Mansour and L. Roditty. [AEEMR] to appear in SODA 2006.

2. Context • The internet has over 20,000 Autonomous Systems (AS) • Everyone picks their own upstream and/or peers • MACHBA wants to be close to everyone else on the network, but doesn’t care about the network at large

3. Question: What is the “penalty” in terms of poor network structure incurred by having the “users” create the network, without centralized control?

4. In this talk we… • Introduce a simple model of network creation by self-interested agents • Briefly review game-theoretic concepts • Talk about related work • Show bounds on the “price of anarchy” in the model – using both papers results • Disprove the tree conjecture • A weighted network creation game • Cost sharing • Discuss extensions and open relevant problems.

5. Pay $a for each link you buy Pay $1 for every hop to every node A Simple Model • N agents, each represented by a vertex and can buy (undirected) links to a set of others (si) • One agent buys a link, but anyone can use it • Undirected graph G is built • Cost to agent: (a may depend on n)

6. 2 1 -1 3 -3 4 2 1 c(i)=2+9 Example +  c(i)=+13 (Convention: arrow from the node buying the link)

8. Definitions • V={1..n} set of players • A strategy for v is a set of vertices Sv  V\{v}, such that v creates an edge to every w Sv. • G(S)=(V,E) is the resulted graph given a combination of strategies S=(S1,..,Sn), V set of plyaers / nodes and E the laid edges. • Social optimum: combination of strategies that minimizes the social cost • “What a dictator would do” • Not necessarily palatable to any given agent • Social cost: • The simplest notion of “global benefit”

9. Definitions: Nash Equilibria • Nash equilibrium: a situation such that no single player can change what he is doing and benefit • Presumes complete rationality and knowledge on behalf of each agent • Not guaranteed to exist, but they do for our model • The cost of player i under s:

10. Definitions: Nash Equilibria • A combination of strategies S forms Nash equilibrium, if for any player i and any other strategy U, such that U differs from S only in i’s component • > • G(S) is the equilibrium graph • Strong Nash equilibrium is when for any i • Otherwise, it is a weak Nash equilibrium, where at least one player can change its strategy without affecting its cost. • Transient Nash equilibria is a weak equilibrium from which there exists a sequence of single-player strategy changes, which do not change the deviator’s cost, leading to a non-equilibrium position.

11. +1 -2 -1 -5 +2 -1 -1 +5 +5 +5 -5 -5 +1 +4 -1 -5 +1 Example ? ! • Set =5, and consider:

12. Definitions: Price of Anarchy • Price of Anarchy (Koutsoupias & Papadimitriou, 1999): the ratio between the worst-case social cost of a Nash equilibrium network and the optimum network over all Nash equilibria S • We bound the worst-case price of anarchy to evaluate “the price we pay” for operating without centralized control

13. The presented papers • On a Network Creation Game by A.Fabrikant, A. Luthra, E. Maneva, C. H. Papadimitriou, and S. Shenker, [FLMPS], PODC 2003 • On Nash Equilibria for a Network Creation Game by Albers, S. Eilts, E. Even-Dar, Y. Mansour and L. Roditty. [AEEMR] to appear in SODA 2006.

14. Related Work • Corbo and Parkes (PODS 05) • Study the P.O.A of the network creation game assuming the edges are bought by both players • Anshelevich, et al. (STOC 2003) • Agents are “global” and pick from a set of links to connect between their own terminals, observed the “price of stability” • A body of similar work on social networks in the econometrics literature (e.g. Bala&Goyal 2000, Dutta&Jackson 2000) • Earning by forming links • Players heterogenity, etc

15. In this talk we… • Introduce a simple model of network creation by self-interested agents • Briefly review game-theoretic concepts • Talk about related work • Show bounds on the “price of anarchy” in the model – using both papers results • Disprove the tree conjecture • A weighted network creation game • Cost sharing • Discuss extensions and open problems we believe to be relevant and potentially tractable.

16. Social optima - clique • When <2, any missing edge can be added at cost  and subtract at least 2 from social cost

17. Social optima - star • When 2, consider a star. Any extra edges are too expensive.

18. Equilibria: very small  (<2) • For <1, the clique is the only N.E. • For 1<<2, clique no longer N.E., but the diameter is at most 2; else: >2 • Then, the star is the worst N.E., can be seen to yield P.o.A. of at most 4/3 -2 +

19. P.O.A for very small  (<2) • The star is also a Nash equilibrium, but there may be worse Nash equilibrium.

20. RESULTS: p.o.a Bounds for different  values Constant P.O.A Upper bounds: [AEEMR] • For • Not larger than 1.5 • Goes to 1 as  increases • For any other , Constant for  Increases for Maximum at =n :

21. Nash Equilibrium Characteristics • [FLMPS] Tree Conjecture: For all >A (A constant), all non-transient Nash equilibria are for trees • [AEEMR] disproves it and show that for any positive integer n0, there exists a graph built by n ≥ n0 players that contains cycles and forms a strong Nash equilibrium, • But If  every Nash equilibrium is a tree

22. p.o.a Upper Bound [AEEMR] Price of Anarchy Upper Bound [AEEMR]

23. Constant p.o.a Upper Bound forα ≥ 12n log n [AEEMR] (1) Theorem 2For α ≥ 12n log n , the price of anarchy is bounded by 1 + (6n log n / α) ≤ 1.5 and any G(S) equilibrium graph is a tree. Proof Based on Proposition 1 where proved that G(S) whose girth ≥ 12log n is a tree whose maximal depth is 6log n, and on Lemma 5 that connects between the girth length and α.

24. Improved Upper Bound forα ≥ 12n log n [AEEMR] (2) Proposition 1If G(S) is an equilibrium graph whose girth ≥ 12 log n then • The diameter of G(S ) ≤ 6 log n • G(S) is a tree In order to prove the above, The following graph analysis Were provided

25. Improved Upper Bound forα ≥ 12n log n [AEEMR] (3) DefinitionG(S) is an equilibrium graph. • T(u) in V shortest path tree rooted at u. and this vertex represents layer 0 of the tree. • Given vertex layers 0 to i − 1, layer i is constructed as follows. • Tree edges A node w belongs to layer i if it is not yet contained in layers 0 to i − 1 and there is a vertex v in layer i−1 such that {v,w} E (only one is added to the the shortest path tree). • Non-tree edges - all remaining edges of E that are added to T(u) • T(u) a layered version of G with distinguished tree edges.

26. Improved Upper Bound forα ≥ 12n log n [AEEMR] (4) A vertex v in V at a depth ≤ 6 log n in T(u), is: • Expanding- If v has at least two children, each with at least one descendent in the Boundary level. • Neutral- If v has exactly one child with at least one descendent in the Boundary level. • Degenerate- If v does not have any descendent in the Boundary level

27. Improved Upper Bound forα ≥ 12n log n [AEEMR] (5) • vT(u) is a Neutral vertex. • Du(v) is the set of its Degenerate children and their descendants at T(u). Lemma 3G(S) an equilibrium graph whose girth ≥12 log n. Every path from xDu(v) to yV\Du(v) in G(S) must go through v neutral vertex. • Neutral edge An edge on the shortest path from u to v that both of its endpoints are Neutral vertices. Lemma 4G(S) an equilibrium graph whose girth ≥ 12 log n. The total number of Neutral edges is 2 log n. ProofIt is more beneficial to buy a link to a neutral node than to a degenerated mode. This decision can be taken no more than log n

28. Improved Upper Bound forα ≥ 12n log n [AEEMR] (6) Proposition 1If G(S) is an equilibrium graph whose girth ≥ 12 log n then • The diameter of G(S ) ≤ 6 log n • G(S) is a tree Proof • By contradiction, assume that the diameter is at least 6 log n. • Let uV on one of the endpoints of the diameter, and look on T(u). Since • U is either Neutral or Expanding vertex (one of the diameter endpoints) . • Goal: show that the number of descendants at the Boundary level is at least n.  • leads to contradiction and implies that the maximal depth is at most 6 log n and that there are no cycles. : see details…

29. Improved Upper Bound forα ≥ 12n log n [AEEMR] (7) Proposition 1 Proof more details • Let v  V , d the depth of v in T(u), b the number of Neutral edges on the path from u to v. • (d, b) is a label per vertex [example: for u it is (0,0)] • Let v be a non-Degenerate vertex whose label is (d, b) • N(d, b) be a lower bound on the number of its descendants at the Boundary level. • Note: two vertices might have the same label, but have different number of descendants at the boundary level. • We claim that • and for the root : • thus proving the claim will lead to the desired contradiction.

30. Improved Upper Bound forα ≥ 12n log n [AEEMR] (8) Lemma 5If G(S) is an equilibrium graph and c be any positive constant. If α>cn log n then • the length of the girth of G(S ) ≥ c log n. Proof Assume by contradiction that minimal girth is clog n. U on the cycle wants to buy an edge: • Benefit by distance reduction: (clog n -1)n • Loss by edge addition:α = cnlog n  It is not an equilibia , contradiction.

31. Improved Upper Bound forany α (α < 12n log n) [AEEMR] (1) • Theorem 3Let α > 0. For any Nash equilibrium N, the price of anarchy is bounded by • Proof • Fix an arbitrary v0V, such that v0 built only tree edges in T(v0). For any vertex vV , let Ev be the number of tree edges built by v in T(v0). • for any vV , v v0, Cost(v) ≤ α(Ev + 1) + Dist(v0) + n − 1 •  Cost(N) ≤ 2α(n − 1) + nDist(v0) + (n − 1)2. • Need to analyze Dist(v0) …. More details…

32. Improved Upper Bound for any α (α < 12n log n) [AEEMR] (2) • For α<1, • Dist(v0)≤n-1 (complete graph) • Cost(N) ≤ 2α(n−1)+2n(n−1), • Cost(OPT) ≥ α(n−1)+n(n−1)  p.o.a ≤ 2 • For α > n2, • Dist(v0)≤(n-1)2 • Cost(N) ≤ 2α(n−1)+2n(n−1)2, • Cost(OPT) > α(n−1)> n2(n−1)  p.o.a ≤ 4

33. Improved Upper Bound for any α (α < 12n log n) [AEEMR] (3) • For 1≤α≤ n2 • Cost(OPT)> α(n−1)+ 2(n-1)2> α(n−1)+ n2 (star), for n ≥ 2 players • For d ≤ 9 , Dist(v0)≤9n, Cost(N) ≤ 2α(n−1)+10n2 • For d ≥ 10 ,Dist(v0)≤(n-1)15α/nc ≤15αn1-c), • Cost(N) ≤ 2α(n−1)+ 15αn2-c+n2, • For α≤ n, nc=(αn)1/3 • For α> n, nc=(αn)1/3 , • [α(n-1)+n2>αn because α ≤ n2]

34. Improved Upper Bound any α (α < 12n log n) [AEEMR] (4) • Theorem 4 In any Nash equilibrium N, the total cost incurred by the players in building edges is bounded by twice the cost of the social optimum. There exists a shortest path tree such that, for any player v, the number of non-tree edges built by v is bounded by 1 + ⌊(n − 1)/α⌋.   The only critical part in bounding the P.O.A is the sum of the shortest path distances between players.

35. Trees [FLMPS] • Conjecture: for >0, some constant, all Nash equilibria are trees • Benefit: a tree has a center (a node that, when removed, yields no components with more than n/2 nodes) • Given a tree N.E., can use the fact that no additional nodes want to link to center to bound the depth and show that the price of anarchy is at most 5

36. Disproving the Tree Conjecture [AEEMR] (1) • Family of graphs construction that form strong Nash equilibria and have induced cycles of length three and five. • To construct these graphs, we have to define affine planes

37. Disproving the Tree Conjecture [AEEMR] (2) DefinitionAn affine plane is a pair (A,L) • A is a set (of points) • L is a family of subsets of A (of lines) satisfying the following four conditions. • For any two points, there is a unique line containing these points. • Each line contains at least two points. • Given a point x and a line L that does not contain x, there is a unique line L′ that contains x and is disjoint (parallel) from L (xL). • There exists a triangle, i.e. there are three distinct points which do not lie on a line. • If A is finite, then the affine plane is called finite. • Equivalence relation on the lines by parallelism • L’s equivalence class [L].

38. Disproving the Tree Conjecture [AEEMR] (3) Affine Plane definition • Set • Where field F=GF(q) (q prime) • Set • AG(2, q): affine plane of order q. • The plane contains: • q2 points • q*(q+1) lines • q+1 equivalence classes • Each has q lines • Each such line has q points

39. Disproving the Tree Conjecture [AEEMR] (4) Graphs represnet Strong Nash Equilibria • G=(V,E) construction • Set of vertices V=A  L : 2q2+q+1 players • The edge set E : • A point and a line are connected by an edge  the line contains the point. • Two lines are connected by an edge  they are parallel : complete subgraph Kq • No two points are connected by an edge.

40. Disproving the Tree Conjecture [AEEMR] (5) • If L[Lq] then the cost of the player representing L is • (2+s)α+(2q−1)+2(2q−1)q = (s+2)α+4q2−1, • s=s(L)=q−1−r. • If L[Lq], then the cost is • sα + 4q2 − 1. • Line L (representative from eq. class iq) builds edges • to points x1,x2 such that x1LLqi and x2LLqi+1 • To L1-Lr (parallel to L) : same equivalence class • Points x3-xqLbuilds edges • to L • to other lines containing x • (from different equivalence class) • r(L) indegree,s(L) outdegree of L in Kq • Line Lq(from eq. class q) does not build edges

41. Disproving the Tree Conjecture [AEEMR] (6) • Point X builds edges • to all lines containing it (different equivalence class) • The cost of the player representing x is • (q−1)α +(q+1)+2(q + 1)(2(q − 1)) = (q − 1)α + 4q2 +q−3.

42. Disproving the Tree Conjecture [AEEMR] (7) Lemma 1Let q>10. For α in the range 1<α<q+1, no player associated with a line L has a different strategy that achieves a cost ≤ L’s original one. For α in the range 1≤ α≤q+1, L has no strategy with a smaller cost. Proof • Fix a line L[Lq]. Consider all possible strategy changes. • L builds l>s+2 edges, • at best there are l−s−2+2q−1 vertices at distance 1 while the other vertices are at distance 2 from L. • In L’s original strategy there are 2q−1 vertices at distance 1 while all other vertices are at distance 2. •  L’s original strategy has a cost at least α(l−s−2)−(l−s−2) < S, and this expression is strictly positive for α > 1.  Thus buying more than s + 2 edges does not pay off. • L builds at most s + 2 edges and show it does not pay off. • There is an edge building cost of α while the shortest path distance costdecreases by at least q + 1. • If α<q+1, there is a net cost saving and S is worse than L’s original strategy given by G. If α = q + 1, then L’s original strategy is at least as good

43. Disproving the Tree Conjecture [AEEMR] (7) • Lemma 2For α in the range 1 < α ≤ q + 1, no player associated with a point x has a different strategy that achieves a cost equal to or smaller than that of x’s original strategy. For α = 1, no player associated with a point has a strategy that achieves a smaller cost.  • Theorem 1Let q > 10. The graph G is a strong Nash equilibrium, for 1 < α < q + 1, and a Nash equilibrium, for 1 ≤ α ≤ q + 1.

44. A Weighted Network Creation Game [AEEMR] (1) • Most agents don’t care to connect closely to everyone else • What if we know the amount of traffic between each pair of nodes and weight the distance terms accordingly? • If n2 parameters is too much, what about restricted traffic matrices?

45. A Weighted Network Creation Game [AEEMR] (2) • Assume at least n ≥ 2 players. • Player u sends a traffic amount of wuv>0 to player v, with u v. • W = (wuv)u,v is nxn traffic matrix. • Cost of player u: • wmin=minuv wuv smallest traffic entry • wmax=maxuv wuv largest traffic entry. • The sum of the traffic values

46. A Weighted Network Creation Game [AEEMR] (3) Theorem 8 (generalization of Theorem 3, up to constant factors, where wmin = 1 • For 0<α≤ wminn2 and any Nash equilibrium N, the price of anarchy is bounded by • For wminn2<α<wmaxn2 , the price of anarchy is bounded by • For wmaxn2 ≤ α. Then the price of anarchy is bounded by 4.

47. Cost Sharing [AEEMR] (1) • What if agents collaborate to create a link? • Each node can pay for a fraction of a link; • Link exists only if total “investment” is α • May yield a wider variety of equilibria

48. Cost Sharing [AEEMR] (2) • Theorem 9 • n the unweighted scenario the bounds of Theorem 3 hold. • In the weighted scenario the bound of Theorem 8 hold. • Theorem 10For n > 6 and α in the range 16n2+n<α<12n2−n, there exist strong Nash equilibria with n players that contain cycle an in which the cost is split evenly among players.

49. Discussion • The price tag of decentralization in network design appears modest • not directly dependent on the size of the network being built • The Internet is not strictly a clique, or a star, or a tree, but often resembles one of these at any given scale • Many possible extensions remain to be explored

50. Directions for Future Work • The network is dynamic : Introduce time? • Network develops in stages as new nodes arrive • Assume equilibrium state is reached at every stage • Other points on the spectrum between dictatorship and anarchy? Agreements,sync • Measurements to assess applicability to existing real systems