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Research Topic

Geometry & Complexity

The following text from the website of the Complex Systems Society addresses the question "What are Complex Systems?" and perfectly describes the main concept that underlies this project: "The most famous quote about Complex Systems comes from Aristole who said that 'The whole is more than the sum of its parts'. Complex systems are systems where the collective behavior of their parts entails emergence of properties that can hardly, if not at all, be inferred from properties of the parts."

We propose a geometric formalization of this concept. The complexity of a system is quantified as its deviation from the sum of its parts which is interpreted as a geometric projection. While our initial approach was based on information geometry only, the current study also applies the theory of hierarchical and, in particular, graphical models and causality theory based on Bayesian networks. This allows for an integrated analysis of the interplay of causal interactions, stochastic dependence, and complexity.

Relations to and among other approaches to complexity are studied. We are particularly interested in understanding how algorithmic notions of complexity correspond to probabilistic ones, similar to the well-known close connection between algorithmic complexity and Shannon entropy. In that context, various complexity measures for stochastic processes are related to corresponding complexities of typical process realizations, thereby identifying similarities of seemingly different concepts.

People

Collaborations

inJournal
2021 Repository Open Access
Ugur Tirnakli, Constantino Tsallis and Nihat Ay

Approaching a large deviation theory for complex systems

In: Nonlinear dynamics : an international journal of nonlinear dynamics and chaos in engineering systems, 106 (2021) 3, pp. 2537-2546
BibTex DOI: 10.1007/s11071-021-06904-3 ArXiv: 2010.09508
inJournal
2021 Repository Open Access
Domenico Felice and Nihat Ay

Towards a canonical divergence within information geometry

In: Information geometry, 4 (2021) 1, pp. 65-130
BibTex DOI: 10.1007/s41884-021-00047-5 ArXiv: 1806.11363 MiS Preprint
inJournal
2020 Journal Open Access
Carlotta Langer and Nihat Ay

Complexity as causal information integration

In: Entropy, 22 (2020) 10, p. 1107
BibTex DOI: 10.3390/e22101107 ArXiv: 2008.11430 MiS Preprint
inJournal
2020 Journal Open Access
Nihat Ay, Daniel Polani and Nathaniel Virgo

Information decomposition based on cooperative game theory

In: Kybernetika, 56 (2020) 5, pp. 979-1014
BibTex DOI: 10.14736/kyb-2020-5-0979 ArXiv: 1910.05979 MiS Preprint
inJournal
2019 Journal Open Access
Domenico Felice and Nihat Ay

Canonical divergence for flat α-connections : classical and quantum

In: Entropy, 21 (2019) 9, p. 831
BibTex DOI: 10.3390/e21090831 ArXiv: 1907.11122
inJournal
2019 Journal Open Access
Domenico Felice, Stefano Mancini and Nihat Ay

Canonical divergence for measuring classical and quantum complexity

In: Entropy, 21 (2019) 4, p. 435
BibTex DOI: 10.3390/e21040435 ArXiv: 1903.09797
inBook
2019 Repository Open Access
Domenico Felice and Nihat Ay

Divergence functions in information geometry

In: Geometric science of information : 4th international conference, GSI 2019, Toulouse, France, August 27-29, 2019, proceedings / Frank Nielsen... (eds.)
Cham : Springer, 2019. - pp. 433-442
(Lecture notes in computer science ; 11712)
BibTex DOI: 10.1007/978-3-030-26980-7_45 ArXiv: 1903.02379
inBook
2018 Repository Open Access
Carlotta Langer and Nihat Ay

Comparison and connection between the joint and the conditional generalized iterative scaling algorithm

In: Proceedings of the 11th workshop on uncertainty processing WUPES '18, June 6-9, 2018 / Václav Kratochvíl (ed.)
Praha : MatfyzPress, 2018. - pp. 105-116
BibTex Link: wupes.utia.cas.cz
inBook
2018 Repository Open Access
Pradeep Kumar Banerjee, Johannes Rauh and Guido Montúfar

Computing the unique information

In: IEEE international symposium on information theory (ISIT) from June 17 to 22, 2018 at the Talisa Hotel in Vail, Colorado, USA
Piscataway, NY : IEEE, 2018. - pp. 141-145
BibTex DOI: 10.1109/ISIT.2018.8437757 ArXiv: 1709.07487 MiS Preprint Code: github.com
Preprint
2018 Repository Open Access
Domenico Felice and Nihat Ay

Dynamical systems induced by canonical divergence in dually flat manifolds

BibTex ArXiv: 1812.04461 MiS Preprint
inJournal
2018 Repository Open Access
Domenico Felice, Carlo Cafaro and Stefano Mancini

Information geometric methods for complexity

In: Chaos, 28 (2018) 3, p. 032101
BibTex DOI: 10.1063/1.5018926 ArXiv: 1801.03026
inJournal
2018 Journal Open Access
Milajiguli Rexiti, Domenico Felice and Stefano Mancini

The volume of two-qubit states by information geometry

In: Entropy, 20 (2018) 2, p. 146
BibTex DOI: 10.3390/e20020146 ArXiv: 1802.08835
inJournal
2017 Journal Open Access
Maxinder S. Kanwal, Joshua A. Grochow and Nihat Ay

Comparing information-theoretic measures of complexity in Boltzmann machines

In: Entropy, 19 (2017) 7, p. 310
BibTex DOI: 10.3390/e19070310 ArXiv: 1706.09667
inJournal
2016 Repository Open Access
Paolo Perrone and Nihat Ay

Hierarchical quantification of synergy in channels

In: Frontiers in robotics and AI, 2 (2016), p. 35
BibTex DOI: 10.3389/frobt.2015.00035 ArXiv: 1512.03614 MiS Preprint
Preprint
2016 Repository Open Access
Paolo Perrone and Nihat Ay

Iterative scaling algorithm for channels

BibTex ArXiv: 1603.07181
inJournal
2016 Repository Open Access
Oliver Pfante, Nils Bertschinger, Eckehard Olbrich, Nihat Ay and Jürgen Jost

Wie findet man eine geeignete Beschreibungsebene für ein komplexes System?

In: Jahrbuch der Max-Planck-Gesellschaft, 2016 (2016), Forschungsbericht - Max-Planck-Institut für Mathematik in den Naturwissenschaften
BibTex Link: www.mpg.de
inBook
2015 Repository Open Access
Paolo Perrone and Nihat Ay

Decomposition of Markov kernels

In: Proceedings of the 10th workshop on uncertainty processing WUPES '15, Moninec, Czech Republic, September 16-19, 2015 / Václav Kratochvíl (ed.)
Praha : Oeconomica, 2015. - pp. 167-178
BibTex Link: wupes.fm.vse.cz
inJournal
2015 Journal Open Access
Nihat Ay

Information geometry on complexity and stochastic interaction

In: Entropy, 17 (2015) 4, pp. 2432-2458
BibTex DOI: 10.3390/e17042432 MiS Preprint
inBook
2015 Repository Open Access
Guido Montúfar and Johannes Rauh

Mode poset probability polytopes

In: Proceedings of the 10th workshop on uncertainty processing WUPES '15, Moninec, Czech Republic, September 16-19, 2015 / Václav Kratochvíl (ed.)
Praha : Oeconomica, 2015. - pp. 147-154
BibTex ArXiv: 1503.00572 MiS Preprint Link: wupes.fm.vse.cz
inJournal
2015 Repository Open Access
Holger Bernigau, Michael J. Kastoryano and Jens Eisert

Mutual information area laws for thermal free fermions

In: Journal of statistical mechanics, 2015 (2015) 2, P02008
BibTex DOI: 10.1088/1742-5468/2015/02/P02008 ArXiv: 1301.5646
inJournal
2015 Repository Open Access
Oliver Pfante and Nihat Ay

Operator-theoretic identification of closed sub-systems of dynamical systems

In: An interdisciplinary journal of discontinuity, nonlinearity, and complexity, 4 (2015) 1, pp. 91-109
BibTex DOI: 10.5890/DNC.2015.03.007 MiS Preprint
inBook
2015 Repository Open Access
Stephan Weis

The MaxEnt extension of a quantum Gibbs family, convex geometry and geodesics

In: Bayesian inference and maximum entropy methods in science and engineering : (MaxEnt 2014) : Clos Lucé, Amboise, France, September 21-26 2014 / Ali Mohammad-Djafari (ed.)
Melville, NY : AIP Publising, 2015. - pp. 173-180
(AIP conference proceedings ; 1641)
BibTex DOI: 10.1063/1.4905976 ArXiv: 1411.0015
inJournal
2014 Repository Open Access
Oliver Pfante, Eckehard Olbrich, Nils Bertschinger, Nihat Ay and Jürgen Jost

Closure measures for coarse-graining of the tent map

In: Chaos, 24 (2014), p. 013136
BibTex DOI: 10.1063/1.4869075 MiS Preprint
inJournal
2014 Repository Open Access
Oliver Pfante, Nils Bertschinger, Eckehard Olbrich, Nihat Ay and Jürgen Jost

Comparison between different methods of level identification

In: Advances in complex systems, 17 (2014) 2, p. 1450007
BibTex DOI: 10.1142/S0219525914500076
inJournal
2014 Journal Open Access
Nils Bertschinger, Johannes Rauh, Eckehard Olbrich, Jürgen Jost and Nihat Ay

Quantifying unique information

In: Entropy, 16 (2014) 4, pp. 2161-2183
BibTex DOI: 10.3390/e16042161 ArXiv: 1311.2852 MiS Preprint Code: github.com
inBook
2014 Repository Open Access
Johannes Rauh, Nils Bertschinger, Eckehard Olbrich and Jürgen Jost

Reconsidering unique information : towards a multivariate information decomposition

In: IEEE international symposium on information theory proceedings (ISIT) 2014 : June 29-July 4, 2014 in Honolulu, USA
Piscataway, NY : IEEE, 2014. - pp. 2232-2236
BibTex DOI: 10.1109/ISIT.2014.6875230 ArXiv: 1404.3146
inBook
2014 Repository Open Access
Nils Bertschinger and Johannes Rauh

The Blackwell relation defines no lattice

In: IEEE international symposium on information theory proceedings (ISIT) 2014 : June 29-July 4, 2014 in Honolulu, USA
Piscataway, NY : IEEE, 2014. - pp. 2479-2483
BibTex DOI: 10.1109/ISIT.2014.6875280 ArXiv: 1401.3146
inBook
2013 Repository Open Access
Nils Bertschinger, Johannes Rauh, Eckehard Olbrich and Jürgen Jost

Shared information : new insights and problems in decomposing information in complex systems

In: Proceedings of the European Conference on Complex Systems 2012 / Thomas Gilbert... (eds.)
Dordrecht : Springer, 2013. - pp. 251-269
(Springer proceedings in complexity)
BibTex DOI: 10.1007/978-3-319-00395-5_35 ArXiv: 1210.5902
inJournal
2012
Wolfgang Löhr

Predictive models and generative complexity

In: Journal of systems science and complexity, 25 (2012) 1, pp. 30-45
BibTex DOI: 10.1007/s11424-012-9173-x
inJournal
2012 Repository Open Access
Wolfgang Löhr, Arleta Szkoła and Nihat Ay

Process dimension of classical and non-commutative processes

In: Open systems and information dynamics, 19 (2012) 1, p. 1250007
BibTex DOI: 10.1142/S1230161212500072 ArXiv: 1108.3984 MiS Preprint
inJournal
2011 Repository Open Access
Nihat Ay, Eckehard Olbrich, Nils Bertschinger and Jürgen Jost

A geometric approach to complexity

In: Chaos, 21 (2011) 3, p. 037103
BibTex DOI: 10.1063/1.3638446 MiS Preprint
Preprint
2011 Repository Open Access
Yuri Cassio Campbell-Borges and Jose Roberto C. Piqueira

Classical hierarchical correlation quantification on tripartite qubit mixed state families

BibTex ArXiv: 1110.6128
inJournal
2011 Journal Open Access
Nihat Ay, Markus Müller and Arleta Szkoła

Effective complexity of stationary process realizations

In: Entropy, 13 (2011) 6, pp. 1200-1211
BibTex DOI: 10.3390/e13061200 ArXiv: 1001.2686 MiS Preprint
inJournal
2010 Repository Open Access
Nihat Ay, Markus Müller and Arleta Szkoła

Effective complexity and its relation to logical depth

In: IEEE transactions on information theory, 56 (2010) 9, pp. 4593-4607
BibTex DOI: 10.1109/TIT.2010.2053892 ArXiv: 0810.5663
Academic
2010 Repository Open Access
Wolfgang Löhr

Models of discrete-time stochastic processes and associated complexity measures

Dissertation, Universität Leipzig, 2010
BibTex Link: nbn-resolving.de
inJournal
2010 Repository Open Access
Eckehard Olbrich, Thomas Kahle, Nils Bertschinger, Nihat Ay and Jürgen Jost

Quantifying structure in networks

In: The European physical journal / B, 77 (2010) 2, pp. 239-247
BibTex DOI: 10.1140/epjb/e2010-00209-0 ArXiv: 0912.4450 MiS Preprint
inJournal
2009 Repository Open Access
Thomas Kahle, Eckehard Olbrich, Jürgen Jost and Nihat Ay

Complexity measures from interaction structures

In: Physical review / E, 79 (2009) 2, pt. 2, p. 026201
BibTex DOI: 10.1103/PhysRevE.79.026201 ArXiv: 0806.2552 MiS Preprint
inBook
2009
Wolfgang Löhr and Nihat Ay

Non-sufficient memories that are sufficient for prediction

In: Complex sciences : first international conference, Complex 2009, Shanghai, China, February 23 - 25, 2009, revised papers. Pt. 1 / Jie Zhou (ed.)
Berlin [u. a.] : Springer, 2009. - pp. 265-276
(Lecture notes of the Institute for Computer Science, Social Informatics and Telecommunications Engineering ; 4)
BibTex DOI: 10.1007/978-3-642-02466-5_25
inJournal
2009 Repository Open Access
Wolfgang Löhr and Nihat Ay

On the generative nature of prediction

In: Advances in complex systems, 12 (2009) 2, pp. 169-194
BibTex DOI: 10.1142/S0219525909002143 MiS Preprint
inJournal
2009 Journal Open Access
Wolfgang Löhr

Properties of the statistical complexity functional and partially deterministic HMMs

In: Entropy, 11 (2009) 3, pp. 385-401
BibTex DOI: 10.3390/e110300385 MiS Preprint
inJournal
2008 Journal Open Access
Eckehard Olbrich, Nils Bertschinger, Nihat Ay and Jürgen Jost

How should complexity scale with system size?

In: The European physical journal / B, 63 (2008) 3, pp. 407-415
BibTex DOI: 10.1140/epjb/e2008-00134-9
inJournal
2007
Thomas Wennekers, Nihat Ay and Peter Andras

High-resolution multiple-unit EEG in cat auditory cortex reveals large spatio-temporal stochastic interactions

In: Biosystems, 89 (2007) 1/3, pp. 190-197
BibTex DOI: 10.1016/j.biosystems.2006.04.017
inJournal
2006
Thomas Wennekers and Nihat Ay

A temporal learning rule in recurrent systems supports high spatio-temporal stochastic interactions

In: Neurocomputing, 69 (2006) 10/12, pp. 1199-1202
BibTex DOI: 10.1016/j.neucom.2005.12.075
inBook
2006
Nihat Ay, Eckehard Olbrich, Nils Bertschinger and Jürgen Jost

A unifying framework for complexity measures of finite systems

In: ECCS'06 : proceedings of the European Conference on Complex Systems 2006 ; towards a science of complex systems / Jürgen Jost... (eds.)
Paris : European Complex Systems Society, 2006. - pp. 80-80
BibTex Link: www.santafe.edu
inJournal
2006 Journal Open Access
Nihat Ay and Andreas Knauf

Maximizing multi-information

In: Kybernetika, 42 (2006) 5, pp. 517-538
BibTex ArXiv: math-ph/0702002 MiS Preprint
inJournal
2005
Thomas Wennekers and Nihat Ay

Finite state automata resulting from temporal information maximization and a temporal learning rule

In: Neural computation, 17 (2005) 10, pp. 2258-2290
BibTex DOI: 10.1162/0899766054615671
inJournal
2005 Repository Open Access
Nihat Ay and James P. Crutchfield

Reductions of hidden information sources

In: Journal of statistical physics, 120 (2005) 3-4, pp. 659-684
BibTex DOI: 10.1007/s10955-005-6797-4 ArXiv: math/0405425
inJournal
2005
Thomas Wennekers and Nihat Ay

Stochastic interaction in associative nets

In: Neurocomputing, 65 (2005), pp. 387-392
BibTex DOI: 10.1016/j.neucom.2004.10.033
inJournal
2004 Repository Open Access
Ionas Erb and Nihat Ay

Multi-information in the thermodynamic limit

In: Journal of statistical physics, 115 (2004) 3-4, pp. 949-976
BibTex DOI: 10.1023/B:JOSS.0000022375.49904.ea MiS Preprint
inJournal
2003 Repository Open Access
Nihat Ay and Thomas Wennekers

Dynamical properties of strongly interacting Markov chains

In: Neural networks, 16 (2003) 10, pp. 1483-1497
BibTex DOI: 10.1016/S0893-6080(03)00190-4 MiS Preprint
inJournal
2003
Thomas Wennekers and Nihat Ay

Spatial and temporal stochastic interaction in neuronal assemblies

In: Theory in biosciences, 122 (2003) 1, pp. 5-18
BibTex DOI: 10.1007/s12064-003-0034-y
inJournal
2003
Nihat Ay and Thomas Wennekers

Temporal infomax leads to almost deterministic dynamical systems

In: Neurocomputing, 52 (2003) 4, pp. 461-466
BibTex DOI: 10.1016/S0925-2312(02)00732-4
inJournal
2003
Thomas Wennekers and Nihat Ay

Temporal Infomax on Markov chains with input leads to finite state automata

In: Neurocomputing, 52 (2003) 4, pp. 431-436
BibTex DOI: 10.1016/S0925-2312(02)00862-7
Preprint
2002 Repository Open Access
Thomas Wennekers and Nihat Ay

Information-theoretic grounding of finite automata in neural systems

BibTex MiS Preprint
Academic
2001
Nihat Ay

Aspekte einer Theorie pragmatischer Informationsstrukturierung

Dissertation, Universität Leipzig, 2001
BibTex