This paper investigates the evolution of cooperation and the emergence of hierarchical leadership structure in random regular graphs. It is found that there exist different learning patterns between cooperators and defectors, and cooperators are able to attract more followers and hence more likely to become leaders. Hence, the heterogeneous distributions of reputation and leadership can emerge from homogeneous random graphs. The important directed game-learning skeleton is then studied, revealing some important structural properties, such as the heavy-tailed degree distribution and the positive in-in degree correlation.
REFERENCES
1.
2.
M.
Doebeli
and C.
Hauert
, “Models of cooperation based on the Prisoner’s Dilemma and the Snowdrift game
,” Ecol. Lett.
8
, 748
–766
(2005
). 3.
M. A.
Nowak
, Evolutionary Dynamics
(Harvard University Press
, 2006
).4.
G.
Szabó
and G.
Fáth
, “Evolutionary games on graphs
,” Phys. Rep.
446
, 97
–216
(2007
). 5.
T.
Gross
and B.
Blasius
, “Adaptive coevolutionary networks: A review
,” J. R. Soc. Interf.
5
, 259
–271
(2008
). 6.
D. G.
Rand
and M. A.
Nowak
, “Human cooperation
,” Trends Cogn. Sci.
17
, 413
–425
(2013
). 7.
M.
Perc
, J. J.
Jordan
, D. G.
Rand
, Z.
Wang
, S.
Boccaletti
, and A.
Szolnoki
, “Statistical physics of human cooperation
,” Phys. Rep.
687
, 1
–51
(2017
). 8.
M. A.
Nowak
, “Five rules for the evolution of cooperation
,” Science
314
, 1560
–1563
(2006
). 9.
M. A.
Nowak
and R. M.
May
, “Evolutionary games and spatial chaos
,” Nature
359
, 826
–829
(1992
). 10.
M. A.
Nowak
and R. M.
May
, “The spatial dilemma of evolution
,” Int. J. Bifurcat. Chaos
3
, 35
–78
(1993
). 11.
B.
Allen
, G.
Lippner
, Y.-T.
Chen
, B.
Fotouhi
, N.
Momeni
, S.-T.
Yau
, and M. A.
Nowak
, “Evolutionary dynamics on any population structure
,” Nature
544
, 227
–230
(2017
). 12.
B.
Fotouhi
, N.
Momeni
, B.
Allen
, and M. A.
Nowak
, “Conjoining uncooperative societies facilitates evolution of cooperation
,” Nat. Hum. Behav.
2
, 492
–499
(2018
). 13.
G.
Szabó
and C.
Tőke
, “Evolutionary prisoner’s dilemma game on a square lattice
,” Phys. Rev. E
58
, 69
–73
(1998
). 14.
P.
Holme
, A.
Trusina
, B.
Kim
, and P.
Minnhagen
, “Prisoners’ dilemma in real-world acquaintance networks: Spikes and quasiequilibria induced by the interplay between structure and dynamics
,” Phys. Rev. E
68
, 30901
(2003
). 15.
C.
Hauert
and G.
Szabó
, “Game theory and physics
,” Am. J. Phys.
73
, 405
–414
(2005
). 16.
G.
Szabó
, J.
Vukov
, and A.
Szolnoki
, “Phase diagrams for an evolutionary prisoner’s dilemma game on two-dimensional lattices
,” Phys. Rev. E
72
, 047107
(2005
). 17.
X.
Qian
, F.
Xu
, J.
Yang
, and J.
Kurths
, “The expansion of neighborhood and pattern formation on spatial prisoner’s dilemma
,” Chaos
25
, 047107
(2015
). 18.
K.
Huang
, Y.
Zhang
, Y.
Li
, C.
Yang
, and Z.
Wang
, “Effects of external forcing on evolutionary games in complex networks
,” Chaos
28
, 093108
(2018
). 19.
F. C.
Santos
and J. M.
Pacheco
, “Scale-free networks provide a unifying framework for the emergence of cooperation
,” Phys. Rev. Lett.
95
, 098104
(2005
). 20.
F. C.
Santos
, J. M.
Pacheco
, and T.
Lenaerts
, “Evolutionary dynamics of social dilemmas in structured heterogeneous populations
,” Proc. Natl. Acad. Sci. U.S.A.
103
, 3490
–3494
(2006
). 21.
J.
Gómez-Gardeñes
, M.
Campillo
, Y.
Moreno
, and L. M.
Floría
, “Dynamical organization of cooperation in complex networks
,” Phys. Rev. Lett.
98
, 108103
(2007
). 22.
Z.
Rong
, X.
Li
, and X.
Wang
, “Roles of mixing patterns in cooperation on a scale-free networked game
,” Phys. Rev. E
76
, 027101
(2007
). 23.
T.
Chadefaux
and D.
Helbing
, “How wealth accumulation can promote cooperation
,” PLoS One
5
, e13471
(2010
). 24.
J.
Poncela
, J.
Gómez-Gardeñes
, and Y.
Moreno
, “Cooperation in scale-free networks with limited associative capacities
,” Phys. Rev. E
83
, 057101
(2011
). 25.
A.
Antonioni
and M.
Tomassini
, “Network fluctuations hinder cooperation in evolutionary games
,” PLoS One
6
, e25555
(2011
). 26.
A.
Traulsen
and J. C.
Claussen
, “Similarity based cooperation and spatial segregation
,” Phys. Rev. E
70
, 046128
(2004
). 27.
Z.-X.
Wu
, X.
Xu
, Z.
Huang
, S.
Wang
, and Y.-H.
Wang
, “Evolutionary prisoner’s dilemma game with dynamic preferential selection
,” Phys. Rev. E
74
, 021107
(2006
). 28.
H.-X.
Yang
, W.
Wang
, Z.-X.
Wu
, Y.
Lai
, and B.
Wang
, “Diversity-optimized cooperation on complex networks
,” Phys. Rev. E
79
, 056107
(2009
). 29.
M.
Perc
and Z.
Wang
, “Heterogeneous aspirations promote cooperation in the prisoner’s dilemma game
,” PLoS One
5
, e15117
(2011
). 30.
X.
Chen
and L.
Wang
, “Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game
,” Phys. Rev. E
77
, 017103
(2008
). 31.
Y.
Liu
, X.
Chen
, L.
Zhang
, L.
Wang
, and M.
Perc
, “Win-stay-lose-learn promotes cooperation in the spatial prisoner’s dilemma game
,” PLoS One
7
, e30689
(2012
). 32.
A.
Szolnoki
and G.
Szabó
, “Cooperation enhanced by inhomogeneous activity of teaching for evolutionary prisoner’s dilemma games
,” Europhys. Lett.
77
, 30004
(2007
). 33.
A.
Szolnoki
and M.
Perc
, “Coevolution of teaching activity promotes cooperation
,” New J. Phys.
10
, 043036
(2008
). 34.
G.
Szabó
and A.
Szolnoki
, “Cooperation in spatial prisoner’s dilemma with two types of players for increasing number of neighbors
,” Phys. Rev. E
79
, 016106
(2009
). 35.
A.
Szolnoki
, Z.
Wang
, and M.
Perc
, “Wisdom of groups promotes cooperation in evolutionary social dilemmas
,” Sci. Rep.
2
, 576
(2012
). 36.
H.
Ohtsuki
, M. A.
Nowak
, and J. M.
Pacheco
, “Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs
,” Phys. Rev. Lett.
98
, 108106
(2007
). 37.
H.
Ohtsuki
, J. M.
Pacheco
, and M. A.
Nowak
, “Evolutionary graph theory: Breaking the symmetry between interaction and replacement
,” J. Theor. Biol.
246
, 681
–694
(2007
). 38.
Z.-X.
Wu
and Y.-H.
Wang
, “Cooperation enhanced by the difference between interaction and learning neighborhoods for evolutionary spatial prisoner’s dilemma games
,” Phys. Rev. E
74
, 041114
(2007
). 39.
C. P.
Roca
, J. A.
Cuesta
, and A.
Sánchez
, “Time scales in evolutionary dynamics
,” Phys. Rev. Lett.
97
, 158701
(2006
). 40.
Z.-X.
Wu
, Z.
Rong
, and P.
Holme
, “Diversity of reproduction time scale promotes cooperation in spatial prisoner’s dilemma games
,” Phys. Rev. E
80
, 036103
(2009
). 41.
Z.
Rong
, Z.-X.
Wu
, and W.-X.
Wang
, “Emergence of cooperation through coevolving time scale in spatial prisoner’s dilemma
,” Phys. Rev. E
82
, 026101
(2010
). 42.
Z.
Rong
, Z.-X.
Wu
, D.
Hao
, M. Z. Q.
Chen
, and T.
Zhou
, “Diversity of timescale promotes the maintenance of extortioners in a spatial prisoner’s dilemma game
,” New J. Phys.
17
, 033032
(2015
). 43.
C. P.
Roca
, J. A.
Cuesta
, and A.
Sánchez
, “Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics
,” Phys. Life Rev.
6
, 208
–249
(2009
). 44.
M.
Perc
and A.
Szolnoki
, “Coevolutionary games—A mini review
,” Biosystems
99
, 109
–125
(2010
). 45.
M.
Perc
, J.
Gómez-Gardeñes
, A.
Szolnoki
, L. M.
Floría
, and Y.
Moreno
, “Evolutionary dynamics of group interactions on structured populations: A review
,” J. R. Soc. Interface
10
, 20120997
(2013
). 46.
J. R. G.
Dyer
, A.
Johansson
, D.
Helbing
, I. D.
Couzin
, and J.
Krause
, “Leadership, consensus decision making and collective behaviour in humans
,” Philos. Trans. R. Soc. B
364
, 781
–789
(2009
). 47.
M.
Nagy
, Z.
Ákos
, D.
Biro
, and T.
Vicsek
, “Hierarchical group dynamics in pigeon flocks
,” Nature
464
, 890
–893
(2010
). 48.
Y.
Tang
, H.
Gao
, W.
Zhang
, and J.
Kurths
, “Leader-following consensus of a class of stochastic delayed networked multi-agent systems with partial mixed impulses
,” Automatica
53
, 346
–354
(2015
). 49.
Q.
Song
, F.
Liu
, J.
Cao
, A. V.
Vasilakos
, and Y.
Tang
, “Leader-following synchronization of coupled homogeneous and heterogeneous harmonic oscillators based on relative position measurements
,” IEEE Trans. Control Network Syst.
6
, 13
–23
(2018
). 50.
X.
Wang
and H.
Su
, “Self-triggered leader-following consensus of multi-agent systems with input time delay
,” Neurocomputing
330
, 70
–77
(2019
). 51.
X.
Wang
and H.
Su
, “Consensus of hybrid multi-agent systems by event-triggered/self-triggered strategy
,” Appl. Math. Comput.
359
, 490
–501
(2019
). 52.
L.
Rendell
, R.
Boyd
, D.
Cownden
, M.
Enquist
, K.
Eriksson
, M. W.
Feldman
, L.
Fogarty
, S.
Ghirlanda
, T.
Lillicrap
, and K. N.
Laland
, “Why copy others? Insights from the social learning strategies tournament
,” Science
328
, 208
–213
(2010
). 53.
A. J.
King
, D. D. P.
Johnson
, and M. V.
Vugt
, “The origins and evolution of leadership
,” Curr. Biol.
19
, 911
–916
(2009
). 54.
M.
Anghel
, Z.
Toroczkai
, K. E.
Bassler
, and G.
Korniss
, “Competition-driven network dynamics: Emergence of a scale-free leadership structure and collective efficiency
,” Phys. Rev. Lett.
92
, 058701
(2004
). 55.
T. S.
Lo
, K. P.
Chan
, P.
Hui
, and N. F.
Johnson
, “Theory of enhanced performance emerging in a sparsely connected competitive population
,” Phys. Rev. E
71
, 050101
(2005
). 56.
L. D. F.
Costa
and G.
Travieso
, “Strength distribution in derivative networks
,” Int. J. Mod. Phys. C
16
, 1097
–1105
(2005
). 57.
S. H.
Lee
and H.
Jeong
, “Effects of substrate network topologies on competition dynamics
,” Phys. Rev. E
74
, 026118
(2006
). 58.
V. M.
Eguíluz
, M. G.
Zimmermann
, C. J.
Cela-Conde
, and M. S.
Miguel
, “Cooperation and the emergence of role differentiation in the dynamics of social networks
,” Am. J. Sociology
110
, 977
–1008
(2005
). 59.
J. M.
Pacheco
, A.
Traulsen
, and M. A.
Nowak
, “Co-evolution of strategy and structure in complex networks with dynamical linking
,” Phys. Rev. Lett.
97
, 258103
(2006
). 60.
A.
Szolnoki
and M.
Perc
, “Emergence of multilevel selection in the prisoner’s dilemma game on coevolving random networks
,” New J. Phys.
11
, 093033
(2009
). 61.
A.
Szolnoki
and M.
Perc
, “Resolving social dilemmas on evolving random networks
,” EPL
86
, 30007
(2009
). 62.
S.
Lee
, P.
Holme
, and Z.-X.
Wu
, “Emergent hierarchical structures in multiadaptive games
,” Phys. Rev. Lett
106
, 028702
(2011
). 63.
B.
Wu
, J.
Arranz
, J.
Du
, D.
Zhou
, and A.
Traulsen
, “Evolving synergetic interactions
,” J. R. Soc. Interface
13
, 20160282
(2016
). 64.
S.
Maslov
and K.
Sneppen
, “Specificity and stability in topology of protein networks
,” Science
296
, 910
–913
(2002
). 65.
A.-L.
Barabási
and R.
Albert
, “Emergence of scaling in random networks
,” Science
286
, 509
–513
(1999
). 66.
M. E. J.
Newman
, “The structure and function of complex networks
,” SIAM Rev.
45
, 167
–256
(2003
).67.
S.
Boccaletti
, V.
Latora
, Y.
Moreno
, M.
Chavez
, and D. U.
Hwang
, “Complex networks: Structure and dynamics
,” Phys. Rep.
424
, 175
–308
(2006
). 68.
J. G.
Foster
, D. V.
Foster
, P.
Grassberger
, and M.
Paczuski
, “Edge direction and the structure of networks
,” Proc. Natl. Acad. Sci. U.S.A.
107
, 10815
–10820
(2010
). 69.
P.
Holme
, “Core-periphery organization of complex networks
,” Phys. Rev. E
72
, 046111
(2005
). 70.
Z.
Toroczkai
and K. E.
Bassler
, “Jamming is limited in scale-free systems
,” Nature
428
, 716
(2004
). 71.
P.
Erdos
and A.
Rényi
, “On the evolution of random graphs,” in Publication of the Mathematical Institute of the Hungarian Academy of Sciences (Hungarian Academy of Sciences, 1960), Vol. 5, pp. 17–61. 72.
M. E. J.
Newman
, “The structure of scientific collaboration networks
,” Proc. Natl. Acad. Sci. U.S.A.
98
, 404
–409
(2001
). © 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
