Discontinuity, Nonlinearity, and Complexity
Perspectives on MultiLevel Dynamics
Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 313339  DOI:10.5890/DNC.2016.09.009
Fatihcan M. Atay$^{1}$, Sven Banisch$^{2}$, Philippe Blanchard$^{3}$, Bruno Cessac$^{4}$, Eckehard Olbrich$^{5}$, Dima Volchenkov$^{6}$
$^{1}$ Max Planck Institute for Mathematics in the Sciences, D04103 Leipzig, Germany
$^{2}$ Max Planck Institute for Mathematics in the Sciences, D04103 Leipzig, Germany
$^{3}$ University of Bielefeld, Department of Physics D33619 Bielefeld, Germany
$^{4}$ Inria Sophia Antipolis Méditerranée, Neuromathcomp projectteam, Sophia Antipolis 06902, France
$^{5}$ Max Planck Institute for Mathematics in the Sciences, D04103 Leipzig, Germany
$^{6}$ University of Bielefeld, Center of Excellence  Cognitive Interaction Technology (CITEC), D33619 Bielefeld, Germany
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Abstract
As Physics did in previous centuries, there is currently a common dream of extracting generic laws of nature in economics, sociology, neuroscience, by focalising the description of phenomena to a minimal set of variables and parameters, linked together by causal equations of evolution whose structure may reveal hidden principles. This requires a huge reduction of dimensionality (number of degrees of freedom) and a change in the level of description. Beyond the mere necessity of developing accurate techniques affording this reduction, there is the question of the correspondence between the initial system and the reduced one. In this paper, we offer a perspective towards a common framework for discussing and understanding multilevel systems exhibiting structures at various spatial and temporal levels. We propose a common foundation and illustrate it with examples from different fields. We also point out the difficulties in constructing such a general setting and its limitations.
Acknowledgments
The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/20072013) under grant agreement no. 318723: Mathematics ofMultiLevel Anticipatory Complex Systems (MatheMACS). S.B. also acknowledges financial support by the Klaus Tschira Foundation. D.V. acknowledges the support from the Cluster of Excellence Cognitive Interaction Technology ’CITEC’ (EXC 277) at Bielefeld University, which is funded by the German Research Foundation (DFG).
References

[1]  Atay, F.M. and Roncoroni, L. (2013), Exact lumpability of linear evolution equations in Banach spaces. MPIMIS Preprint Series, 109. http://www.mis.mpg.de/publications/preprints/2013/prepr2013109.html. 

[2]  Horstmeyer, L. and Atay, F.M. (2015), Characterization of exact lumpability of smooth dynamics on manifolds, MPIMIS Preprint Series, 70. http://www.mis.mpg.de/publications/preprints/2015/prepr201570.html. 

[3]  Cover, T. and Thomas, J. (1991), Elements of Information Theory, WileyInterscience, New York. 

[4]  Geiger, D., Verma, T., and Pearl, J. Identifying independence in bayesian networks, Networks, 20(5), 507534. 

[5]  Pfante, O., Olbrich, E., Bertschinger, N., Ay, N., and Jost, J. (2014), Comparison between different methods of level identification, Advances in Complex Systems, 17(2), 1450007. 

[6]  Pfante, O., Olbrich, E., Bertschinger, N., Ay, N., and Jost, J. (2014), Closure measures for coarsegraining of the tent map, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24(1), 013136. 

[7]  Schelling, T. (1971), Dynamic Models of Segregation, Journal of Mathematical Sociology, 1(2), 143186. 

[8]  Axelrod, R. (1997), The Dissemination of Culture: A Model with Local Convergence and Global Polarization, The Journal of Conflict Resolution, 41(2), 203226. 

[9]  Kemeny, J.G. and Snell, J.L. (1976), Finite Markov Chains., Springer. 

[10]  Banisch, S., Lima, R., and Araújo, T. (2012), Agent based models and opinion dynamics as Markov chains. Social Networks, 34, 549561. 

[11]  Gillespie, N.I. and Praeger, C.E. (2013), Neighbour transitivity on codes in hamming graphs, Designs, codes and cryptography, 67(3), 385393. 

[12]  Banisch, S.(2014), The probabilistic structure of discrete agentbased models, Discontinuity, Nonlinearity, and Complexity, 3(3), 281292. http://arxiv.org/abs/1410.6277. 

[13]  Banisch, S. and Lima, R. (2015), Markov Chain Aggregation for Simple AgentBased Models on Symmetric Networks: The Voter Model, Advances in Complex Systems, 18(03n04), 1550011. arxiv.org/abs/1209.3902. 

[14]  Banisch, S. (2015)(in press), Markov Chain Aggregation for AgentBased Models, Understanding Complex Systems, Springer. 

[15]  Amari, S. (1972), Characteristics of random nets of analoglike elements, IEEE Trans. Syst. Man and Cybernetics., SMC2(5), 643657. 

[16]  Wilson, H. and Cowan, J.(1972), Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J., 12, 124. 

[17]  Faugeras, O., Touboul, J., and Cessac, B. (2015), Asymptotic Description of Neural Networks with Correlated Synaptic Weights, Entropy, 17(7), 4701. 

[18]  Jansen, B.H. and Rit, V.G. (1995), Electroencephalogram and visual evoked potential generation in a mathematical model of coupledcortical columns, Biological Cybernetics, 73, 357366. 

[19]  Freeman,W. (1975), Mass Action in the Nervous System, Academic Press, New York. 

[20]  Geman, S. (1982), Almost sure stable oscillations in a large system of randomly coupled equations, SIAM J. Appl. Math., 42(4), 695703. 

[21]  Faugeras, O. and Laurin, J.M. (2013), A large deviation principle for networks of rate neurons with correlated synaptic weights, arXiv, 1302.1029. 

[22]  Moynot, O. and Samuelides, M. (2002), Large deviations and meanfield theory for asymmetric random recurrent neural networks, Probability Theory and Related Fields, 123(1), 4175. 

[23]  Cessac, B. (1994), Occurence of chaos and AT line in random neural networks, Europhys. Lett., 26(8), 577582. 

[24]  Cessac, B. (1995), Increase in complexity in random neural networks, J. de Physique, 5, 409432. 

[25]  Cessac, B., Doyon, B., Quoy, M., and Samuelides, M. (1994), Meanfield equations, bifurcation map, and route to chaos in discrete time neural networks, Physica D, 2444. 

[26]  Sompolinsky, H., Crisanti, A., and Sommers,H. (1988), Chaos in random neural networks, Physical Review Letters, 61(3), 259262. 

[27]  Molgedey, L., Schuchardt, J., and Schuster, H. (1992), Supressing chaos in neural networks by noise, Physical Review Letters, 69(26), 37173719. 

[28]  BenArous, G.and Guionnet, A. (1995), Large deviations for langevin spin glass dynamics, Probability Theory and Related Fields, 102(4), 455509. 

[29]  Bouchaud, J.P., Cugliandolo, L.F., Kurchan, J., and Mézard, M. (1996), Modecoupling approximations, glass theory and disordered systems, Physica A, 226, 243273. 

[30]  Sompolinsky, H. and Zippelius, A. (1982), Relaxational dynamics of the EdwardsAnderson model and the meanfield theory ofspinglasses, Physical Review B, 25(11), 68606875. 

[31]  Daucé, E., Quoy, M., Cessac, B., Doyon, B., and Samuelides, M. (1998), Selforganization and dynamics reduction in recurrent networks: stimulus presentation and learning, Neural Networks, 11, 521533. 

[32]  Naudé, J., Cessac, B., Berry, H., and Delord, B. (2013), Effects of cellular homeostatic intrinsic plasticity on dynamical and computational properties of biological recurrent neural networks, Journal of Neuroscience, 33(38), 1503215043. 

[33]  Adzhemyan, A. V. L.Ts. and Antonov, N.V. (1999), The Field Theoretic Renormalization Group in Fully Developed Turbulence, Gordon and Breach. 

[34]  Vasiliev, A.(2004), The field theoretic renormalization group in critical behavior theory and stochastic dynamics, Chapman and Hall / CRC. 

[35]  ZinnJustin, J. (1999), Renormalization and renormalization group: From the discovery of uv divergences to the concept of effective field theories, Proceedings of the NATO ASI on Quantum Field Theory: Perspective and Prospective. In: de WittMorette C., Zuber J.B. (eds), Kluwer Academic Publishers, NATO ASI Series C, 530, 375388. 

[36]  ZinnJustin, J. (2002), Quantum field theory and critical phenomena, Oxford, Clarendon Press. 

[37]  Bellac, M.L. (1991), Quantum and Statistical Field Theory, Oxford, Clarendon Press. 

[38]  Volchenkov, D. (2009), Renormalization group and instantons in stochastic nonlinear dynamics: From selforganized criticality to thermonuclear reactors, European Physical Journal  Special Topics 19516355, 170(1), 1142. 

[39]  Blanchard, P. and Olkiewicz, R. (2003), Decoherence induced transition from quantum to classical dynamics, Rev. Math. Phys., 15, 217243. 

[40]  Blanchard, P. and Hellmich, M. (2012), Decoherence in infinite quantum systems, Quantum Africa 2010: Theoretical and Experimental Foundations of Recent Quantum Technology, AIP Conf. Proc. 1469: 215. 

[41]  Araki, H. (1999), Mathematical Theory of Quantum Fields, Oxford University Press. 

[42]  Barber, M., Blanchard, P., Buchinger, E., Cessac, B., and Streit, L. (2006), A luhmannbased model of communication, learning and innovation, Journal of Artificial Societies and Social Simulation, 9(4). 

[43]  LamarchePerrin, R., Banisch, S., and Olbrich, E.(2015), The Information Bottleneck Method for Optimal Prediction of Multilevel Agentbased Systems, submitted to Advances in Complex Systems, 2015. published in the MPI MIS Preprint Series (2015). 

[44]  Bertschinger, N. and Pfante, O. (2015), Inferring volatility in the heston model and its relatives: An information theoretical approach,in Society for Computational Economics Computing in Economics and Finance. 