American Institute of Mathematical Sciences

Advanced
2013, 10(5&6): 1381-1398. doi: 10.3934/mbe.2013.10.1381

Prisoner's Dilemma on real social networks: Revisited

Sharon M. Cameron 1, and  Ariel Cintrón-Arias 2,

1. 

East Tennessee State University, Department of Mathematics and Statistics, Box 70663, Johnson City, TN 37614-1701, United States
2. 

East Tennessee State University, Department of Mathematics and Statistics, Institute for Quantitative Biology, Box 70663, Johnson City, TN 37614-1701, United States

Received  August 2012 Revised  October 2012 Published  August 2013

Prisoner's Dilemma is a game theory model used to describe altruistic behavior seen in various populations. This theoretical game is important in understanding why a seemingly selfish strategy does persist and spread throughout a population that is mixing homogeneously at random. For a population with structure determined by social interactions, Prisoner's Dilemma brings to light certain requirements for the altruistic strategy to become established. Monte Carlo simulations of Prisoner's Dilemma are carried out using both simulated social networks and a dataset of a real social network. In both scenarios we confirm the requirements for the persistence of altruism in a population.
Keywords: social networks, Prisoner's Dilemma., Game theory, small-world networks.
Mathematics Subject Classification: Primary: 91A40; Secondary: 91D3.
Citation: Sharon M. Cameron, Ariel Cintrón-Arias. Prisoner's Dilemma on real social networks: Revisited. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1381-1398. doi: 10.3934/mbe.2013.10.1381
References:
[1]

G. Abramson and M. Kuperman, Social games in a social network, Phys. Rev. E, 63 (2001), 030901. doi: 10.1103/PhysRevE.63.030901.  Google Scholar

[2]

E. Ahmed and A.S. Elgazzar, On local prisoner's dilemma game with Pareto updating rule, Int. J. Mod. Phys. C, 11 (2000), 1539-1544. doi: 10.1142/S0129183100001334.  Google Scholar

[3]

R. Axelrod, "The Evolution of Cooperation," Revised Edition, Basic Books, New York, 2006. doi: 10.1126/science.7466396.  Google Scholar

[4]

C. T. Bauch, A. P. Galvani and D. J. Earn, Group interest versus self-interest in smallpox vaccination policy, P. Natl. Acad. Sci. USA, 100 (2003), 10564-10567. doi: 10.1073/pnas.1731324100.  Google Scholar

[5]

D. M. Boyd and N. B. Ellison, Social network sites: Definition, history, and scholarship, J. Computer-Mediated Comm., 13 (2008), 210-230. doi: 10.1109/EMR.2010.5559139.  Google Scholar

[6]

A. Cassar, Coordination and cooperation in local, random and small world networks: Experimental evidence, Game Econ. Behav., 58 (2007), 209-230. doi: 10.1016/j.geb.2006.03.008.  Google Scholar

[7]

Y. Chen, S. M. Qin, L. C. Yu and S. L. Zhang, Emergence of synchronization induced by the interplay between two prisoner's dilemma games with volunteering in small-world networks, Phys. Rev. E, 77 (2008), 032103. doi: 10.1103/PhysRevE.77.032103.  Google Scholar

[8]

X. J. Chen and L. Wang, Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game, Phys. Rev. E, 77 (2008), 017103. doi: 10.1103/PhysRevE.77.017103.  Google Scholar

[9]

F. Chen, A mathematical analysis of public avoidance behavior during epidemics using game theory, J. Theor. Biol., 302 (2012), 18-28. doi: 10.1016/j.jtbi.2012.03.002.  Google Scholar

[10]

X. H. Deng, Y. Liu and Z. G. Chen, Memory-based evolutionary game on small-world network with tunable heterogeneity, Physica A, 389 (2010), 5173-5181. doi: 10.1016/j.physa.2010.08.004.  Google Scholar

[11]

L. R. Dong, Dynamic evolution with limited learning information on a small-world network, Commun. Theor. Phys., 54 (2010), 578-582. Google Scholar

[12]

W. B. Du, X. B. Cao, L. Zhao and H. Zhou, Evolutionary games on weighted Newman-Watts small-world networks, Chinese Phys. Lett., 26 (2009), 058701. Google Scholar

[13]

R. Durrett, "Random Graph Dynamics," Cambridge University Press, Cambridge, 2007.  Google Scholar

[14]

D. Easley and J. Kleinberg, " Networks, Crowds, and Markets: Reasoning about a Highly Connected World," Cambridge University Press, New York, 2010.  Google Scholar

[15]

V. M. Eguiluz, 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. Sociol., 110 (2005), 977-1008. doi: 10.1086/428716.  Google Scholar

[16]

F. Fu, L. H. Liu and L. Wang, Evolutionary prisoner's dilemma on heterogeneous Newman-Watts small-world network, Eur. Phys. J. B, 56 (2007), 367-372. doi: 10.1140/epjb/e2007-00124-5.  Google Scholar

[17]

M. Granovetter, The strength of weak ties, Am. J. Sociol., 78 (1973), 1360-1380. Google Scholar

[18]

J. Y. Guan, Z. X. Wu, Z. G. Huang and Y. H. Wang, Prisoner's dilemma game with nonlinear attractive effect on regular small-world networks, Chinese Phys. Lett., 23 (2006), 2874-2877. Google Scholar

[19]

C. Hauert and G. Szabo, Game theory and physics, Am. J. Phys., 73 (2005), 405-414. doi: 10.1119/1.1848514.  Google Scholar

[20]

C. Hauert and L. A. Imhof, Evolutionary games in deme structured, finite populations, J. Theor. Biol., 299 (2012), 106-112. doi: 10.1016/j.jtbi.2011.06.010.  Google Scholar

[21]

B. J. Kim, A. Trusina, P. Holme, P. Minnhagen, J. S. Chung and M. Y. Choi, Dynamic instabilities induced by asymmetric influence: Prisoner's dilemma game in small-world networks, Phys. Rev. E, 66 (2002), 021907. Google Scholar

[22]

K. Lewis, J. Kaufman, M. Gonzalez, M. Wimmer and N. A. Christakis, Tastes, ties, and time: A new (cultural, multiplex, and longitudinal) social network dataset using Facebook.com, Social Networks, 30 (2008), 330-342. Google Scholar

[23]

T. Lenaerts, J. M. Pacheco and F. C. Santos, Evolutionary dynamics of social dilemmas in structured heterogenous populations, P. Natl. Acad. Sci. USA, 109 (2006), 3490-3494. Google Scholar

[24]

N. Masuda and K. Aihara, Spatial prisoner's dilemma optimally played in small-world networks, Phys. Lett. A, 313 (2003), 55-61. doi: 10.1016/S0375-9601(03)00693-5.  Google Scholar

[25]

A. Mayer and S. L. Puller, The old boy (and girl) network: Social network formation on university campuses, J. Public Econom., 92 (2008), 328-347. doi: 10.1016/j.jpubeco.2007.09.001.  Google Scholar

[26]

J. Maynard-Smith, "Evolution and the Theory of Games," Cambridge University Press, Cambridge, 1982. doi: 10.1016/0022-5193(74)90110-6.  Google Scholar

[27]

S. Milgram, The small-world problem, Psychol. Today, 2 (1967), 60-67. doi: 10.1037/e400002009-005.  Google Scholar

[28]

J. F. Nash, Equilibrium points in $n$-person games, Proc. Natl. Acad. Sci. USA, 36 (1950), 48-49. doi: 10.1073/pnas.36.1.48.  Google Scholar

[29]

M. E. J. Newman, "Networks: An Introduction," Oxford University Press, Oxford, 2010. doi: 10.1093/acprof:oso/9780199206650.001.0001.  Google Scholar

[30]

M. A. Nowak and R. M. May, Evolutionary games and spatial chaos, Nature, 359 (1992), 826-829. doi: 10.1038/359826a0.  Google Scholar

[31]

M. A. Nowak, "Evolutionary Dynamics: Exploring the Equations of Life," Harvard University Press, Cambridge, 2006.  Google Scholar

[32]

M. A. Nowak, "Super Cooperators: Altruism, Evolution, and Why We Need Each Other to Succeed," Free Press, New York, 2012. Google Scholar

[33]

H. Ohtsuki, C. Hauert, E. Lieberman and M. A. Nowak, A simple rule for the evolution of cooperation on graphs and social networks, Nature, 441 (2006), 502-505. doi: 10.1038/nature04605.  Google Scholar

[34]

M. Perc, Double resonance in cooperation induced by noise and network variation for an evolutionary prisoner's dilemma, New J. Phys., 8 (2006), 183. doi: 10.1088/1367-2630/8/9/183.  Google Scholar

[35]

E. Shim, L. A. Meyers and A. P. Galvani, Optimal H1N1 vaccination strategies based on self-interest versus group interest, BMC Public Health, 11, Suppl. 1, 25 February, 2011. Google Scholar

[36]

G. Szabo and J. Vukov, Cooperation for volunteering and partially random partnerships, Phys. Rev. E, 69 (2004), 036107. doi: 10.1103/PhysRevE.69.036107.  Google Scholar

[37]

C. L. Tang, W. X. Wang, X. Wu and B. H. Wang, Effects of average degree on cooperation in networked evolutionary game, Eur. Phys. J. B, 53 (2006), 411-415. doi: 10.1140/epjb/e2006-00395-2.  Google Scholar

[38]

M. Tomochi, Defectors' niches: Prisoner's dilemma game on disordered networks, Soc. Networks, 26 (2004), 309-321. doi: 10.1016/j.socnet.2004.08.003.  Google Scholar

[39]

X. Thibert-Plante and L. Parrott, Prisoner's dilemma and clusters on small-world networks, Complexity, 12 (2007), 22-36. doi: 10.1002/cplx.20182.  Google Scholar

[40]

A. L. Traud, E. D. Kelsic, P. J. Mucha and M. A. Porter, Comparing community structure to characteristics in online collegiate social networks, SIAM Rev., 53 (2011), 526-543. doi: 10.1137/080734315.  Google Scholar

[41]

A. L. Traud, P. J. Mucha and M. A. Porter, Social structure of Facebook networks, Physica A, 391 (2012), 4165-4180. Google Scholar

[42]

J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior," Princeton University Press, Princeton, NJ, 1944.  Google Scholar

[43]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. Google Scholar

[44]

D. J. Watts, "Small Worlds: The Dynamics of Networks, Between Order and Randomness," Princeton University Press, Princeton, NJ, 1999.  Google Scholar

[45]

G. S. Wilkinson, Reciprocal food sharing in the vampire bat, Nature, 308 (1984), 181-184. Google Scholar

[46]

Z. X. Wu, X. J. Xu, Y. Chen and Y. H. Wang, Spatial prisoner's dilemma game with volunteering in Newman-Watts small-world networks, Phys. Rev. E, 71 (2005), 037103. doi: 10.1103/PhysRevE.71.037103.  Google Scholar

[47]

Z. X. Wu, X. J. Xu and Y. H. Wang, Prisoner's dilemma game with heterogeneous influential effect on regular small-world networks, Chinese Phys. Lett., 23 (2006), 531-534. Google Scholar

[48]

Q. Z. Xia, X. H. Liao, W. Li and G. Hu, Enhance cooperation by catastrophic collapses of rich cooperators in coevolutionary networks, EPL-Europhys. Lett., 92 (2010), 40009. doi: 10.1209/0295-5075/92/40009.  Google Scholar

[49]

L. X. Zhong, D. F. Zheng, B. Zheng, C. Xu and P. M. Hui, Networking effects on cooperation in evolutionary snowdrift game, Europhys. Lett., 76 (2006), 724-730. doi: 10.1209/epl/i2006-10323-2.  Google Scholar

show all references

References:
[1]

G. Abramson and M. Kuperman, Social games in a social network, Phys. Rev. E, 63 (2001), 030901. doi: 10.1103/PhysRevE.63.030901.  Google Scholar

[2]

E. Ahmed and A.S. Elgazzar, On local prisoner's dilemma game with Pareto updating rule, Int. J. Mod. Phys. C, 11 (2000), 1539-1544. doi: 10.1142/S0129183100001334.  Google Scholar

[3]

R. Axelrod, "The Evolution of Cooperation," Revised Edition, Basic Books, New York, 2006. doi: 10.1126/science.7466396.  Google Scholar

[4]

C. T. Bauch, A. P. Galvani and D. J. Earn, Group interest versus self-interest in smallpox vaccination policy, P. Natl. Acad. Sci. USA, 100 (2003), 10564-10567. doi: 10.1073/pnas.1731324100.  Google Scholar

[5]

D. M. Boyd and N. B. Ellison, Social network sites: Definition, history, and scholarship, J. Computer-Mediated Comm., 13 (2008), 210-230. doi: 10.1109/EMR.2010.5559139.  Google Scholar

[6]

A. Cassar, Coordination and cooperation in local, random and small world networks: Experimental evidence, Game Econ. Behav., 58 (2007), 209-230. doi: 10.1016/j.geb.2006.03.008.  Google Scholar

[7]

Y. Chen, S. M. Qin, L. C. Yu and S. L. Zhang, Emergence of synchronization induced by the interplay between two prisoner's dilemma games with volunteering in small-world networks, Phys. Rev. E, 77 (2008), 032103. doi: 10.1103/PhysRevE.77.032103.  Google Scholar

[8]

X. J. Chen and L. Wang, Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game, Phys. Rev. E, 77 (2008), 017103. doi: 10.1103/PhysRevE.77.017103.  Google Scholar

[9]

F. Chen, A mathematical analysis of public avoidance behavior during epidemics using game theory, J. Theor. Biol., 302 (2012), 18-28. doi: 10.1016/j.jtbi.2012.03.002.  Google Scholar

[10]

X. H. Deng, Y. Liu and Z. G. Chen, Memory-based evolutionary game on small-world network with tunable heterogeneity, Physica A, 389 (2010), 5173-5181. doi: 10.1016/j.physa.2010.08.004.  Google Scholar

[11]

L. R. Dong, Dynamic evolution with limited learning information on a small-world network, Commun. Theor. Phys., 54 (2010), 578-582. Google Scholar

[12]

W. B. Du, X. B. Cao, L. Zhao and H. Zhou, Evolutionary games on weighted Newman-Watts small-world networks, Chinese Phys. Lett., 26 (2009), 058701. Google Scholar

[13]

R. Durrett, "Random Graph Dynamics," Cambridge University Press, Cambridge, 2007.  Google Scholar

[14]

D. Easley and J. Kleinberg, " Networks, Crowds, and Markets: Reasoning about a Highly Connected World," Cambridge University Press, New York, 2010.  Google Scholar

[15]

V. M. Eguiluz, 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. Sociol., 110 (2005), 977-1008. doi: 10.1086/428716.  Google Scholar

[16]

F. Fu, L. H. Liu and L. Wang, Evolutionary prisoner's dilemma on heterogeneous Newman-Watts small-world network, Eur. Phys. J. B, 56 (2007), 367-372. doi: 10.1140/epjb/e2007-00124-5.  Google Scholar

[17]

M. Granovetter, The strength of weak ties, Am. J. Sociol., 78 (1973), 1360-1380. Google Scholar

[18]

J. Y. Guan, Z. X. Wu, Z. G. Huang and Y. H. Wang, Prisoner's dilemma game with nonlinear attractive effect on regular small-world networks, Chinese Phys. Lett., 23 (2006), 2874-2877. Google Scholar

[19]

C. Hauert and G. Szabo, Game theory and physics, Am. J. Phys., 73 (2005), 405-414. doi: 10.1119/1.1848514.  Google Scholar

[20]

C. Hauert and L. A. Imhof, Evolutionary games in deme structured, finite populations, J. Theor. Biol., 299 (2012), 106-112. doi: 10.1016/j.jtbi.2011.06.010.  Google Scholar

[21]

B. J. Kim, A. Trusina, P. Holme, P. Minnhagen, J. S. Chung and M. Y. Choi, Dynamic instabilities induced by asymmetric influence: Prisoner's dilemma game in small-world networks, Phys. Rev. E, 66 (2002), 021907. Google Scholar

[22]

K. Lewis, J. Kaufman, M. Gonzalez, M. Wimmer and N. A. Christakis, Tastes, ties, and time: A new (cultural, multiplex, and longitudinal) social network dataset using Facebook.com, Social Networks, 30 (2008), 330-342. Google Scholar

[23]

T. Lenaerts, J. M. Pacheco and F. C. Santos, Evolutionary dynamics of social dilemmas in structured heterogenous populations, P. Natl. Acad. Sci. USA, 109 (2006), 3490-3494. Google Scholar

[24]

N. Masuda and K. Aihara, Spatial prisoner's dilemma optimally played in small-world networks, Phys. Lett. A, 313 (2003), 55-61. doi: 10.1016/S0375-9601(03)00693-5.  Google Scholar

[25]

A. Mayer and S. L. Puller, The old boy (and girl) network: Social network formation on university campuses, J. Public Econom., 92 (2008), 328-347. doi: 10.1016/j.jpubeco.2007.09.001.  Google Scholar

[26]

J. Maynard-Smith, "Evolution and the Theory of Games," Cambridge University Press, Cambridge, 1982. doi: 10.1016/0022-5193(74)90110-6.  Google Scholar

[27]

S. Milgram, The small-world problem, Psychol. Today, 2 (1967), 60-67. doi: 10.1037/e400002009-005.  Google Scholar

[28]

J. F. Nash, Equilibrium points in $n$-person games, Proc. Natl. Acad. Sci. USA, 36 (1950), 48-49. doi: 10.1073/pnas.36.1.48.  Google Scholar

[29]

M. E. J. Newman, "Networks: An Introduction," Oxford University Press, Oxford, 2010. doi: 10.1093/acprof:oso/9780199206650.001.0001.  Google Scholar

[30]

M. A. Nowak and R. M. May, Evolutionary games and spatial chaos, Nature, 359 (1992), 826-829. doi: 10.1038/359826a0.  Google Scholar

[31]

M. A. Nowak, "Evolutionary Dynamics: Exploring the Equations of Life," Harvard University Press, Cambridge, 2006.  Google Scholar

[32]

M. A. Nowak, "Super Cooperators: Altruism, Evolution, and Why We Need Each Other to Succeed," Free Press, New York, 2012. Google Scholar

[33]

H. Ohtsuki, C. Hauert, E. Lieberman and M. A. Nowak, A simple rule for the evolution of cooperation on graphs and social networks, Nature, 441 (2006), 502-505. doi: 10.1038/nature04605.  Google Scholar

[34]

M. Perc, Double resonance in cooperation induced by noise and network variation for an evolutionary prisoner's dilemma, New J. Phys., 8 (2006), 183. doi: 10.1088/1367-2630/8/9/183.  Google Scholar

[35]

E. Shim, L. A. Meyers and A. P. Galvani, Optimal H1N1 vaccination strategies based on self-interest versus group interest, BMC Public Health, 11, Suppl. 1, 25 February, 2011. Google Scholar

[36]

G. Szabo and J. Vukov, Cooperation for volunteering and partially random partnerships, Phys. Rev. E, 69 (2004), 036107. doi: 10.1103/PhysRevE.69.036107.  Google Scholar

[37]

C. L. Tang, W. X. Wang, X. Wu and B. H. Wang, Effects of average degree on cooperation in networked evolutionary game, Eur. Phys. J. B, 53 (2006), 411-415. doi: 10.1140/epjb/e2006-00395-2.  Google Scholar

[38]

M. Tomochi, Defectors' niches: Prisoner's dilemma game on disordered networks, Soc. Networks, 26 (2004), 309-321. doi: 10.1016/j.socnet.2004.08.003.  Google Scholar

[39]

X. Thibert-Plante and L. Parrott, Prisoner's dilemma and clusters on small-world networks, Complexity, 12 (2007), 22-36. doi: 10.1002/cplx.20182.  Google Scholar

[40]

A. L. Traud, E. D. Kelsic, P. J. Mucha and M. A. Porter, Comparing community structure to characteristics in online collegiate social networks, SIAM Rev., 53 (2011), 526-543. doi: 10.1137/080734315.  Google Scholar

[41]

A. L. Traud, P. J. Mucha and M. A. Porter, Social structure of Facebook networks, Physica A, 391 (2012), 4165-4180. Google Scholar

[42]

J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior," Princeton University Press, Princeton, NJ, 1944.  Google Scholar

[43]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. Google Scholar

[44]

D. J. Watts, "Small Worlds: The Dynamics of Networks, Between Order and Randomness," Princeton University Press, Princeton, NJ, 1999.  Google Scholar

[45]

G. S. Wilkinson, Reciprocal food sharing in the vampire bat, Nature, 308 (1984), 181-184. Google Scholar

[46]

Z. X. Wu, X. J. Xu, Y. Chen and Y. H. Wang, Spatial prisoner's dilemma game with volunteering in Newman-Watts small-world networks, Phys. Rev. E, 71 (2005), 037103. doi: 10.1103/PhysRevE.71.037103.  Google Scholar

[47]

Z. X. Wu, X. J. Xu and Y. H. Wang, Prisoner's dilemma game with heterogeneous influential effect on regular small-world networks, Chinese Phys. Lett., 23 (2006), 531-534. Google Scholar

[48]

Q. Z. Xia, X. H. Liao, W. Li and G. Hu, Enhance cooperation by catastrophic collapses of rich cooperators in coevolutionary networks, EPL-Europhys. Lett., 92 (2010), 40009. doi: 10.1209/0295-5075/92/40009.  Google Scholar

[49]

L. X. Zhong, D. F. Zheng, B. Zheng, C. Xu and P. M. Hui, Networking effects on cooperation in evolutionary snowdrift game, Europhys. Lett., 76 (2006), 724-730. doi: 10.1209/epl/i2006-10323-2.  Google Scholar

[1]

Qingyun Wang, Xia Shi, Guanrong Chen. Delay-induced synchronization transition in small-world Hodgkin-Huxley neuronal networks with channel blocking. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 607-621. doi: 10.3934/dcdsb.2011.16.607
[2]

Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1927-1941. doi: 10.3934/jimo.2019036
[3]

Kashi Behrstock, Michel Benaïm, Morris W. Hirsch. Smale strategies for network prisoner's dilemma games. Journal of Dynamics & Games, 2015, 2 (2) : 141-155. doi: 10.3934/jdg.2015.2.141
[4]

Ethan Akin. Good strategies for the Iterated Prisoner's Dilemma: Smale vs. Markov. Journal of Dynamics & Games, 2017, 4 (3) : 217-253. doi: 10.3934/jdg.2017014
[5]

Werner Creixell, Juan Carlos Losada, Tomás Arredondo, Patricio Olivares, Rosa María Benito. Serendipity in social networks. Networks & Heterogeneous Media, 2012, 7 (3) : 363-371. doi: 10.3934/nhm.2012.7.363
[6]

Yuki Kumagai. Social networks and global transactions. Journal of Dynamics & Games, 2019, 6 (3) : 211-219. doi: 10.3934/jdg.2019015
[7]

Robin Cohen, Alan Tsang, Krishna Vaidyanathan, Haotian Zhang. Analyzing opinion dynamics in online social networks. Big Data & Information Analytics, 2016, 1 (4) : 279-298. doi: 10.3934/bdia.2016011
[8]

Almerima Jamakovic, Steve Uhlig. On the relationships between topological measures in real-world networks. Networks & Heterogeneous Media, 2008, 3 (2) : 345-359. doi: 10.3934/nhm.2008.3.345
[9]

Serap Ergün, Sirma Zeynep Alparslan Gök, Tuncay Aydoǧan, Gerhard Wilhelm Weber. Performance analysis of a cooperative flow game algorithm in ad hoc networks and a comparison to Dijkstra's algorithm. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1085-1100. doi: 10.3934/jimo.2018086
[10]

Anne Shiu, Timo de Wolff. Nondegenerate multistationarity in small reaction networks. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2683-2700. doi: 10.3934/dcdsb.2018270
[11]

Sourabh Bhattacharya, Abhishek Gupta, Tamer Başar. Jamming in mobile networks: A game-theoretic approach. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 1-30. doi: 10.3934/naco.2013.3.1
[12]

Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609
[13]

Meng Zhao. The longtime behavior of the model with nonlocal diffusion and free boundaries in online social networks. Electronic Research Archive, 2020, 28 (3) : 1143-1160. doi: 10.3934/era.2020063
[14]

Lea Ellwardt, Penélope Hernández, Guillem Martínez-Cánovas, Manuel Muñoz-Herrera. Conflict and segregation in networks: An experiment on the interplay between individual preferences and social influence. Journal of Dynamics & Games, 2016, 3 (2) : 191-216. doi: 10.3934/jdg.2016010
[15]

Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240
[16]

Yi Xu, Qing Yang, Dianhui Chu. Exploring timeliness for accurate recommendation in location-based social networks. Mathematical Foundations of Computing, 2018, 1 (1) : 11-48. doi: 10.3934/mfc.2018002
[17]

Xiaoshuang Xing, Gaofei Sun, Yong Jin, Wenyi Tang, Xiuzhen Cheng. Relay selection based on social relationship prediction and information leakage reduction for mobile social networks. Mathematical Foundations of Computing, 2018, 1 (4) : 369-382. doi: 10.3934/mfc.2018018
[18]

Zhuwei Qin, Fuxun Yu, Chenchen Liu, Xiang Chen. How convolutional neural networks see the world --- A survey of convolutional neural network visualization methods. Mathematical Foundations of Computing, 2018, 1 (2) : 149-180. doi: 10.3934/mfc.2018008
[19]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295
[20]

Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences