1. On a Network Creation Game PoA Seminar Presenting: Oren Gilon Based on an article by Fabrikant et al 1

2. Background Attempt to model the creation of a joint infrastructure ▫Internet ▫Cell towers What costs people money? ▫Creating connections ▫Slow communication Selfish-Routing inaccurate 2

3. Outline Define a model for our problem Prove upper and lower bounds on the PoA Present a conjecture concerning structure of NE Prove tighter bounds on PoA if conjecture is true State later results that improve bounds on PoA 3

4. The Model N players, each represented by a vertex Each player i chooses a set of “links”, or edges (i,j) he wishes to add to the graph The price of each edge is α A graph G is created by the union of all edges Each player also pays the sum of all distances to other players in G 4

5. More Formally Player i’s strategy is some, the set of vertices he wishes to connect to. For some strategy profile, the resulting graph is Player i’s cost under s is 5

6. Example 1 Assume that the price of a link is 3 NE? 6 12 6 4 3 5

7. Example 1 No! 7 12 6 4 3 5

8. Example 2 Assume that the price of a link is 3 Star NE? 8 1 2 6 4 3 5

9. Example 3 Assume that the price of a link is 4 NE? Cost(6) = 8+15=23 9 13 45 2 7 8 90 6

10. Example 3 Cost(6) = 23 Cost(4) = 4+3+12=19 10 13 45 2 7 8 90 6

11. Example 3 Cost(4) = 4+3+12=19 Cost(7) = 8+2+10+6=26 11 13 45 2 7 8 90 6

12. Example 3 Cost(7) = 8+2+10+6=26 Cost(6) = 8+3+12=23 12 13 45 2 7 8 90 6

13. Example 3 Cost(6) = 4+2+12+3 = 21 < 23 “Transient” 13 13 45 2 7 8 90 6

14. Best-Responses It is NP-hard to calculate a player’s best- response to a given strategy profile Reduction from dominating-set ▫We take the price of a link to be 2 ▫A player’s best response to a given graph is clearly a minimal dominating-set of the graph 14

15. Social Cost Sum of all costs of all players, i.e. social welfare In NE because no edge is purchased twice 15

16. Social Cost Distance between unconnected vertices is at least 2, so If diameter is at most 2, bound is achieved 16

17. Special Cases In a NE, the graph may not be missing any edge that would decrease the sum of a vertex’s inter- node distances by more than α α<1 ▫Only NE is complete graph ▫Social optimum is complete graph ▫PoA=1 17

18. Special Cases cont. If 1<α<2 ▫Social optimum is still the complete graph ▫Every NE is of diameter of at-most 2 ▫Worst when |E| is minimal, i.e. |E|=n-1 ▫Worst NE is the star 18

19. Special Cases cont. 19

20. Special Cases cont. 2<α ▫Social optimum’s diameter is 2 ▫Social optimum has minimal |E| ▫Star is the social optimum! ▫It is also a NE, but there may be others… 20

21. Upper Bound on PoA, A strategy profile s is a NE iff G[s] is a tree ▫Otherwise, a player could remove one of his links and profit Again, social optimum is the star PoA is O(1) 21

22. Upper Bound on PoA, Theorem: PoA is Proof: The social optimum is a star For any G that is the result of a NE 22

23. Upper Bound on PoA, Clearly ▫Otherwise i would profit by connecting to j ▫Saves at least 23 152364

24. Upper Bound on PoA, Lemma: for any graph with diameter d, the cost is at most O(d) times that of the optimum. Proof: We need to bound the cost of edges 24

25. Upper Bound on PoA, Two types of edges: ▫Cut edges – removing them disconnects the graph ▫Non-cut edges – all others Cut edges cost at most What do non-cut edges cost? 25

26. Upper Bound on PoA, Consider the edges out of a vertex v For a non-cut edge out of v, e, let These groups must be disjoint 26

27. Upper Bound on PoA, Before adding e, the shortest path between u and v is of length at-most 2d If and v are connected after removing e, then for any and the total improvement is 27

28. Upper Bound on PoA, Thus v has at most non-cut edges. Using this, the total cost of non-cut edges is And we get 28

29. Upper Bound on PoA, As we have seen, the diameter of a NE is Using the lemma, this means that the PoA is also as desired 29

30. Tighter Upper Bound on PoA The Lemma we proved gives us a method of bounding the PoA We simply need to bound the diameter of NE graphs Lin proved a tighter bound on the diameter, giving a tighter bound on the PoA 30

31. Tighter Upper Bound on PoA Theorem: the PoA is Proof: suffices to show the bound on the graph’s diameter Let G be a NE graph with diameter d Let u, v be two nodes at distance d Let Let B the set of nodes at distance at most d’ from u 31

32. Tighter Upper Bound on PoA 32 v u w G B

33. Tighter Upper Bound on PoA For some node w in B, we look at how d(v,w) changes after adding the edge (v,u) Before adding the edge: After adding the edge: And in total, v saves at least 33

34. Tighter Upper Bound on PoA For some node w in B, let be all nodes t such that the u-t shortest path leaves B after w If is nonempty then d(u,w)=d’ Therefore, if u would connect to w it would save There must exist a node w such that 34

35. Tighter Upper Bound on PoA Combining these we get Which implies 35

36. Tighter Upper Bound on PoA And since Recalling We get 36

37. Tighter Upper Bound on PoA Which means And finally as promised 37

38. Tree Conjecture Experimentally, the only non-tree equilibrium found is the Petersen graph. Conjecture – for some constant A, if the price of a link is greater than A then all NE are trees. Could give a very strong bound on the PoA 38

39. Tree NE Theorem: for any tree NE, the approximation ratio to the social optimum is at most 5 Proof: Let s be the tree NE, and T be G[s] Let L(i) be the largest connected component after removing node i from the graph Let z be T’s central node Let d be the tree’s depth when rooted at z 39

40. Tree NE Proof If d is 1, T is a star Otherwise, there is some leaf l at depth d that decided not to link to z Connecting to z would yield a profit of at least And so we get 40

41. Tree NE Proof Since the distance between two nodes is at most the diameter of T, and the approximation ratio 41

42. Tree Conjecture Sadly the conjecture was disproven in an article by Albers et al Proven: if then all NE are trees Proven: the PoA is also bounded by 1.5 in this case! Tighter bound was shown in an article by Mihalak et al, showing NE are trees for all 42

43. A Lower Bound Theorem: For any there is an instance of the network connection game where the PoA is at least Proof: For any we look at the family of complete k-ary trees of depth d,. We call the number of nodes in this tree n, and set 43

44. k-ary trees of depth d 44 1 1k … d 1k … 1k …

45. A Lower Bound Lemma: If all links are purchased by the parent node, this is a NE Proof: The graph is a tree, every node must connect at least once to its subtree (otherwise an infinite penalty would be incurred) If only one link is sent to the subtree, clearly linking to the direct child is optimal 45

46. A Lower Bound The node will not connect more than one link to any of its subtrees ▫Additional links would shorten paths to at most n nodes by at most d-1, but would cost A node will not add more than 1 link A node will not send a link to the root A node will not send a link to a sibling 46

47. A Lower Bound For any other node i and another node j not in its subtree and not i’s parent (trivially wasteful) we will see i does not link to j We look at the node that is an ancestor of j and a sibling of i, or the root if no such node exists i does not want to link to that node 47

48. A Lower Bound 48 i … d … j …

49. A Lower Bound In conclusion a node’s best response is: ▫Connect exactly once to every subtree, at the root. ▫Don’t connect to any vertex outside your subtrees The strategy is a NE! 49

50. A Lower Bound The cost of the social optimum (the star) is By counting only distances between leaves, the cost of is at least 50

51. A Lower Bound Thus, we get that the approximation ratio 51

52. Interesting Expansions Players buy fractions of links. ▫More room for cooperation between the players ▫More room for inefficiency Edges are directed ▫Less earned by being a “central node” ▫Building edges is less worthwhile Weighted data ▫Each player suffers differently from QoS ▫Players with more data build more links 52

53. Summary 53 Known BoundProven Boundα 11α<1 4/3 1<α<2 α<273n 5273n<α<12nlogn 1.512nlogn<α< O(1) <α

54. Summary We presented a new game We saw it is hard to compute best response sequences in this game We proved an upper bound on the PoA for different parameters of the game We showed a conjecture, and explained why it would give us a good upper bound on the PoA We proved a lower bound on the PoA 54

55. Questions? 55