Master equations in cooperative and social phenomena

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Master equations when applied to model the behaviour and/or attitudes of interacting living beings, resulting in supra-individual patterns of behaviour and collective effects. Master equations give frequently a good framework for the development of the mechanistic models needed in social sciences when dealing with probabilistic human behaviour. They have been applied in some models of public-opinion dynamics and other social processes in [a26] and [a24], in the dynamics of urban systems in [a2], in the dynamics of vehicular traffic in [a19], in the migration of populations in [a25], in informational processes in [a11], and in networks of cognitive systems in [a15] and [a16].

In these models, the transition probabilities are, in general, obtained either experimentally, by means of some theoretical model or by means of ad hoc hypotheses.

In many of the most complex social processes one finds that:

i) the process involves many individual behaviours;

ii) individual behaviour is not deterministically predictable, but it can be considered as a (probabilistic) function of the internal attitude and the external context (or scenario);

iii) the individual interactions and synchronized actions bring about social regularities (e.g., mobilizations [a22]) as well as institutions;

iv) the macro-social patterns obtained in this manner are projected in mental meanings which are negotiated between groups and individuals; during the whole process these meanings have an influence on the individuals attitudes;

v) any change in these attitudes must imply a change in the probability of persistence and no-persistence of the individual contributions to the specific synchronization that is viewed as the origin of the macro pattern.

Models in which the transition probabilities depend on the form of the distribution $p(x,t)$ are appropriate when dealing with these social and cooperative self-organizing processes, [a17], [a9], [a11], [a20], with feedbacks between the micro and the macro levels. In this kind of problems, a set of micro processes synchronize themselves in such a way that they produce a macroscopic space-temporal structure; subsequently, this structure probabilistically facilitates the maintenance of the synchronization that is able to produce it.

Bifurcation phenomena and other non-linear behaviours obtainable from master and Fokker–Planck equations (cf. Fokker–Planck equation) when the transitions probabilities are non-linear functions of the state or depend on the whole distribution solution are given in [a7], [a8].

Probabilistic models can also be used in more complex social processes, involving many self-organizing systems.

In this regard, technological systems of production, distribution and control can be considered as "networks" of self-organizing systems, and the economic, technological and social change in human societies can be considered as an evolutive dynamics of cooperation between those lower-level systems. This kind of non-linear network models is increasingly used in thermodynamics [a18], evolutive biology [a5], [a21], [a6], physiology [a4], ecology [a23], [a1], and receives an increasing attention in social systems [a12], [a3], [a14], [a13]. In this context, one hypercycle of fluxes between self-organizing systems can define the higher-level organization by means of some probabilistic equation, e.g. one master equation for the relevant variables in the new macro level, contributing a good framework to the research of the system's deterministic and stochastic dynamics.


[a1] P.M. Allen, "Evolution: why the whole is greater than the sum of the parts" W. Wolff (ed.) , Ecodynamics , Springer (1988)
[a2] P.M. Allen, M. Sanglier, G. Engelen, "Chance and necessity in urban systems" P. Schuster (ed.) , Stochastic phenomena and chaotic behaviour in complex systems , Springer (1984) pp. 231–249
[a3] M. Callon, "Society in the making: the study of technology as a tool for sociological analysis" W.E. Bijker (ed.) T.P. Hughes (ed.) T. Pinch (ed.) , The Social Construction of the Technological Systems , MIT (1989)
[a4] G. Chauvet, "Traité de physiologie théorique. Physiologie intégrative-champ et organization fonctionnelle" , III , Masson (1990)
[a5] M. Eigen, P. Schuster, "The hypercycle: a principle of natural self-organization" , Springer (1979)
[a6] R. Feistel, W. Ebeling, "Evolution of complex systems" , Kluwer Acad. Publ. (1989)
[a7] A. García-Olivares, "Self-organization and intermittency in social sciences: towards a science of complexity" Kybernetes , 22 : 3 (1993) pp. 14–24
[a8] A. García-Olivares, A. Muñoz, "Fokker–Planck equations in the simulation of complex systems" Mathematics and Computers in Simulation , 36 (1994) pp. 17–48
[a9] H. Haken, "Synergetics: an introduction" , Springer (1983)
[a10] H. Haken, "Advanced synergetics" , Springer (1987)
[a11] H. Haken, "Information and self-organization" , Springer (1988)
[a12] P.M. Hejl, "Towards a theory of social systems: self-organization and self-maintenance, self-reference and syn-reference" , Self-Organization and Management of Social Systems , Springer (1984)
[a13] B. Latour, "Nous n'avons jamais été modernes" , Ed. La Découverte (1991)
[a14] J. Law, "Technology and heterogeneous engineering: the case of Portuguese expansion" W.E. Bijker (ed.) T.P. Hughes (ed.) T.Pinch (ed.) , The Social Construction of the Technological Systems , MIT (1989)
[a15] J.S. Nicolis, "Bifurcations in cognitive networks: a paradigm of self-organization via desyncronization" H. Haken (ed.) , Dynamics of Synergetic Systems , Springer (1980) pp. 220–234
[a16] J.S. Nicolis, "Dynamics of hierarchical systems, an evolutionary approach" , Springer (1986)
[a17] G. Nicolis, I. Prigogine, "Self-organization in non-equilibrium systems" , Wiley (1977) pp. Chapt. 9.3; 10
[a18] G. Oster, A. Perelson, A. Katchalsky, "Network thermodynamics" Nature , 234 (1971) pp. 393–399
[a19] I. Prigogine, R. Herman, "Kinetic theory of vehicular traffic" , Amer. Elsevier (1971)
[a20] P. Schuster, "Introductory remarks" P. Schuster (ed.) , Stochastic phenomena and chaotic behaviour in complex systems , Ser. in Synergetics , Springer (1984)
[a21] P. Schuster, "Polynucleotide replication and biological evolution" Frehland (ed.) , Synergetics: From Microscopic to Macroscopic Order , Springer (1984)
[a22] Ch. Tilly, "From mobylisation to revolution" , Addison-Wesley (1978)
[a23] R.E. Ullanowicz, "Growth and development. Ecosystems phenomenology" , Springer (1986)
[a24] W. Weidlich, "Physics and social science: the approach of synergetics" Phys. Rep. , 204 : 1 (1991) pp. 1–163
[a25] W. Weidlich, G. Haag, "Dynamics of interacting groups in society with application to the migration of population" H. Haken (ed.) , Dynamics of Synergetic Systems , Springer (1980) pp. 235–243
[a26] W. Weidlich, G. Haag, "Concepts and models of a quantitative sociology" , Springer (1983)
How to Cite This Entry:
Master equations in cooperative and social phenomena. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A. García-Olivares (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article