Prisoner's Dilemma on real social networks: Revisited

Previous Article
Metering effects in population systems
MBE Home
This Issue
Next Article
Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection

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

Abstract
Related Papers

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). doi: 10.1103/PhysRevE.63.030901.
[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. doi: 10.1142/S0129183100001334.
[3]	R. Axelrod, "The Evolution of Cooperation,", Revised Edition, (2006). doi: 10.1126/science.7466396.
[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. doi: 10.1073/pnas.1731324100.
[5]	D. M. Boyd and N. B. Ellison, Social network sites: Definition, history, and scholarship,, J. Computer-Mediated Comm., 13 (2008), 210. doi: 10.1109/EMR.2010.5559139.
[6]	A. Cassar, Coordination and cooperation in local, random and small world networks: Experimental evidence,, Game Econ. Behav., 58 (2007), 209. doi: 10.1016/j.geb.2006.03.008.
[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). doi: 10.1103/PhysRevE.77.032103.
[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). doi: 10.1103/PhysRevE.77.017103.
[9]	F. Chen, A mathematical analysis of public avoidance behavior during epidemics using game theory,, J. Theor. Biol., 302 (2012), 18. doi: 10.1016/j.jtbi.2012.03.002.
[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. doi: 10.1016/j.physa.2010.08.004.
[11]	L. R. Dong, Dynamic evolution with limited learning information on a small-world network,, Commun. Theor. Phys., 54 (2010), 578.
[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).
[13]	R. Durrett, "Random Graph Dynamics,", Cambridge University Press, (2007).
[14]	D. Easley and J. Kleinberg, " Networks, Crowds, and Markets: Reasoning about a Highly Connected World,", Cambridge University Press, (2010).
[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. doi: 10.1086/428716.
[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. doi: 10.1140/epjb/e2007-00124-5.
[17]	M. Granovetter, The strength of weak ties,, Am. J. Sociol., 78 (1973), 1360.
[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.
[19]	C. Hauert and G. Szabo, Game theory and physics,, Am. J. Phys., 73 (2005), 405. doi: 10.1119/1.1848514.
[20]	C. Hauert and L. A. Imhof, Evolutionary games in deme structured, finite populations,, J. Theor. Biol., 299 (2012), 106. doi: 10.1016/j.jtbi.2011.06.010.
[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).
[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.
[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.
[24]	N. Masuda and K. Aihara, Spatial prisoner's dilemma optimally played in small-world networks,, Phys. Lett. A, 313 (2003), 55. doi: 10.1016/S0375-9601(03)00693-5.
[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. doi: 10.1016/j.jpubeco.2007.09.001.
[26]	J. Maynard-Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982). doi: 10.1016/0022-5193(74)90110-6.
[27]	S. Milgram, The small-world problem,, Psychol. Today, 2 (1967), 60. doi: 10.1037/e400002009-005.
[28]	J. F. Nash, Equilibrium points in $n$-person games,, Proc. Natl. Acad. Sci. USA, 36 (1950), 48. doi: 10.1073/pnas.36.1.48.
[29]	M. E. J. Newman, "Networks: An Introduction,", Oxford University Press, (2010). doi: 10.1093/acprof:oso/9780199206650.001.0001.
[30]	M. A. Nowak and R. M. May, Evolutionary games and spatial chaos,, Nature, 359 (1992), 826. doi: 10.1038/359826a0.
[31]	M. A. Nowak, "Evolutionary Dynamics: Exploring the Equations of Life,", Harvard University Press, (2006).
[32]	M. A. Nowak, "Super Cooperators: Altruism, Evolution, and Why We Need Each Other to Succeed,", Free Press, (2012).
[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. doi: 10.1038/nature04605.
[34]	M. Perc, Double resonance in cooperation induced by noise and network variation for an evolutionary prisoner's dilemma,, New J. Phys., 8 (2006). doi: 10.1088/1367-2630/8/9/183.
[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 (2011).
[36]	G. Szabo and J. Vukov, Cooperation for volunteering and partially random partnerships,, Phys. Rev. E, 69 (2004). doi: 10.1103/PhysRevE.69.036107.
[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. doi: 10.1140/epjb/e2006-00395-2.
[38]	M. Tomochi, Defectors' niches: Prisoner's dilemma game on disordered networks,, Soc. Networks, 26 (2004), 309. doi: 10.1016/j.socnet.2004.08.003.
[39]	X. Thibert-Plante and L. Parrott, Prisoner's dilemma and clusters on small-world networks,, Complexity, 12 (2007), 22. doi: 10.1002/cplx.20182.
[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. doi: 10.1137/080734315.
[41]	A. L. Traud, P. J. Mucha and M. A. Porter, Social structure of Facebook networks,, Physica A, 391 (2012), 4165.
[42]	J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior,", Princeton University Press, (1944).
[43]	D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440.
[44]	D. J. Watts, "Small Worlds: The Dynamics of Networks, Between Order and Randomness,", Princeton University Press, (1999).
[45]	G. S. Wilkinson, Reciprocal food sharing in the vampire bat,, Nature, 308 (1984), 181.
[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). doi: 10.1103/PhysRevE.71.037103.
[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.
[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). doi: 10.1209/0295-5075/92/40009.
[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. doi: 10.1209/epl/i2006-10323-2.

show all references

References:

[1]	G. Abramson and M. Kuperman, Social games in a social network,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.030901.
[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. doi: 10.1142/S0129183100001334.
[3]	R. Axelrod, "The Evolution of Cooperation,", Revised Edition, (2006). doi: 10.1126/science.7466396.
[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. doi: 10.1073/pnas.1731324100.
[5]	D. M. Boyd and N. B. Ellison, Social network sites: Definition, history, and scholarship,, J. Computer-Mediated Comm., 13 (2008), 210. doi: 10.1109/EMR.2010.5559139.
[6]	A. Cassar, Coordination and cooperation in local, random and small world networks: Experimental evidence,, Game Econ. Behav., 58 (2007), 209. doi: 10.1016/j.geb.2006.03.008.
[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). doi: 10.1103/PhysRevE.77.032103.
[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). doi: 10.1103/PhysRevE.77.017103.
[9]	F. Chen, A mathematical analysis of public avoidance behavior during epidemics using game theory,, J. Theor. Biol., 302 (2012), 18. doi: 10.1016/j.jtbi.2012.03.002.
[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. doi: 10.1016/j.physa.2010.08.004.
[11]	L. R. Dong, Dynamic evolution with limited learning information on a small-world network,, Commun. Theor. Phys., 54 (2010), 578.
[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).
[13]	R. Durrett, "Random Graph Dynamics,", Cambridge University Press, (2007).
[14]	D. Easley and J. Kleinberg, " Networks, Crowds, and Markets: Reasoning about a Highly Connected World,", Cambridge University Press, (2010).
[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. doi: 10.1086/428716.
[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. doi: 10.1140/epjb/e2007-00124-5.
[17]	M. Granovetter, The strength of weak ties,, Am. J. Sociol., 78 (1973), 1360.
[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.
[19]	C. Hauert and G. Szabo, Game theory and physics,, Am. J. Phys., 73 (2005), 405. doi: 10.1119/1.1848514.
[20]	C. Hauert and L. A. Imhof, Evolutionary games in deme structured, finite populations,, J. Theor. Biol., 299 (2012), 106. doi: 10.1016/j.jtbi.2011.06.010.
[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).
[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.
[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.
[24]	N. Masuda and K. Aihara, Spatial prisoner's dilemma optimally played in small-world networks,, Phys. Lett. A, 313 (2003), 55. doi: 10.1016/S0375-9601(03)00693-5.
[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. doi: 10.1016/j.jpubeco.2007.09.001.
[26]	J. Maynard-Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982). doi: 10.1016/0022-5193(74)90110-6.
[27]	S. Milgram, The small-world problem,, Psychol. Today, 2 (1967), 60. doi: 10.1037/e400002009-005.
[28]	J. F. Nash, Equilibrium points in $n$-person games,, Proc. Natl. Acad. Sci. USA, 36 (1950), 48. doi: 10.1073/pnas.36.1.48.
[29]	M. E. J. Newman, "Networks: An Introduction,", Oxford University Press, (2010). doi: 10.1093/acprof:oso/9780199206650.001.0001.
[30]	M. A. Nowak and R. M. May, Evolutionary games and spatial chaos,, Nature, 359 (1992), 826. doi: 10.1038/359826a0.
[31]	M. A. Nowak, "Evolutionary Dynamics: Exploring the Equations of Life,", Harvard University Press, (2006).
[32]	M. A. Nowak, "Super Cooperators: Altruism, Evolution, and Why We Need Each Other to Succeed,", Free Press, (2012).
[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. doi: 10.1038/nature04605.
[34]	M. Perc, Double resonance in cooperation induced by noise and network variation for an evolutionary prisoner's dilemma,, New J. Phys., 8 (2006). doi: 10.1088/1367-2630/8/9/183.
[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 (2011).
[36]	G. Szabo and J. Vukov, Cooperation for volunteering and partially random partnerships,, Phys. Rev. E, 69 (2004). doi: 10.1103/PhysRevE.69.036107.
[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. doi: 10.1140/epjb/e2006-00395-2.
[38]	M. Tomochi, Defectors' niches: Prisoner's dilemma game on disordered networks,, Soc. Networks, 26 (2004), 309. doi: 10.1016/j.socnet.2004.08.003.
[39]	X. Thibert-Plante and L. Parrott, Prisoner's dilemma and clusters on small-world networks,, Complexity, 12 (2007), 22. doi: 10.1002/cplx.20182.
[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. doi: 10.1137/080734315.
[41]	A. L. Traud, P. J. Mucha and M. A. Porter, Social structure of Facebook networks,, Physica A, 391 (2012), 4165.
[42]	J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior,", Princeton University Press, (1944).
[43]	D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440.
[44]	D. J. Watts, "Small Worlds: The Dynamics of Networks, Between Order and Randomness,", Princeton University Press, (1999).
[45]	G. S. Wilkinson, Reciprocal food sharing in the vampire bat,, Nature, 308 (1984), 181.
[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). doi: 10.1103/PhysRevE.71.037103.
[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.
[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). doi: 10.1209/0295-5075/92/40009.
[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. doi: 10.1209/epl/i2006-10323-2.

[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, 2017, 13 (5) : 1-15. 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]	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
[7]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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
[17]	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
[18]	Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257
[19]	M. D. König, Stefano Battiston, M. Napoletano, F. Schweitzer. On algebraic graph theory and the dynamics of innovation networks. Networks & Heterogeneous Media, 2008, 3 (2) : 201-219. doi: 10.3934/nhm.2008.3.201
[20]	Manuela A. D. Aguiar, Ana Paula S. Dias, Martin Golubitsky, Maria Conceição A. Leite. Homogeneous coupled cell networks with s₃-symmetric quotient. Conference Publications, 2007, 2007 (Special) : 1-9. doi: 10.3934/proc.2007.2007.1

2018 Impact Factor: 1.313

American Institute of Mathematical Sciences