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On the Tree Conjecture for the Network Creation Game

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Abstract

Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al. (2003) is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of α per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all α and that for α ≥ n all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edge-price α and employ it to improve on the best known bound for the latter conjecture. In particular we show that for α > 4n − 13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of α. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.

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Notes

  1. No edge can have two owners in any equilibrium network. Hence, we will assume throughout the paper that each edge in E(s) has a unique owner.

  2. A swap of edge (a, b) to edge (a, c) by agent a who owns edge (a, b) consists of deleting edge (a, b) and buying edge (a, c).

  3. In a cycle of length ℓ, two vertices of the cycle are antipodal if their distance is ⌊ℓ/2⌋.

References

  1. Albers, S., Eilts, S., Even-Dar, E., Mansour, Y., Roditty, L.: On Nash equilibria for a network creation game. ACM Trans. Econ. Comput. (TEAC) 2(1), 2 (2014)

    MATH  Google Scholar 

  2. Alon, N., Demaine, E.D., Hajiaghayi, M.T., Leighton, T.: Basic network creation games. SIAM J. Discret. Math. 27(2), 656–668 (2013)

    Article  MathSciNet  Google Scholar 

  3. Àlvarez, C., Blesa, M.J., Duch, A., Messegué, A., Serna, M.: Celebrity games. Theor. Comput. Sci. 648, 56–71 (2016)

    Article  MathSciNet  Google Scholar 

  4. Àlvarez, C., Messegué, A.: Max celebrity games. In: International workshop on algorithms and models for the web-graph (WAW), pp. 88–99. Springer (2016)

  5. Ȧlvarez, C., Messeguė, A.: Network creation games: Structure vs anarchy. arXiv:1706.09132 (2017)

  6. Àlvarez, C., Messegué, A.: On the constant price of anarchy conjecture. arXiv:1809.08027 (2018)

  7. Andelman, N., Feldman, M., Mansour, Y.: Strong price of anarchy. Games Econom. Behav. 65(2), 289–317 (2009)

    Article  MathSciNet  Google Scholar 

  8. Anshelevich, E., Bhardwaj, O., Usher, M.: Friend of my friend: Network formation with two-hop benefit. Theory Comput. Syst. 57(3), 711–752 (2015)

    Article  MathSciNet  Google Scholar 

  9. Bala, V., Goyal, S.: A noncooperative model of network formation. Econometrica 68(5), 1181–1229 (2000)

    Article  MathSciNet  Google Scholar 

  10. Barabási, A.: Network science. Cambridge University Press, Cambridge (2016)

    MATH  Google Scholar 

  11. Bilò, D., Gualà, L., Leucci, S., Proietti, G.: Network creation games with traceroute-based strategies. In: International colloquium on structural information and communication complexity (SIROCCO), pp. 210–223. Springer (2014)

  12. Bilò, D., Gualà, L., Leucci, S., Proietti, G.: The max-distance network creation game on general host graphs. Theor. Comput. Sci. 573, 43–53 (2015)

    Article  MathSciNet  Google Scholar 

  13. Bilò, D., Gualà, L., Leucci, Stefano, Proietti, G.: Locality-based network creation games. ACM Transactions on Parallel Computing (TOPC) 3(1), 6 (2016)

    MATH  Google Scholar 

  14. Bilò, D., Gualà, L., Proietti, G.: Bounded-distance network creation games. ACM Trans. Econ. Comput. (TEAC) 3(3), 16 (2015)

    MathSciNet  Google Scholar 

  15. Bilȯ, D., Lenzner, P.: On the tree conjecture for the network creation game. In: 35th symposium on theoretical aspects of computer science (STACS), pp. 14:1–14:15 (2018)

  16. Brandes, U., Hoefer, M., Nick, B.: Network creation games with disconnected equilibria. In: International workshop on internet and network economics (WINE), pp. 394–401. Springer (2008)

  17. Chauhan, A., Lenzner, P., Melnichenko, A., Molitor, L.: Selfish network creation with non-uniform edge cost. In: International symposium on algorithmic game theory (SAGT), pp. 160–172. Springer (2017)

  18. Chauhan, A., Lenzner, P., Melnichenko, A., Münn, M.: On selfish creation of robust networks. In: International symposium on algorithmic game theory (SAGT), pp. 141–152. Springer (2016)

  19. Corbo, J., Parkes, D.: The price of selfish behavior in bilateral network formation. In: 24th ACM symposium on principles of distributed computing (PODC), pp. 99–107. ACM (2005)

  20. Cord-Landwehr, A., Lenzner, P.: Network creation games: think global–act local. In: International symposium on mathematical foundations of computer science (MFCS), pp. 248–260. Springer (2015)

  21. Cord-Landwehr, A., Mäcker, A., auf der Heide, F.M.: Quality of service in network creation games. In: International conference on web and internet economics (WINE), pp. 423–428. Springer (2014)

  22. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms. MIT Press (2009)

  23. de Keijzer, B., Janus, T.: On strong equilibria and improvement dynamics in network creation games. Internet Mathematics, 2019 (2019)

  24. Demaine, E.D., Hajiaghayi, M.T., Mahini, H., Zadimoghaddam, M.: The price of anarchy in network creation games. ACM Trans. Algorithms (TALG) 8(2), 13 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Demaine, E.D, Hajiaghayi, M., Mahini, H., Zadimoghaddam, M.: The price of anarchy in cooperative network creation games. ACM SIGecom Exchanges 8(2), 2 (2009)

    Article  Google Scholar 

  26. Ehsani, S., Fadaee, S.S., Fazli, M., Mehrabian, A., Sadeghabad, S.S., Safari, M., Saghafian, M.: A bounded budget network creation game. ACM Trans. Algorithms (TALG) 11(4), 34 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C., Shenker, S.: On a network creation game. In: 22nd symposium on principles of distributed computing (PODC), pp. 347–351. ACM (2003)

  28. Friedrich, T., Ihde, S., Keßler, C., Lenzner, P., Neubert, S., Schumann, D.: Efficient best response computation for strategic network formation under attack. In: International symposium on algorithmic game theory (SAGT), pp. 199–211. Springer (2017)

  29. Garey, M.R., Johnson, D.S.: Computers and intractability, vol. 29. wh Freeman, New York (2002)

    Google Scholar 

  30. Goyal, S., Jabbari, S., Kearns, M., Khanna, S., Morgenstern, J.: Strategic network formation with attack and immunization. In: International conference on web and internet economics (WINE), pp. 429–443. Springer (2016)

  31. Graham, R., Hamilton, L., Levavi, A., Loh, P.-S.: Anarchy is free in network creation. In: International workshop on algorithms and models for the web-graph (WAW), pp. 220–231. Springer (2013)

  32. Jackson, M.O.: A survey of network formation models: stability and efficiency. Group Formation in Economics: Networks Clubs, and Coalitions, pp. 11–49 (2005)

  33. Jackson, M.O.: Social and economic networks. Princeton University Press, Princeton (2010)

    Book  Google Scholar 

  34. Jackson, M.O, Wolinsky, A.: A strategic model of social and economic networks. J. Econ. Theory 71(1), 44–74 (1996)

    Article  MathSciNet  Google Scholar 

  35. Janus, T., de Keijzer, B.: On strong equilibria and improvement dynamics in network creation games. In: International conference on web and internet economics (WINE), pp. 161–176. Springer (2017)

  36. Johnson, D.S., Lenstra, J.K., Kan, R.AHG.: The complexity of the network design problem. Networks 8(4), 279–285 (1978)

    Article  MathSciNet  Google Scholar 

  37. Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems. ii: The p-medians. SIAM J Appl Math 37(3), 539–560 (1979)

    Article  MathSciNet  Google Scholar 

  38. Kawald, B., Lenzner, P.: On dynamics in selfish network creation. In: 25th ACM symposium on parallelism in algorithms and architectures (SPAA), pp. 83–92. ACM (2013)

  39. Kleinberg, J.: The small-world phenomenon: An algorithmic perspective. In: 32nd ACM symposium on theory of computing (STOC), pp. 163–170. ACM (2000)

  40. Kliemann, L.: The price of anarchy for network formation in an adversary model. Games 2(3), 302–332 (2011)

    Article  MathSciNet  Google Scholar 

  41. Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: 16th symposium on theoretical aspects of computer science (STACS), pp. 404–413. Springer (1999)

  42. Lenzner, P.: On dynamics in basic network creation games. In: International symposium on algorithmic game theory (SAGT), pp. 254–265. Springer (2011)

  43. Lenzner, P.: Greedy selfish network creation. In: International workshop on internet and network economics (WINE), pp. 142–155. Springer (2012)

  44. Lenzner, P.: On selfish network creation. PhD thesis, Humboldt University of Berlin (2014)

    Google Scholar 

  45. Mamageishvili, A., Mihalák, M., Müller, D.: Tree Nash equilibria in the network creation game. Internet Math. 11(4-5), 472–486 (2015)

    Article  MathSciNet  Google Scholar 

  46. Meirom, E.A., Mannor, S., Orda, A.: Network formation games with heterogeneous players and the internet structure. In: 15th ACM conference on economics and computation (EC), pp. 735–752. ACM (2014)

  47. Mihalák, M., Schlegel, J.C.: Asymmetric swap-equilibrium: a unifying equilibrium concept for network creation games. In: International symposium on mathematical foundations of computer science (MFCS), pp. 693–704. Springer (2012)

  48. Mihalák, M., Schlegel, J.C.: The price of anarchy in network creation games is (mostly) constant. Theory Comput. Syst. 53(1), 53–72 (2013)

    Article  MathSciNet  Google Scholar 

  49. Papadimitriou, C.: Algorithms, games, and the internet. In: 33rd ACM symposium on theory of computing (STOC), pp. 749–753. ACM (2001)

  50. Travers, J., Milgram, S.: The small world problem. Phys. Today 1, 61–67 (1967)

    Google Scholar 

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Authors and Affiliations

  1. Department of Humanities and Social Sciences, University of Sassari, Sassari, Italy

    Davide Bilò

  2. Algorithm Engineering Group, Hasso Plattner Institute, University of Potsdam, Potsdam, Germany

    Pascal Lenzner

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  1. Davide Bilò
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  2. Pascal Lenzner
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Correspondence to Davide Bilò.

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This article is part of the Topical Collection on Special Issue on Theoretical Aspects of Computer Science (2018)

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Bilò, D., Lenzner, P. On the Tree Conjecture for the Network Creation Game. Theory Comput Syst 64, 422–443 (2020). https://doi.org/10.1007/s00224-019-09945-9

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