Special Issue on Mathematical Models and Methods for Complex SystemsNo Access

ON A MATHEMATICAL THEORY OF COMPLEX SYSTEMS ON NETWORKS WITH APPLICATION TO OPINION FORMATION

D. KNOPOFF

Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Medina Allende s/n, 5000 Córdoba, Argentina

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https://doi.org/10.1142/S0218202513400137Cited by:34 (Source: Crossref)

Abstract

This paper presents a development of the so-called kinetic theory for active particles to the modeling of living, hence complex, systems localized in networks. The overall system is viewed as a network of interacting nodes, mathematical equations are required to describe the dynamics in each node and in the whole network. These interactions, which are nonlinearly additive, are modeled by evolutive stochastic games. The first conceptual part derives a general mathematical structure, to be regarded as a candidate towards the derivation of models, suitable to capture the main features of the said systems. An application on opinion formation follows to show how the theory can generate specific models.

Keywords:

References

D. Acemoglu, K. Bimpikis and A. Ozdaglar, National Bureau Econ. Res. 16410 (2010), http://dx.doi.org/10.2139/ssrn.1682243. Google Scholar
D. Acemoglu and J. A. Robinson , Economic Origins of Dictatorship and Democracy ( Cambridge Univ. Press , 2006 ) . Google Scholar
G. Ajmone Marsan, N. Bellomo and M. Egidi, Kinet. Relat. Models 1, 249 (2008). Crossref, Web of Science, Google Scholar
G. Ajmone Marsan , N. Bellomo and A. Tosin , Complex Systems and Society — Modeling and Simulation ( Springer , 2013 ) . Google Scholar
L. A. N. Amaralet al., Proc. Natl. Acad. Sci. USA 97, 11149 (2000), DOI: 10.1073/pnas.200327197. Crossref, Web of Science, Google Scholar
L. Arlotti and E. De Angelis, Appl. Math. Lett. 24, 257 (2011), DOI: 10.1016/j.aml.2010.09.015. Crossref, Web of Science, Google Scholar
L. Arlottiet al., Appl. Math. Lett. 25, 490 (2012), DOI: 10.1016/j.aml.2011.09.043. Crossref, Web of Science, Google Scholar
F. Bagarello, Quantum Dynamics for Classical Systems: With Applications of the Number Operator (Wiley, 2013) pp. 109–128. Google Scholar
F. Bagarello and F. Oliveri, Math. Models Methods Appl. Sci. 23, 471 (2013), DOI: 10.1142/S0218202512500534. Link, Web of Science, Google Scholar
M. Balleriniet al., Proc. Natl. Acad. Sci. USA 105, 1232 (2008), DOI: 10.1073/pnas.0711437105. Crossref, Web of Science, Google Scholar
A. L. Barabási , Linked: The New Science of Networks ( Perseus , 2002 ) . Google Scholar
A. L. Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999). Crossref, Web of Science, Google Scholar
S. Barbera, M. Maschler and J. Shalev, Game. Econ. Behav. 37, 40 (2001). Crossref, Web of Science, Google Scholar
L. R. Bartels, The Oxford Handbook of American Elections and Political Behavior (Oxford Univ. Press, 2010) pp. 239–261. Crossref, Google Scholar
U. Bastollaet al., Nature 458, 1018 (2009), DOI: 10.1038/nature07950. Crossref, Web of Science, Google Scholar
N. Bellomo, Appl. Math. Lett. 23, 907 (2010), DOI: 10.1016/j.aml.2010.04.008. Crossref, Web of Science, Google Scholar
N. Bellomo and F. Brezzi, Math. Models Methods Appl. Sci. 22, 1103001 (2012), DOI: 10.1142/S0218202511030011. Link, Web of Science, Google Scholar
N. Bellomo and B. Carbonaro, Phys. Life Rev. 8, 1 (2011), DOI: 10.1016/j.plrev.2010.12.001. Crossref, Web of Science, Google Scholar
N. Bellomo and V. Coscia, Contemp. Math. 594, 99 (2013), DOI: 10.1090/conm/594/11783. Crossref, Google Scholar
N. Bellomo, B. Firmani and L. Guerri, Appl. Math. Lett. 12, 39 (1999), DOI: 10.1016/S0893-9659(98)00146-3. Crossref, Web of Science, Google Scholar
N. Bellomo, M. A. Herrero and A. Tosin, Kinet. Relat. Models 6, 459 (2013). Crossref, Web of Science, Google Scholar
N. Bellomo, D. Knopoff and J. Soler, Math. Models Methods Appl. Sci. 23, 1861 (2013), DOI: 10.1142/S021820251350053X. Link, Web of Science, Google Scholar
N. Bellomo, B. Piccoli and A. Tosin, Math. Models Methods Appl. Sci. 22, 1230004 (2012), DOI: 10.1142/S0218202512300049. Link, Web of Science, Google Scholar
N. Bellomo and J. Soler, Math. Models Methods Appl. Sci. 22, 1140006 (2012), DOI: 10.1142/S0218202511400069. Link, Web of Science, Google Scholar
A. Bellouquid, E. De Angelis and L. Fermo, Math. Models Methods Appl. Sci. 22, 1140003 (2012), DOI: 10.1142/S0218202511400033. Link, Web of Science, Google Scholar
A. Bellouquid, E. De Angelis and D. Knopoff, Math. Models Methods Appl. Sci. 23, 949 (2013), DOI: 10.1142/S0218202512500650. Link, Web of Science, Google Scholar
A. Benfenati and V. Coscia, Appl. Math. Lett. 26, 979 (2013), DOI: 10.1016/j.aml.2013.04.007. Crossref, Web of Science, Google Scholar
M. L. Bertotti and M. Delitala, Math. Models Methods Appl. Sci. 18, 913 (2008), DOI: 10.1142/S0218202508002905. Link, Web of Science, Google Scholar
M. L. Bertotti and M. Delitala, Math. Comput. Model. 48, 1107 (2008), DOI: 10.1016/j.mcm.2007.12.021. Crossref, Web of Science, Google Scholar
L. Boudin, A. Mercier and F. Salvarani, Physica A: Statist. Mech. Appl. 391, 5672 (2012), DOI: 10.1016/j.physa.2012.05.070. Crossref, Web of Science, Google Scholar
C. F. Camerer , Behavioral Game Theory: Experiments in Strategic Interaction ( Princeton Univ. Press , 2003 ) . Google Scholar
C. F. Camerer , G. Loewenstein and M. Rabin , Advances in Behavioral Economics ( Princeton Univ. Press , 2003 ) . Google Scholar
A. Cavagnaet al., Math. Models Methods Appl. Sci. 20, 1491 (2010), DOI: 10.1142/S0218202510004660. Link, Web of Science, Google Scholar
A. Cavagnaet al., Proc. Natl. Acad. Sci. USA 107, 11865 (2010), DOI: 10.1073/pnas.1005766107. Crossref, Web of Science, Google Scholar
D. Chong and J. N. Druckman, J. Commun. 57, 99 (2007). Web of Science, Google Scholar
V. Coscia, Math. Comput. Model. 54, 1919 (2011), DOI: 10.1016/j.mcm.2011.04.031. Crossref, Web of Science, Google Scholar
V. Coscia, L. Fermo and N. Bellomo, Comput. Math. Appl. 62, 3902 (2011), DOI: 10.1016/j.camwa.2011.09.043. Crossref, Web of Science, Google Scholar
E. De Angeliset al., Math. Comput. Model. 37, 1131 (2003), DOI: 10.1016/S0895-7177(03)00125-0. Crossref, Web of Science, Google Scholar
P. M. DeMarzo, D. Vayanos and J. Zwiebel, Quart. J. Econ. 118, 909 (2003), DOI: 10.1162/00335530360698469. Crossref, Web of Science, Google Scholar
P. Markowich, J. F. Pietschmann and M. T. Wolfram, Proc. Roy. Soc. London A Mat. 465, 3687 (2009). Web of Science, Google Scholar
R. Eftimie, J. Math. Biol. 65, 35 (2012), DOI: 10.1007/s00285-011-0452-2. Crossref, Web of Science, Google Scholar
N. Ganguly , A. Deutsch and A. Mukherjee , Dynamics on and of Complex Networks. Applications to Biology, Computer Science, and the Social Sciences ( Birkhäuser , 2009 ) . Google Scholar
A. Gerber, D. S. Karlan and D. Bergan, Does the media matter? A field experiment measuring the effect of newspapers on voting behavior and political opinions, Yale Economic Applications and Policy Discussion Paper(12), (2006), available at http://ssrn.com/abstract=903812 . Google Scholar
H. Gintis , Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction ( Princeton Univ. Press , 2009 ) . Crossref, Google Scholar
S. Goyal and F. Vega-Redondo, Game Econ. Behav. 50, 178 (2005), DOI: 10.1016/j.geb.2004.01.005. Crossref, Web of Science, Google Scholar
D. Helbing, Physica A 193, 241 (1993), DOI: 10.1016/0378-4371(93)90028-3. Crossref, Web of Science, Google Scholar
D. Helbing , Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes ( Springer , 2010 ) . Crossref, Google Scholar
D. Helbing and A. Johansson, Phys. Rev. E 81, 016112 (2010), DOI: 10.1103/PhysRevE.81.016112. Crossref, Web of Science, Google Scholar
C. Kaiser, S. Schlick and F. Bodendorf, Knowledge Discovery, Knowledge Engineering and Knowledge Management (Springer, 2013) pp. 107–121. Crossref, Google Scholar
S. Y. Kim, C. H. Park and K. Kim, Int. J. Mod. Phys. C 18, 1429 (2007), DOI: 10.1142/S0129183107011431. Link, Web of Science, Google Scholar
D. Knopoff, Math. Models Methods Appl. Sci. 23, 541 (2013), DOI: 10.1142/S0218202512500558. Link, Web of Science, Google Scholar
P. Langer, M. A. Nowak and C. Hauert, J. Theor. Biol. 250, 634 (2008), DOI: 10.1016/j.jtbi.2007.11.002. Crossref, Web of Science, Google Scholar
R. M. May, Science 303, 790 (2004), DOI: 10.1126/science.1094442. Crossref, Web of Science, Google Scholar
E. Mayr, Science 305, 46 (2004), DOI: 10.1126/science.1100561. Crossref, Web of Science, Google Scholar
N. McCarty and A. Meirowitz , Political Game Theory ( Cambridge Univ. Press , 2007 ) . Crossref, Google Scholar
M. A. Nowak , Evolutionary Dynamics. Exploring the Equations of Life ( Harvard Univ. Press , 2006 ) . Crossref, Google Scholar
M. A. Nowak and K. Sigmund, Science 303, 793 (2004), DOI: 10.1126/science.1093411. Crossref, Web of Science, Google Scholar
F. C. Santos, J. M. Pacheco and T. Lenaerts, Proc. Natl. Acad. Sci. USA 103, 3490 (2006), DOI: 10.1073/pnas.0508201103. Crossref, Web of Science, Google Scholar
F. C. Santoset al., Math. Models Methods Appl. Sci. 22, 1140004 (2012), DOI: 10.1142/S0218202511400045. Link, Web of Science, Google Scholar
M. Schefferet al., Nature 461, 53 (2009), DOI: 10.1038/nature08227. Crossref, Web of Science, Google Scholar
N. N. Taleb , The Black Swan: The Impact of the Highly Improbable ( Random House , 2007 ) . Google Scholar
C. E. Tarnitaet al., Proc. Natl. Acad. Sci. USA 106, 8601 (2009), DOI: 10.1073/pnas.0903019106. Crossref, Web of Science, Google Scholar
G. Toscani, Commun. Math. Sci. 4, 481 (2006), DOI: 10.4310/CMS.2006.v4.n3.a1. Crossref, Web of Science, Google Scholar
G. Toscani and C. Brugna, Econophysics and Economics of Games, Social Choices and Quantitative Techniques (Springer, 2010) pp. 71–82. Crossref, Google Scholar
G. Toscani, C. Brugna and S. Demichelis, J. Statist. Phys. 151, 549 (2013), DOI: 10.1007/s10955-012-0653-0. Crossref, Web of Science, Google Scholar
F. Vega-Redondo , Complex Social Networks ( Cambridge Univ. Press , 2007 ) . Crossref, Google Scholar
R. Wang and Q. A. Wang, Phys. Rev. E 84, 036108 (2011), DOI: 10.1103/PhysRevE.84.036108. Crossref, Web of Science, Google Scholar
D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998), DOI: 10.1038/30918. Crossref, Web of Science, Google Scholar