System Complexity and Its Measures: How Complex Is Complex

  • Witold Kinsner
Part of the Studies in Computational Intelligence book series (SCI, volume 323)


The last few decades of physics, chemistry, biology, computer science, engineering, and social sciences have been marked by major developments of views on cognitive systems, dynamical systems, complex systems, complexity, self-organization, and emergent phenomena that originate from the interactions among the constituent components (agents) and with the environment, without any central authority. How can measures of complexity capture the intuitive sense of pattern, order, structure, regularity, evolution of features, memory, and correlation? This chapter describes several key ideas, including dynamical systems, complex systems, complexity, and quantification of complexity. As there is no single definition of a complex system, its complexity and complexity measures too have many definitions. As a major contribution, this chapter provides a new comprehensive taxonomy of such measures. This chapter also addresses some practical aspects of acquiring the observables properly.


Dynamical systems complex systems cognitive systems complexity complexity measures monoscale and multiscale measures 


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  1. [Abar96]
    Abarbanel, H.D.I.: Analysis of Observed Chaotic Data, p. 272. Springer, New York (1996)Google Scholar
  2. [Addi97]
    Addison, P.S.: Fractals and Chaos: An Illustrated Course, p. 256. Institute of Physics Publishing, Philadelphia (1997)MATHGoogle Scholar
  3. [AlBa02]
    Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Reviews of Modern Physics 74(1), 47–97 (2002)MathSciNetCrossRefGoogle Scholar
  4. [Alek89]
    Aleksander, I. (ed.): Neural Computing Architectures: The Design of Brain-Like Machines, p. 401. MIT Press, Cambridge (1989)Google Scholar
  5. [AlSY96]
    Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems, p. 603. Springer, New York (1996)MATHGoogle Scholar
  6. [AOBJ06]
    Ay, N., Olbrich, E., Bertschinger, N., Jost, J.: A unifying framework for complexity measures of finite systems. Working Paper 06-08-028.pdf, p. 15. Santa Fe Institute, Santa Fe (2006)Google Scholar
  7. [ABDG08]
    Ay, N., Bertschinger, N., Der, R., Güntler, F., Olbrich, E.: Predictive information and explorative behavior of autonomous robots. Working Paper 08-02-006.pdf, p. 22. Santa Fe Institute, Santa Fe (2008)Google Scholar
  8. [AtSc91]
    Atmanspacher, H., Scheingraber, H. (eds.): Information Dynamics, p. 380. Springer, New York (1991)MATHGoogle Scholar
  9. [Bara02]
    Barabási, A.-L.: Linked: The New Science of Networks, p. 280. Perseus Publishing, Cambridge (2002)Google Scholar
  10. [BaBo03]
    Barabási, A.-L., Bonabeau, E.: Scale-free networks. American Scientist 288(5), 60–69 (1989)Google Scholar
  11. [BCNV08]
    Baraniuk, R.G., Candes, E., Novak, R., Vetterli, M.: Compressive sampling. IEEE Signal Processing 25(2), 12–20 (2008)CrossRefGoogle Scholar
  12. [Barn88]
    Barnsley, M.: Fractals Everywhere, p. 396. Academic, San Diego (1988)MATHGoogle Scholar
  13. [Barn06]
    Barnsley, M.: Superfractals, p. 453. Cambridge University Press, Cambridge (2006)MATHGoogle Scholar
  14. [BeSc93]
    Beck, C., Schlögl, F.: Thermodynamics of Chaotic Systems: An Introduction, p. 286. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  15. [Benn08]
    Ben-Naim, A.: A Farewell to Entropy: Statistical Thermodynamics Based on Information, p. 384. World Scientific, Singapore (2008)MATHCrossRefGoogle Scholar
  16. [Benn82]
    Bennett, C.H.: The thermodynamics of computation: A review. Intern. J. Theoretical. Phys. 22(12), 905–940 (1982)CrossRefGoogle Scholar
  17. [Benn86]
    Bennett, C.H.: On the nature and origin of complexity in discrete, homogeneous, locally interacting systems. Found. Phys. 16(5), 585–592 (1986)MathSciNetCrossRefGoogle Scholar
  18. [Benn90]
    Bennett, C.H.: How to define complexity in physics, and why. In: [Zure 1990], pp. 137–148 (1990)Google Scholar
  19. [BiNT01]
    Bialek, W., Nemenman, I., Tishby, N.: Predictability, complexity, and learning. Neural Computation 13(11), 2409–2463 (2001)MATHCrossRefGoogle Scholar
  20. [Bish06]
    Bishop, C.M.: Pattern Recognition and Machine Learning, 2nd edn., p. 738. Springer Science, Cambridge (2004)Google Scholar
  21. [BoPo99]
    Bodii, R., Politi, A.: Complexity: Hierarchical Structures and Scaling in Physics, p. 332. Cambridge University Press, Cambridge (1999)Google Scholar
  22. [BoSc03]
    Bornholdt, S., Schuster, H.G.: Handbook of Graphs and Networks: Form the Genome to the Internet, p. 417. Wiley-VCH, New York (2003)Google Scholar
  23. [Buch02]
    Buchanan, M.: Nexus: Small Worlds and the Groundbreaking Science of Networks, p. 235. W.W. Norton, New York (2002)Google Scholar
  24. [CDFS01]
    Camazine, S., Deneubourg, J.-L., Franks, N.R., Sneyd, J., Theraulaz, G., Bonabeau, E.: Self-Organization in Biological Systems, p. 538. Princeton Univ. Press, Princeton (2001)Google Scholar
  25. [Chai66]
    Chaitin, G.J.: On the length of programs for computing finite binary sequences. J. Assoc. Comp. Mach. 13(4), 547–569 (1966)MATHMathSciNetGoogle Scholar
  26. [Chai75a]
    Chaitin, G.J.: Randomness and mathematical proof. Scientific American 232(5), 47–52 (1975)CrossRefGoogle Scholar
  27. [Chai75b]
    Chaitin, G.J.: A theory of program size formally identical to information theory. J. Assoc. Comp. Mach. 22(3), 329–340 (1975)MATHMathSciNetGoogle Scholar
  28. [Chai87]
    Chaitin, G.J.: Algorithmic Information Theory, p. 175. Cambridge University Press, Cambridge (1987)CrossRefGoogle Scholar
  29. [Chat04]
    Chatfield, C.: The Analysis of Time Series: An Introduction, p. 333. Chapman & Hall, CRC, Boca Raton (2004)MATHGoogle Scholar
  30. [CNN08]
    CNN, Government unveils world’s fastest computer. (June 9, 2008),
  31. [CoSt94]
    Cohen, J., Stewart, I.: The Collapse of Chaos: Discovering Simplicity in Complex World, p. 495. Penguin, New York (1994)Google Scholar
  32. [CoLR91]
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms, p. 1028. MIT Press, Cambridge (1991)Google Scholar
  33. [CPGH03]
    Costa, M., Peng, C.-K., Goldberger, A.L., Hausdorff, J.M.: Multiscale entropy analysis of human gait dynamics. Physica A 330(1), 53–60 (2003)MATHCrossRefGoogle Scholar
  34. [Cots07]
    Cotsaftis, M.: What makes a system complex? An approach to self-organization and emergence. Presented at the Emergent Properties in Natural and Artificial Complex Systems, EPNACS 2007, Dresden, GE, October 4-5 (2007), A satellite to European Conference on Complex Systems, ECCS (2007), (June 2008)
  35. [Cout07]
    Couture, M.: Complexity and chaos: State-of-the-art formulations and measures of complexity, Defence R&D Canada-Valcartier, ON: Technical Note TN 2006-451, p. 62 (September 2007), (May 2008)
  36. [CoTh91]
    Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn., p. 776. Wiley-Interscience, New York (2006)MATHGoogle Scholar
  37. [CoPM99]
    Cowen, G.A., Pines, D., Meltzer, D. (eds.): Complexity, Metaphors, Models, and Reality, p. 731. Westview Press, Boulder (1999)Google Scholar
  38. [Crut92]
    Crutchfield, J.P.: Knowledge and meaning: Chaos and complexity. In: Lam, L., Morris, H.C. (eds.) Modeling Complex Systems, pp. 66–101. Springer, Heidelberg (1992)Google Scholar
  39. [Crut94]
    Crutchfield, J.P.: The calculi of emergence. Physica D 75(1-3), 11–54 (1994) (Also SFI 94-03-016)MATHCrossRefGoogle Scholar
  40. [CrYo89]
    Crutchfield, J.P., Young, K.: Inferring statistical complexity. Phys. Rev. Lett. 63(2), 295–324 (1989)MathSciNetCrossRefGoogle Scholar
  41. [CrYo90]
    Crutchfield, J.P., Young, K.: Computation at the onset of chaos. In: [Zure 1990], pp. 223–269 (1990)Google Scholar
  42. [Deva86]
    Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, p. 320. The Benjamin-Cummings Publishing, Menlo Park (1986)MATHGoogle Scholar
  43. [Deva92]
    Devaney, R.L.: A First Course in Chaotic Dynamical Systems: Theory and Experiment, p. 302. Addison-Wesley, Reading (1992)MATHGoogle Scholar
  44. [Davi90]
    Davies, P.C.W.: Why is the physical world so comprehensive? In: [Zure 1990], pp. 61–70 (1990)Google Scholar
  45. [Edmo97]
    Edmonds, B.: Bibliography of Measures of Complexity, p. 386. University of Manchester, Manchester (1997), (May 2008) Google Scholar
  46. [Edmo99a]
    Edmonds, B.: What is complexity? The philosophy of complexity per se with applications to some examples in evolution. In: Heylighen, F., Bollen, J., Riegler, A. (eds.) The Evolution of Complexity, Kindle edition, p. 296. Springer, New York (1999)Google Scholar
  47. [Edmo99b]
    Edmonds, B.: Syntactic Measures of Complexities, Ph.D. Thesis, p. 254. University of Manchester, Manchester (1999), (May 2008)Google Scholar
  48. [Erdi08]
    Érdi, P.: Complexity Explained, p. 397. Springer, New York (2008)MATHGoogle Scholar
  49. [Feld05]
    David, P.F.: Some foundations in complex systems: Tools and Concepts. viewgraphs from the SFI, Complex Systems Summer School, Beijing (July 15, 2005), (June 2008)
  50. [FeCr98a]
    Feldman, D.P., Crutchfield, J.P.: Measures of statistical complexity: Why? Physics Letters A 238(4-5), 244–252 (1998)MATHMathSciNetCrossRefGoogle Scholar
  51. [FeCr98b]
    Feldman, D.P., Crutchfield, J.P.: A survey of complexity measures. viewgraphs from the SFI 1998 Complex Systems Summer School (June 11, 1998), (June 2008)
  52. [Forr91]
    Forrest, S. (ed.): Emergent Computation, p. 452. MIT Press, Cambridge (1991)Google Scholar
  53. [Gell99]
    Gell-Mann, M.: Complex adaptive systems. In: [CoPM 1999], pp. 17–46 (1999)Google Scholar
  54. [GiLe07]
    Gilmore, R., Letellier, C.: The Symmetry of Chaos, p. 545. Oxford Univ. Press, Oxford (2007)MATHGoogle Scholar
  55. [GlMa88]
    Glass, L., Mackey, M.: From Clocks to Chaos: The Rhythms of Life, p. 248. Princeton Univ. Press, Princeton (1988)MATHGoogle Scholar
  56. [GrAt06]
    Graben, P.B., Atmanspachen, H.: “Editorial,” Mind and Matter, vol. 4(2), pp. 131–139 (2006)Google Scholar
  57. [Gree97]
    Greenberg, J.: Characterization of emergent computation using entropy-based fractal measures. B.Sc. Thesis, p. 229. Department of Electrical & Computer Eng., University of Manitoba, Winnipeg, MB (September 1997)Google Scholar
  58. [Gras86]
    Grassberger, P.: Towards a quantitative theory of self-generated complexity. Intern. J. Theoretical Physics 25(9), 907–938 (1986)MATHMathSciNetCrossRefGoogle Scholar
  59. [Gras91]
    Grassberger, P.: Information and complexity measures in dynamical systems. In: [AtSc 1991], pp. 15–33Google Scholar
  60. [Haik03]
    Haikonen, P.O.A.: The Cognitive Approach to Conscious Machines, p. 294. Academic, New York (2003)Google Scholar
  61. [Hake04]
    Haken, H.: Synergetic Computers and Cognition: A Top-Down Approach to Neural Nets, 2nd edn., p. 245. Springer, New York (2004)MATHGoogle Scholar
  62. [Have96]
    Havel, I.M.: Scale dimensions in nature. Intern. J. General Systems 24(3), 295–324 (1996)MATHMathSciNetCrossRefGoogle Scholar
  63. [HaKo01]
    Haykin, S., Kosko, B. (eds.): Intelligent Signal Processing, p. 573. IEEE Press, Piscataway (2001)Google Scholar
  64. [HPSM07]
    Haykin, S., Principe, J.C., Sejnowski, T.J., McWhirter, J. (eds.): New Directions in Statistical Signal Processing, p. 514. MIT Press, Cambridge (2007)MATHGoogle Scholar
  65. [Heyl96]
    Heylighen, F.: What is complexity?, (May 2008)
  66. [Heyl99]
    Heylighen, F.: The growth of structural and functional complexity during evolution. In: Heylighen, F., Bollen, J., Riegler, A. (eds.) The Evolution of Complexity, p. 296. Springer, New York (1999) (Kindle Edition)Google Scholar
  67. [Holl95]
    Holland, J.H.: Hidden Order: How Adaptation Builds Complexity, p. 185. Addison-Wesley, Reading (1995)Google Scholar
  68. [HoBh07]
    Hossein, E., Bhargava, V.K. (eds.): Cognitive Wireless Communications Networks, p. 440. Springer, New York (2007)Google Scholar
  69. [HuHo86]
    Huberman, B.A., Hogg, T.: Complexity and adaptation. Physica D 22(1-3), 376–384 (1986)MathSciNetGoogle Scholar
  70. [HyKO01]
    Hyvärinen, A., Karhunen, J., Oja, E.: Independent Component Analysis, p. 481. Wiley-Interscience, New York (2001)CrossRefGoogle Scholar
  71. [Jack91]
    Atlee Jackson, E.: Perspective on Nonlinear Dynamics, vol. 1, p. 496, vol. 2, p. 633. Cambridge University Press, Cambridge (1991)Google Scholar
  72. [Jen03]
    Jen, E.: Stable or robust? What is the difference? Complexity 8(3), 12–18 (2003), Also available from Santa Fe, N.M: Santa Fe Institute, Working Paper 02-120069.pdf, p. 13, December 17 (2002)MathSciNetCrossRefGoogle Scholar
  73. [Jeon01]
    Jeong, H.: Biological networks: A map of protein-protein interactions (2001), (July 7, 2008)
  74. [JMBO01]
    Jeong, H., Mason, S.P., Barabasi, A.-L., Oltvai, Z.N.: Lethality and centrality in protein networks. Nature 411, 41–42 (2001), (June 2008)CrossRefGoogle Scholar
  75. [Jost05]
    Jost, J.: Dynamical Systems: Examples of Complex Behavior, p. 189. Springer, New York (2005)Google Scholar
  76. [KaAS02]
    Kadanoff, L.P., Aldana, M., Coppersmith, S.: Boolean dynamics with random couplings, p. 69 (April 2002), http://arXiv:nlin.A0/0204062Google Scholar
  77. [Kais00]
    Kaiser, F.: External signals and internal oscillation dynamics: Principal aspects and response of simulated rhythmic processes. In: [Wall 2000], pp. 15–43Google Scholar
  78. [KaSc04]
    Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn., p. 369. Cambridge Univ. Press, Cambridge (2004)MATHGoogle Scholar
  79. [Kauf93]
    Kauffman, S.: The Origins of Order: Self-Organization and Selection in Evolution, p. 734. Oxford Univ. Press, Oxford (1993)Google Scholar
  80. [King98]
    Kingston, J.H.: Algorithms and data structures: Design, Correctness, Analysis, 2nd edn., p. 380. Addison-Wesley, Harlow (1998)MATHGoogle Scholar
  81. [Kins91]
    Kinsner, W.: Review of data compression methods, including Shannon-Fano, Huffman, arithmetic, Storer, Lempel-Ziv-Welch, fractal, neural network, and wavelet algorithms. Technical Report DEL91-1. Winnipeg, MB: DE&CE, University of Manitoba, p. 157 (January 1991)Google Scholar
  82. [Kins94a]
    Kinsner, W.: Fractal dimensions: Morphological, entropy, spectra, and variance classes. Technical Report, DEL94-4, Dept. Electrical & Computer Eng., University of Manitoba, Winnipeg, Manitoba, Canada, p. 146 (May 1994)Google Scholar
  83. [Kins94b]
    Kinsner, W.: Batch and real-time computation of a fractal dimension based on variance of a time series,” Technical Report, DEL94-6, ibid, p. 22 (June 15, 1994)Google Scholar
  84. [Kins03]
    Kinsner, W.: Characterizing chaos through Lyapunov metrics. In: Proc. IEEE 2003 Intern. Conf. Cognitive Informatics, ICCI 2003, London, UK, August 18-20, pp. 189–201 (2003) ISBN: 0-7803-1986-5Google Scholar
  85. [Kins04]
    Kinsner, W.: Fractal and Chaos Engineering. Lecture Notes, Dept. Electrical & Computer Eng., University of Manitoba, Winnipeg, p. 941 (2004)Google Scholar
  86. [Kins07a]
    Kinsner, W.: Towards cognitive machines: Multiscale measures and analysis. Intern. J. Cognitive Informatics and Natural Intelligence 1(1), 28–38 (2007)Google Scholar
  87. [Kins07b]
    Kinsner, W.: Is entropy suitable to characterize data and signals for cognitive informatics? Intern. J. Cognitive Informatics and Natural Intelligence 1(2), 34–57 (2007)Google Scholar
  88. [Kins07c]
    Kinsner, W.: A unified approach to fractal dimensions. Intern. J. Cognitive Informatics and Natural Intelligence 1(4), 26–46 (2007)Google Scholar
  89. [Kins07d]
    Kinsner, W.: Single-scale measures for randomness and complexity. In: Zhang, D., Wang, Y., Kinsner, W. (eds.) Proc. IEEE 6th Intern. Conf. Cognitive Informatics, ICCI 2007, Lake Tahoe, CA, August 6-8, pp. 554–568 (2007) ISBN 1-4244-1327-3Google Scholar
  90. [Kins07e]
    Kinsner, W.: Challenges in the design of adaptive, intelligent and cognitive systems. In: Zhang, D., Wang, Y., Kinsner, W. (eds.) Proc. IEEE 6th Intern. Conf. Cognitive Informatics, ICCI 2007, Lake Tahoe, CA, August 6-8, pp. 13–25 (2007) ISBN 1-4244-1327-3Google Scholar
  91. [Kins08]
    Kinsner, W.: Complexity and its measures in cognitive and other complex systems. In: Wang, Y., Zhang, D., Latombe, J.-C., Kinsner, W. (eds.) Proc. IEEE 2008 Intern. Conf. Cognitive Informatics, ICCI 2008, Stanford University, Palo Alto, CA, August 14-16, pp. 13–29 (2008) ISBN: 978-1-4244-2538-9 Google Scholar
  92. [KCCP06]
    Kinsner, W., Cheung, V., Cannons, K., Pear, J., Martin, T.: Signal classification through multifractal analysis and complex domain neural networks. IEEE Trans. Systems, Man, and Cybernetics, Part C 36(2), 196–203 (2006)CrossRefGoogle Scholar
  93. [KiGr08]
    Kinsner, W., Grieder, W.: Speech segmentation using multifractal measure and amplification of signal features. In: Proc. IEEE 7th Intern. Conf. Cognitive Informatics, ICCI 2008, Palo Alto, CA, August 14-16 (2008) (this issue)Google Scholar
  94. [KiDa06]
    Kinsner, W., Dansereau, R.: A relative fractal dimension spectrum as a complexity measure. In: Yao, Y., Shi, Z., Wang, Y., Kinsner, W. (eds.) Proc. IEEE 5th Intern. Conf. Cognitive Informatics, ICCI 2006, Beijing, China, July 17-19, vol. 1, pp. 200–208 (2006)Google Scholar
  95. [Klir91]
    Klir, G.J.: Facets of Systems Science, p. 684. Springer, New York (1991) (2nd ed., p. 748 (2001))Google Scholar
  96. [KlPN07]
    Klyubin, A.S., Polani, D., Nehaniv, C.L.: Representations of space and time in the maximization of information flow in the perception-action loop. Neural Computation 19(9), 2387–2432 (2007)MATHMathSciNetCrossRefGoogle Scholar
  97. [Koho88]
    Kohonen, T.: Self-Organization and Associative Memory, 2nd edn., p. 312. Springer, New York (1988)MATHGoogle Scholar
  98. [Koho97]
    Kohonen, T.: Self-Organizing Maps, 2nd edn., p. 426. Springer, New York (1997)MATHGoogle Scholar
  99. [Kolm65]
    Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Problems of Information Transmission 1(1), 4–7 (1965) (Russian: Probl. Peredachi Inf., vol. 1(1), pp. 3–11, (1965))Google Scholar
  100. [Kurz05]
    Kurzweil, R.: The Singularity is Near, p. 652. Penguin, New York (2005)Google Scholar
  101. [LaEl05]
    Land, B., Elias, D.: Measuring the complexity of time series (2005), (June 2008)
  102. [LeZi76]
    Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Info. Theory IT-22(1), 75–81 (1996)MathSciNetCrossRefGoogle Scholar
  103. [Lind68]
    Lindenmayer, A.: Mathematical models for cellular interaction in development: Parts I and II. J. Theoretical Biology 18(3), 280–315 (1968)CrossRefGoogle Scholar
  104. [Lloy00]
    Lloyd, S.: Ultimate physical limit to computation. Nature 406(6799), 1047–1054 (2000), (May 2008) (version 3, Feburary 14, 2000) CrossRefGoogle Scholar
  105. [Lloy08]
    Lloyd, S.: Measures of complexity: A non-exhaustive list (2008), (May 2008)
  106. [LlPa88]
    Lloyd, S., Pagels, H.R.: Complexity as thermodynamic depth. Annals of Physics 188(1), 186–213 (1988)MathSciNetCrossRefGoogle Scholar
  107. [LoMC95]
    López-Ruiz, R., Mancini, H.L., Calbet, X.: A statistical measure of complexity. Phys. Lett. A 209(5), 321–326 (1995); See also López-Ruiz, R.: Shannon information, LMC complexity, and Rényi entropies: A straightforward approach December 22 (2003),, See also [FeCr98a] for a critiqueGoogle Scholar
  108. [Mack92]
    Mackey, M.C.: Time’s Arrow: The Origins of Thermodynamic Behavior, p. 175. Springer, New York (1992)Google Scholar
  109. [Main04]
    Mainzer, K.: Thinking in Complexity: The Computational Dynamics of Matter, Mind, and Mankind, 4th edn., p. 456. Springer, New York (2004)MATHGoogle Scholar
  110. [Main05]
    Mainzer, K.: Symmetry and Complexity: The Spirit of Beauty of Nonlinear Science, p. 437. World Scientific, Singapore (2005)MATHCrossRefGoogle Scholar
  111. [Mall98]
    Mallat, S.: A Wavelet Tour of Signal Processing, p. 577. Academic, San Diego (1998)MATHGoogle Scholar
  112. [Mcca94]
    McCauley, J.L.: Chaos, Dynamics, and Fractals: An Algorithmic Approach to Deterministic Chaos, p. 323. Cambridge University Press, Cambridge (1994)Google Scholar
  113. [NaBa95]
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, p. 685. Wiley-Interscience, New York (1995)MATHCrossRefGoogle Scholar
  114. [NiPr89]
    Nicolis, G., Prigogine, I.: Exploring Complexity: An Introduction, p. 313. W.H. Freeman, New York (1989)Google Scholar
  115. [Oresk03]
    Oreskes, N.: The role of quantitative models in science. In: Conham, C.D., Cole, J.J., Lauenroth, W.K. (eds.) Models of Ecosystems Science, pp. 13–31. Princeton Univ. Press, Princeton (2003)Google Scholar
  116. [OrSK94]
    Oreskes, N., Shrader-Frechette, K., Belitz, K.: Verification, validation and confirmation of numerical models in the Earth sciences. Science 263(5147), 641–646 (1994)CrossRefGoogle Scholar
  117. [Ott93]
    Ott, E.: Chaos in Dynamical Systems, p. 385. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  118. [OtSY94]
    Ott, E., Sauer, T.D., Yorke, J.A. (eds.): Chaos:Analysis of Chaotic Data and the Exploration of Chaotic Systems, p. 418. Wiley, New York (1994)Google Scholar
  119. [Papa94]
    Papadimitriou, C.H.: Computational Complexity, p. 523. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  120. [PeJS04]
    Peitgen, H.-O., Jürgens, H., Saupe, D.: Chaos and Fractals, 2nd edn., p. 964. Springer, New York (2004)MATHGoogle Scholar
  121. [PrLi90]
    Prusinkiewicz, P., Lindenmayer, A.: The Algorithmic Beutiy of Plants, p. 228. Springer, New York (1990)Google Scholar
  122. [Riss89]
    Rissanen, J.: Stochastic Complexity and Statistical Inquiry, p. 250. World Scientific, Singapore (1989)Google Scholar
  123. [Riss07]
    Rissanen, J.: Information and Complexity in Statistical Modeling, p. 144. Springer, New York (2007)MATHGoogle Scholar
  124. [Ruel89]
    Ruelle, D.: Chaotic Evolution and Strange Attractors, p. 112. Cambridge Univ. Press, Cambridge (1989)MATHCrossRefGoogle Scholar
  125. [Ruel93]
    Ruelle, D.: Chance and Chaos, p. 214. Princeton Univ. Press, Princeton (1993)Google Scholar
  126. [Ruel93]
    Sayood, K.: Introduction to Data Compression, 2nd edn., p. 636. Morgan Kaufmann, San Francisco (2000)Google Scholar
  127. [SaLo05]
    Sánchez, J.R., López-Ruiz, R.: A method to discern complexity in two-dimensional pattern generated by coupled map lattices. Physica A 355(2-4), 633–640 (2005)CrossRefGoogle Scholar
  128. [Schr91]
    Schroeder, M.: Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, p. 429. W.H. Freedman, New York (1991)MATHGoogle Scholar
  129. [Shal08]
    Shalizi, C.R.: Complexity measures (2008), (June 2008)
  130. [Smal05]
    Small, M.: Applied Nonlinear Time Series Analysis: Applications in Physics, Physiology and Finance, p. 245. World Scientific, Singapore (2005)MATHCrossRefGoogle Scholar
  131. [Solo07]
    Solomon, D.: Data Compression: The Complete Reference, 4th edn., p. 1092. Springer, New York (2007)Google Scholar
  132. [Solo60]
    Solomonoff, R.J.: A Preliminary Report on a General Theory of Inductive Inference. Report V-131. Zator Co., Cambridge (1960) Revised ZTB-138, p. 21 (November 1960) (March 2007)Google Scholar
  133. [Solo64a]
    Solomonoff, R.J.: A formal theory of inductive inference: Part 1. Inform. and Control 7(1), 1–22 (1964)MATHMathSciNetCrossRefGoogle Scholar
  134. [Solo64b]
    Solomonoff, R.J.: A formal theory of inductive inference: Part 2. Inform. and Control 7(2), 224–254 (1964)MATHMathSciNetCrossRefGoogle Scholar
  135. [Spect08]
    Spectrum, The rapture of the geeks: Special issue. IEEE Spectrum 45(6) (June 2008)Google Scholar
  136. [Spro03]
    Sprott, J.C.: Chaos and Time-Series Analysis, p. 507. Oxford Univ. Press, Oxford (2003)MATHGoogle Scholar
  137. [Stud04]
    Studeny, M.: Probabilistic Conditional Interdependence Structures, p. 285. Springer, New York (2004)Google Scholar
  138. [Take81]
    Takens, F.: Detecting strange attractors in turbulence. In: Dynamical Systems and Turbulence Warwick 1980, Lecture Notes in Mathematics, vol. 898, pp. 366–381. Springer, New York (1981)Google Scholar
  139. [Teic02]
    Teichmann, S.A.: The constraints protein–protein interactions place on sequence divergence. J. Mol. Biol. 324, 399–407 (2002)CrossRefGoogle Scholar
  140. [ThSt86]
    Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists, p. 376. Wiley, New York (1986)MATHGoogle Scholar
  141. [ToSE94]
    Tononi, G., Sporns, O., Edelman, G.M.: A measure for brain complexity: Relating functional segregation and integration in the nervous systems. Proc. Natl. Acad. Sci. USA 91(11), 5033–5037 (1994)CrossRefGoogle Scholar
  142. [ToSE99]
    Tononi, G., Sporns, O., Edelman, G.M.: Measures of degeneracy and redundancy in biological networks. Proc. Natl. Acad. Sci. USA 96(6), 3257–3267 (1999)CrossRefGoogle Scholar
  143. [WWAK94]
    Wackerbauer, R., Witt, A., Atmanspacher, H., Kurths, J., Scheingraber, H.: A comparative classification of complexity measures. Chaos, Solitons, and Fractals 4(1), 133–173 (1994)MATHMathSciNetCrossRefGoogle Scholar
  144. [Wall00]
    Wallaczek, J. (ed.): Self-Organized Biological Dynamics and Nonlinear Control, p. 428. Cambridge Univ. Press, Cambridge (2000)Google Scholar
  145. [Wang02]
    Wang, Y.: On cognitive informatics. In: Proc. 1st IEEE Intern. Conf. Cognitive Informatics, Calgary, AB, August 19-20, pp. 34–42 (2002)Google Scholar
  146. [Watt03]
    Watts, D.J.: Six Degrees: The Science of Connected Age, p. 368. W.W. Norton, New York (2003)Google Scholar
  147. [WaSt98]
    Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 400–442 (1989)Google Scholar
  148. [Weav68]
    Weaver, W.: Science and complexity. American Scientist 36(948), 536–544 (1968); (reprinted in [Klir91], pp. 449-456)Google Scholar
  149. [WeKD07]
    Wen, L., Kirk, D., Dromey, R.G.: Software systems as complex networks. In: Zhang, D., Wang, Y., Kinsner, W. (eds.) IEEE 6th Intern. Conf. Cognitive Informatics, ICCI 2007, Lake Tahoe, CA, August 6-8, pp. 106–115 (2007)Google Scholar
  150. [Weyl52]
    Weyl, H.: Symmetry, p. 168. Princeton Univ. Press, Princeton (1952)MATHGoogle Scholar
  151. [Will97]
    Williams, G.P.: Chaos Theory Tamed, p. 499. Joseph Henry Press, Washington (1997)MATHGoogle Scholar
  152. [Winf06]
    Winfree, A.T.: The Geometry of Biological Time, 2nd edn., p. 777. Springer, New York (2006)Google Scholar
  153. [Wolf85]
    Wolfram, S.: Origins of randomness in physical systems. Phys. Rev. Lett. 55(5), 449–452 (1985)MathSciNetCrossRefGoogle Scholar
  154. [Wolf02]
    Wolfram, S.: A New Kind of Science, p. 1264. Wolfram Media, Champain (2002)MATHGoogle Scholar
  155. [WoMa97]
    Wolpert, D.H., Macready, W.G.: Self-similarity: An empirical measure of complexity. Working Paper 97-12-087.pdf, p. 12. Santa Fe Institute, Santa Fe (1997)Google Scholar
  156. [Worn96]
    Wornell, G.W.: Signal Processing with Fractals: A Wavelet-Based Approach, p. 177. Prentice-Hall, Upper Saddle River (1996)Google Scholar
  157. [Xing04]
    Xing, J.: “Measures of information complexity and the implications for automation design,” Technical Report DOT/FAA/AM-04/17. Offic3e of the Aerospace Medicine, Washington, p. 16 (October 2004)Google Scholar
  158. [Zak03]
    Zak, S.H.: Systems and Control, p. 704. Oxford Univ. Press, Oxford (2003)Google Scholar
  159. [Zeh92]
    Zeh, H.-D.: The Physical Basis for the Direction of Time, 2nd edn., p. 188. Springer, New York (1992)Google Scholar
  160. [Zure90]
    Zurek, W.H.: Complexity, Entropy, and the Physics of Information. Santa Fe Institute Studies in Sciences of Complexity, vol. VIII, p. 530. Addison-Wesley, Redwood City (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Witold Kinsner
    • 1
    • 2
  1. 1.Cognitive Systems Laboratory, Department of Electrical and Computer EngineeringUniversity of ManitobaWinnipegCanada
  2. 2.The Institute of Industrial Mathematical Sciences, and Telecommunications Research LaboratoriesTRLabsWinnipegCanada

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