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Abstract: Among complex network models, the hierarchical network model is the one most close to such real networks as world trade web, metabolic network, WWW, actor network, and so on. It has not only the property of power-law degree distribution, but also the scaling clustering coefficient property which barabá;si-Albert (BA) model does not have. BA model is a model of network growth based on growth and preferential attachment, showing the scale-free degree distribution property. In this paper, we study the evolution of cooperation on a hierarchical network model, adopting the prisoner’s dilemma (PD) game and snowdrift game (SG) as metaphors of the interplay between connected nodes. BA model provides a unifying framework for the emergence of cooperation. But interestingly, we found that on hierarchical model, there is no sign of cooperation for PD game, while the frequency of cooperation decreases as the common benefit decreases for SG. By comparing the scaling clustering coefficient properties of the hierarchical network model with that of BA model, we found that the former amplifies the effect of hubs. Considering different performances of PD game and SG on complex network, we also found that common benefit leads to cooperation in the evolution. Thus our study may shed light on the emergence of cooperation in both natural and social environments.

[1] Abramson, G., Kuperman, M., 2001. Social games in a social network. Phys. Rev. E, 63:030901-030904.

[2] Albert, R., Barabási, A.L., 2002. Statistical mechanics of complex networks. Rev. Mod. Phys., 74:47-97.

[3] Albert, R., Jeong, H., Barabási, A.L., 1999. Internet: Diameter of the World-Wide Web. Nature, 401:130-131.

[4] Axelrod, R., Hamilton, W.D., 1981. The evolution of cooperation. Science, 211:1390-1396.

[5] Barabási, A.L., Albert, R., 1999. Emergence of scaling in random networks. Science, 286:509-512.

[6] Barabási, A.L., Ravasz, E., Vicsek, T., 2001. Deterministic scale-free networks. Physica A., 299:559-564.

[7] Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U., 2006. Complex networks: structure and dynamics. Physics Reports, 424:175-308.

[8] Chen, X., Fu, F., Wang, L., 2007. Prisoner’s dilemma on community networks. Physica A., 378:512-518.

[9] Clutton-Brock, T.H., O′Riain, M.J., Brotherton, P.N.M., Gaynor, D., Kansky, R., Griffin, A.S., Manser, M., 1999. Selfish sentinels in cooperative mammals. Science, 284:1640-1644.

[10] Dorogovtsev, S.N., Mendes, J.F.F., 2002. Evolution of networks. Adv. Phys., 51:1079-1187.

[11] Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F., 2002. Pseudofractal scale-free web. Phys. Rev. E, 65:066122-066125.

[12] Ebel, H., Bornholdt, S., 2002a. Coevolutionary games on networks. Phys. Rev. E, 66:056118-056125.

[13] Ebel, H., Bornholdt, S., 2002b. Evolutionary Games and the Emergence of Complex Networks. E-print arXiv:cond-mat/0211666v1.

[14] Fu, F., Chen, X., Liu, L., Wang, L., 2007. Social Dilemmas in an Online Social Network: The Structure and Evolution of Cooperation. E-print arXiv:physics/0701323v1.

[15] Gintis, H., 2000. Game Theory Evolving. Princeton University, Princeton, USA.

[16] Hauert, C., Doebeli, M., 2004. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature, 428:643-646.

[17] Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, UK.

[18] Holme, P., Trusina, A., Kim, B.J., Minnhagen, P., 2003. Prisoners’ dilemma in real-world acquaintance networks: spikes and quasiequilibria induced by the interplay between structure and dynamics. Phys. Rev. E, 68:030901-030904.

[19] Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabási, A.L., 2000. The large-scale organization of metabolic networks. Nature, 407:651-654.

[20] Kim, B.J., Trusina, A., Holme, P., Minnhagen, P., Chung, J.S., Choi, M.Y., 2002. Dynamic instabilities induced by asymmetric influence: prisoners’ dilemma game in small-world networks. Phys. Rev. E, 66:021907-021910.

[21] Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature, 433:312-316.

[22] Maynard Smith, J., 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge, UK.

[23] Newman, M.E.J., 2003. The structure and function of complex networks. SIAM Review, 45:167-256.

[24] Nowak, M.A., May, R.M., 1992. Evolutionary games and spatial chaos. Nature, 359:826-829.

[25] Osborne, M.J., 2002. An Introduction to Game Theory. Oxford University Press, Oxford, UK.

[26] Pastor-Satorras, R., Vespignani, A., 2004. Evolution and Structure of the Internet: A Statistical Physics Approach. Cambridge University Press, Cambridge, UK.

[27] Pastor-Satorras, R., Vazquez, A., Vespignani, A., 2001. Dynamical and correlation properties of the Internet. Phys. Rev. Lett., 87:258701-258704.

[28] Ravasz, E., Barabási, A., 2003. Hierarchical organization in complex networks. Phys. Rev. E, 67:026112-026118.

[29] Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z.N., Barabási, A., 2002. Hierarchical organization of modularity in metabolic networks. Science, 297:1551-1555.

[30] Santos, F.C., Pacheco, J.M., 2005. Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett., 95:098104-098107.

[31] Serrano, M.A., Boguna, M., 2003. Topology of the world trade web. Phys. Rev. E, 68:015101-015104.

[32] Seyfarth, R.M., Cheney, D.L., 1984. Grooming, alliances and reciprocal altruism in vervet monkeys. Nature, 308:541-543.

[33] Sugden, R., 1986. The Economics of Rights, Co-operation and Welfare. Blackwell Press, Oxford, UK.

[34] Vazquez, A., Pastor-Satorras, R., Vespignani, A., 2002. Large-scale topological and dynamical properties of the Internet. Phys. Rev. E, 65:066130-066141.

[35] Von Neumann, J., Morgenstern, O., 1947. Theory of Games and Economic Behavior (2nd Ed.). Princeton University Press, Princeton, USA.

[36] Vukov, J., Szabό, G., 2005. Evolutionary prisoner’s dilemma game on hierarchical lattices. Phys. Rev. E, 71:036133-036139.

[37] Wagner, A., 2001. The yeast protein interaction network evolves rapidly and contains few redundant duplicate genes. Mol. Biol. Evol., 18:1283-1292.

[38] Wasserman, S., Faust, K., 1994. Social Networks Analysis. Cambridge University Press, Cambridge, UK.

[39] Watts, D.J., Strogatz, S.H., 1998. Collective dynamics of ‘small-world’ networks. Nature, 393:440-442.

[40] Wilkinson, G.S., 1984. Reciprocal food sharing in the vampire bat. Nature, 308:181-184.