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Advances in Cognitive Informatics and Cognitive Computing pp 265–295Cite as

System Complexity and Its Measures: How Complex Is Complex

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Part of the book series: Studies in Computational Intelligence ((SCI,volume 323))

Abstract

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.

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References

  1. Abarbanel, H.D.I.: Analysis of Observed Chaotic Data, p. 272. Springer, New York (1996)

    Google Scholar 

  2. Addison, P.S.: Fractals and Chaos: An Illustrated Course, p. 256. Institute of Physics Publishing, Philadelphia (1997)

    MATH  Google Scholar 

  3. Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Reviews of Modern Physics 74(1), 47–97 (2002)

    Article  MathSciNet  Google Scholar 

  4. Aleksander, I. (ed.): Neural Computing Architectures: The Design of Brain-Like Machines, p. 401. MIT Press, Cambridge (1989)

    Google Scholar 

  5. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems, p. 603. Springer, New York (1996)

    MATH  Google Scholar 

  6. 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. 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. Atmanspacher, H., Scheingraber, H. (eds.): Information Dynamics, p. 380. Springer, New York (1991)

    MATH  Google Scholar 

  9. Barabási, A.-L.: Linked: The New Science of Networks, p. 280. Perseus Publishing, Cambridge (2002)

    Google Scholar 

  10. Barabási, A.-L., Bonabeau, E.: Scale-free networks. American Scientist 288(5), 60–69 (1989)

    Google Scholar 

  11. Baraniuk, R.G., Candes, E., Novak, R., Vetterli, M.: Compressive sampling. IEEE Signal Processing 25(2), 12–20 (2008)

    Article  Google Scholar 

  12. Barnsley, M.: Fractals Everywhere, p. 396. Academic, San Diego (1988)

    MATH  Google Scholar 

  13. Barnsley, M.: Superfractals, p. 453. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  14. Beck, C., Schlögl, F.: Thermodynamics of Chaotic Systems: An Introduction, p. 286. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  15. Ben-Naim, A.: A Farewell to Entropy: Statistical Thermodynamics Based on Information, p. 384. World Scientific, Singapore (2008)

    Book  MATH  Google Scholar 

  16. Bennett, C.H.: The thermodynamics of computation: A review. Intern. J. Theoretical. Phys. 22(12), 905–940 (1982)

    Article  Google Scholar 

  17. Bennett, C.H.: On the nature and origin of complexity in discrete, homogeneous, locally interacting systems. Found. Phys. 16(5), 585–592 (1986)

    Article  MathSciNet  Google Scholar 

  18. Bennett, C.H.: How to define complexity in physics, and why. In: [Zure 1990], pp. 137–148 (1990)

    Google Scholar 

  19. Bialek, W., Nemenman, I., Tishby, N.: Predictability, complexity, and learning. Neural Computation 13(11), 2409–2463 (2001)

    Article  MATH  Google Scholar 

  20. Bishop, C.M.: Pattern Recognition and Machine Learning, 2nd edn., p. 738. Springer Science, Cambridge (2004)

    Google Scholar 

  21. Bodii, R., Politi, A.: Complexity: Hierarchical Structures and Scaling in Physics, p. 332. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  22. 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. Buchanan, M.: Nexus: Small Worlds and the Groundbreaking Science of Networks, p. 235. W.W. Norton, New York (2002)

    Google Scholar 

  24. 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. Chaitin, G.J.: On the length of programs for computing finite binary sequences. J. Assoc. Comp. Mach. 13(4), 547–569 (1966)

    MATH  MathSciNet  Google Scholar 

  26. Chaitin, G.J.: Randomness and mathematical proof. Scientific American 232(5), 47–52 (1975)

    Article  Google Scholar 

  27. Chaitin, G.J.: A theory of program size formally identical to information theory. J. Assoc. Comp. Mach. 22(3), 329–340 (1975)

    MATH  MathSciNet  Google Scholar 

  28. Chaitin, G.J.: Algorithmic Information Theory, p. 175. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  29. Chatfield, C.: The Analysis of Time Series: An Introduction, p. 333. Chapman & Hall, CRC, Boca Raton (2004)

    MATH  Google Scholar 

  30. CNN, Government unveils world’s fastest computer. CNN.com (June 9, 2008), http://www.cnn.com/2008/TECH/06/09/fastest.computer.ap

  31. Cohen, J., Stewart, I.: The Collapse of Chaos: Discovering Simplicity in Complex World, p. 495. Penguin, New York (1994)

    Google Scholar 

  32. Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms, p. 1028. MIT Press, Cambridge (1991)

    Google Scholar 

  33. 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)

    Article  MATH  Google Scholar 

  34. 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), http://arXiv.org/pdf/0706.0440 (June 2008)

  35. 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), pubs.drdc.gc.ca/PDFS/unc65/p528160.pdf (May 2008)

  36. Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn., p. 776. Wiley-Interscience, New York (2006)

    MATH  Google Scholar 

  37. Cowen, G.A., Pines, D., Meltzer, D. (eds.): Complexity, Metaphors, Models, and Reality, p. 731. Westview Press, Boulder (1999)

    Google Scholar 

  38. 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. Crutchfield, J.P.: The calculi of emergence. Physica D 75(1-3), 11–54 (1994) (Also SFI 94-03-016)

    Article  MATH  Google Scholar 

  40. Crutchfield, J.P., Young, K.: Inferring statistical complexity. Phys. Rev. Lett. 63(2), 295–324 (1989)

    Article  MathSciNet  Google Scholar 

  41. Crutchfield, J.P., Young, K.: Computation at the onset of chaos. In: [Zure 1990], pp. 223–269 (1990)

    Google Scholar 

  42. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, p. 320. The Benjamin-Cummings Publishing, Menlo Park (1986)

    MATH  Google Scholar 

  43. Devaney, R.L.: A First Course in Chaotic Dynamical Systems: Theory and Experiment, p. 302. Addison-Wesley, Reading (1992)

    MATH  Google Scholar 

  44. Davies, P.C.W.: Why is the physical world so comprehensive? In: [Zure 1990], pp. 61–70 (1990)

    Google Scholar 

  45. Edmonds, B.: Bibliography of Measures of Complexity, p. 386. University of Manchester, Manchester (1997), http://bruce.edmonds.name/combib/ (May 2008)

    Google Scholar 

  46. 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. Edmonds, B.: Syntactic Measures of Complexities, Ph.D. Thesis, p. 254. University of Manchester, Manchester (1999), http://bruce.edmonds.name/thesis/ (May 2008)

    Google Scholar 

  48. Érdi, P.: Complexity Explained, p. 397. Springer, New York (2008)

    MATH  Google Scholar 

  49. David, P.F.: Some foundations in complex systems: Tools and Concepts. viewgraphs from the SFI, Complex Systems Summer School, Beijing (July 15, 2005), http://hornacek.coa.edu/dave/csss/ (June 2008)

  50. Feldman, D.P., Crutchfield, J.P.: Measures of statistical complexity: Why? Physics Letters A 238(4-5), 244–252 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  51. Feldman, D.P., Crutchfield, J.P.: A survey of complexity measures. viewgraphs from the SFI 1998 Complex Systems Summer School (June 11, 1998), http://cse.ucdavis.edu/~cmg/compmech/tutorials/ComplexityMeasures.pdf (June 2008)

  52. Forrest, S. (ed.): Emergent Computation, p. 452. MIT Press, Cambridge (1991)

    Google Scholar 

  53. Gell-Mann, M.: Complex adaptive systems. In: [CoPM 1999], pp. 17–46 (1999)

    Google Scholar 

  54. Gilmore, R., Letellier, C.: The Symmetry of Chaos, p. 545. Oxford Univ. Press, Oxford (2007)

    MATH  Google Scholar 

  55. Glass, L., Mackey, M.: From Clocks to Chaos: The Rhythms of Life, p. 248. Princeton Univ. Press, Princeton (1988)

    MATH  Google Scholar 

  56. Graben, P.B., Atmanspachen, H.: “Editorial,” Mind and Matter, vol. 4(2), pp. 131–139 (2006)

    Google Scholar 

  57. 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. Grassberger, P.: Towards a quantitative theory of self-generated complexity. Intern. J. Theoretical Physics 25(9), 907–938 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  59. Grassberger, P.: Information and complexity measures in dynamical systems. In: [AtSc 1991], pp. 15–33

    Google Scholar 

  60. Haikonen, P.O.A.: The Cognitive Approach to Conscious Machines, p. 294. Academic, New York (2003)

    Google Scholar 

  61. Haken, H.: Synergetic Computers and Cognition: A Top-Down Approach to Neural Nets, 2nd edn., p. 245. Springer, New York (2004)

    MATH  Google Scholar 

  62. Havel, I.M.: Scale dimensions in nature. Intern. J. General Systems 24(3), 295–324 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  63. Haykin, S., Kosko, B. (eds.): Intelligent Signal Processing, p. 573. IEEE Press, Piscataway (2001)

    Google Scholar 

  64. Haykin, S., Principe, J.C., Sejnowski, T.J., McWhirter, J. (eds.): New Directions in Statistical Signal Processing, p. 514. MIT Press, Cambridge (2007)

    MATH  Google Scholar 

  65. Heylighen, F.: What is complexity?, http://pespmc1.vub.ac.be/COMPLEXI.html (May 2008)

  66. 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. Holland, J.H.: Hidden Order: How Adaptation Builds Complexity, p. 185. Addison-Wesley, Reading (1995)

    Google Scholar 

  68. Hossein, E., Bhargava, V.K. (eds.): Cognitive Wireless Communications Networks, p. 440. Springer, New York (2007)

    Google Scholar 

  69. Huberman, B.A., Hogg, T.: Complexity and adaptation. Physica D 22(1-3), 376–384 (1986)

    MathSciNet  Google Scholar 

  70. Hyvärinen, A., Karhunen, J., Oja, E.: Independent Component Analysis, p. 481. Wiley-Interscience, New York (2001)

    Book  Google Scholar 

  71. Atlee Jackson, E.: Perspective on Nonlinear Dynamics, vol. 1, p. 496, vol. 2, p. 633. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  72. 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)

    Article  MathSciNet  Google Scholar 

  73. Jeong, H.: Biological networks: A map of protein-protein interactions (2001), http://www.nd.edu/~networks/Image%20Gallery/gallery_old.htm (July 7, 2008)

  74. Jeong, H., Mason, S.P., Barabasi, A.-L., Oltvai, Z.N.: Lethality and centrality in protein networks. Nature 411, 41–42 (2001), http://www.nd.edu/~networks/Image%20Gallery/gallery_old.htm (June 2008)

    Article  Google Scholar 

  75. Jost, J.: Dynamical Systems: Examples of Complex Behavior, p. 189. Springer, New York (2005)

    Google Scholar 

  76. Kadanoff, L.P., Aldana, M., Coppersmith, S.: Boolean dynamics with random couplings, p. 69 (April 2002), http://arXiv:nlin.A0/0204062

    Google Scholar 

  77. Kaiser, F.: External signals and internal oscillation dynamics: Principal aspects and response of simulated rhythmic processes. In: [Wall 2000], pp. 15–43

    Google Scholar 

  78. Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn., p. 369. Cambridge Univ. Press, Cambridge (2004)

    MATH  Google Scholar 

  79. Kauffman, S.: The Origins of Order: Self-Organization and Selection in Evolution, p. 734. Oxford Univ. Press, Oxford (1993)

    Google Scholar 

  80. Kingston, J.H.: Algorithms and data structures: Design, Correctness, Analysis, 2nd edn., p. 380. Addison-Wesley, Harlow (1998)

    MATH  Google Scholar 

  81. 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. 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. 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. 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: 0-7803-1986-5

    Google Scholar 

  85. Kinsner, W.: Fractal and Chaos Engineering. Lecture Notes, Dept. Electrical & Computer Eng., University of Manitoba, Winnipeg, p. 941 (2004)

    Google Scholar 

  86. Kinsner, W.: Towards cognitive machines: Multiscale measures and analysis. Intern. J. Cognitive Informatics and Natural Intelligence 1(1), 28–38 (2007)

    Google Scholar 

  87. 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. Kinsner, W.: A unified approach to fractal dimensions. Intern. J. Cognitive Informatics and Natural Intelligence 1(4), 26–46 (2007)

    Google Scholar 

  89. 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 1-4244-1327-3

    Google Scholar 

  90. 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 1-4244-1327-3

    Google Scholar 

  91. 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. 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)

    Article  Google Scholar 

  93. 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. 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. Klir, G.J.: Facets of Systems Science, p. 684. Springer, New York (1991) (2nd ed., p. 748 (2001))

    Google Scholar 

  96. 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)

    Article  MATH  MathSciNet  Google Scholar 

  97. Kohonen, T.: Self-Organization and Associative Memory, 2nd edn., p. 312. Springer, New York (1988)

    MATH  Google Scholar 

  98. Kohonen, T.: Self-Organizing Maps, 2nd edn., p. 426. Springer, New York (1997)

    MATH  Google Scholar 

  99. 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. Kurzweil, R.: The Singularity is Near, p. 652. Penguin, New York (2005)

    Google Scholar 

  101. Land, B., Elias, D.: Measuring the complexity of time series (2005), http://web.nbb.cornel.edu/neurobio/land/PROJECTS/complexity/ (June 2008)

  102. Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Trans. Info. Theory IT-22(1), 75–81 (1996)

    Article  MathSciNet  Google Scholar 

  103. Lindenmayer, A.: Mathematical models for cellular interaction in development: Parts I and II. J. Theoretical Biology 18(3), 280–315 (1968)

    Article  Google Scholar 

  104. Lloyd, S.: Ultimate physical limit to computation. Nature 406(6799), 1047–1054 (2000), http://arXiv.org/abs/quant-ph/9908043 (May 2008) (version 3, Feburary 14, 2000)

    Article  Google Scholar 

  105. Lloyd, S.: Measures of complexity: A non-exhaustive list (2008), http://web.mit.edu/esd.83/www/notebook/Complexity.pdf (May 2008)

  106. Lloyd, S., Pagels, H.R.: Complexity as thermodynamic depth. Annals of Physics 188(1), 186–213 (1988)

    Article  MathSciNet  Google Scholar 

  107. 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), http://arxiv.org/abs/nlin/0312056 , See also [FeCr98a] for a critique

    Google Scholar 

  108. Mackey, M.C.: Time’s Arrow: The Origins of Thermodynamic Behavior, p. 175. Springer, New York (1992)

    Google Scholar 

  109. Mainzer, K.: Thinking in Complexity: The Computational Dynamics of Matter, Mind, and Mankind, 4th edn., p. 456. Springer, New York (2004)

    MATH  Google Scholar 

  110. Mainzer, K.: Symmetry and Complexity: The Spirit of Beauty of Nonlinear Science, p. 437. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  111. Mallat, S.: A Wavelet Tour of Signal Processing, p. 577. Academic, San Diego (1998)

    MATH  Google Scholar 

  112. McCauley, J.L.: Chaos, Dynamics, and Fractals: An Algorithmic Approach to Deterministic Chaos, p. 323. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  113. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, p. 685. Wiley-Interscience, New York (1995)

    Book  MATH  Google Scholar 

  114. Nicolis, G., Prigogine, I.: Exploring Complexity: An Introduction, p. 313. W.H. Freeman, New York (1989)

    Google Scholar 

  115. 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. Oreskes, N., Shrader-Frechette, K., Belitz, K.: Verification, validation and confirmation of numerical models in the Earth sciences. Science 263(5147), 641–646 (1994)

    Article  Google Scholar 

  117. Ott, E.: Chaos in Dynamical Systems, p. 385. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  118. 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. Papadimitriou, C.H.: Computational Complexity, p. 523. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  120. Peitgen, H.-O., Jürgens, H., Saupe, D.: Chaos and Fractals, 2nd edn., p. 964. Springer, New York (2004)

    MATH  Google Scholar 

  121. Prusinkiewicz, P., Lindenmayer, A.: The Algorithmic Beutiy of Plants, p. 228. Springer, New York (1990)

    Google Scholar 

  122. Rissanen, J.: Stochastic Complexity and Statistical Inquiry, p. 250. World Scientific, Singapore (1989)

    Google Scholar 

  123. Rissanen, J.: Information and Complexity in Statistical Modeling, p. 144. Springer, New York (2007)

    MATH  Google Scholar 

  124. Ruelle, D.: Chaotic Evolution and Strange Attractors, p. 112. Cambridge Univ. Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  125. Ruelle, D.: Chance and Chaos, p. 214. Princeton Univ. Press, Princeton (1993)

    Google Scholar 

  126. Sayood, K.: Introduction to Data Compression, 2nd edn., p. 636. Morgan Kaufmann, San Francisco (2000)

    Google Scholar 

  127. 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)

    Article  Google Scholar 

  128. Schroeder, M.: Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, p. 429. W.H. Freedman, New York (1991)

    MATH  Google Scholar 

  129. Shalizi, C.R.: Complexity measures (2008), http://www.cscs.umich.edu/~crshalizi/notebooks/complexity-measures.html (June 2008)

  130. Small, M.: Applied Nonlinear Time Series Analysis: Applications in Physics, Physiology and Finance, p. 245. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  131. Solomon, D.: Data Compression: The Complete Reference, 4th edn., p. 1092. Springer, New York (2007)

    Google Scholar 

  132. 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) http://world.std.com/~rjs/pubs.html (March 2007)

    Google Scholar 

  133. Solomonoff, R.J.: A formal theory of inductive inference: Part 1. Inform. and Control 7(1), 1–22 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  134. Solomonoff, R.J.: A formal theory of inductive inference: Part 2. Inform. and Control 7(2), 224–254 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  135. Spectrum, The rapture of the geeks: Special issue. IEEE Spectrum 45(6) (June 2008)

    Google Scholar 

  136. Sprott, J.C.: Chaos and Time-Series Analysis, p. 507. Oxford Univ. Press, Oxford (2003)

    MATH  Google Scholar 

  137. Studeny, M.: Probabilistic Conditional Interdependence Structures, p. 285. Springer, New York (2004)

    Google Scholar 

  138. 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. Teichmann, S.A.: The constraints protein–protein interactions place on sequence divergence. J. Mol. Biol. 324, 399–407 (2002)

    Article  Google Scholar 

  140. Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists, p. 376. Wiley, New York (1986)

    MATH  Google Scholar 

  141. 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)

    Article  Google Scholar 

  142. 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)

    Article  Google Scholar 

  143. 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)

    Article  MATH  MathSciNet  Google Scholar 

  144. Wallaczek, J. (ed.): Self-Organized Biological Dynamics and Nonlinear Control, p. 428. Cambridge Univ. Press, Cambridge (2000)

    Google Scholar 

  145. 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. Watts, D.J.: Six Degrees: The Science of Connected Age, p. 368. W.W. Norton, New York (2003)

    Google Scholar 

  147. Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 400–442 (1989)

    Google Scholar 

  148. Weaver, W.: Science and complexity. American Scientist 36(948), 536–544 (1968); (reprinted in [Klir91], pp. 449-456)

    Google Scholar 

  149. 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. Weyl, H.: Symmetry, p. 168. Princeton Univ. Press, Princeton (1952)

    MATH  Google Scholar 

  151. Williams, G.P.: Chaos Theory Tamed, p. 499. Joseph Henry Press, Washington (1997)

    MATH  Google Scholar 

  152. Winfree, A.T.: The Geometry of Biological Time, 2nd edn., p. 777. Springer, New York (2006)

    Google Scholar 

  153. Wolfram, S.: Origins of randomness in physical systems. Phys. Rev. Lett. 55(5), 449–452 (1985)

    Article  MathSciNet  Google Scholar 

  154. Wolfram, S.: A New Kind of Science, p. 1264. Wolfram Media, Champain (2002)

    MATH  Google Scholar 

  155. 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. Wornell, G.W.: Signal Processing with Fractals: A Wavelet-Based Approach, p. 177. Prentice-Hall, Upper Saddle River (1996)

    Google Scholar 

  157. 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. Zak, S.H.: Systems and Control, p. 704. Oxford Univ. Press, Oxford (2003)

    Google Scholar 

  159. Zeh, H.-D.: The Physical Basis for the Direction of Time, 2nd edn., p. 188. Springer, New York (1992)

    Google Scholar 

  160. 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 

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Authors and Affiliations

  1. Cognitive Systems Laboratory, Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, R3T 5V6

    Witold Kinsner

  2. The Institute of Industrial Mathematical Sciences, and Telecommunications Research Laboratories, TRLabs, Winnipeg, MB, Canada, R3T 5V6

    Witold Kinsner

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  1. Dept. of Electrical and Computer Engineering Schulich Schools of Engineering, University of Calgary, 2500 University Drive, NW Calgary, T2N 1N4, Alberta, Canada

    Yingxu Wang

  2. Department of Computer Science, California State University, 95819-6021, Sacramento, CA, USA

    Du Zhang

  3. Dept. of Electrical and Computer Engineering, University of Manitoba, R3T 5V6, Winnipeg, Manitoba, Canada

    Witold Kinsner

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Kinsner, W. (2010). System Complexity and Its Measures: How Complex Is Complex. In: Wang, Y., Zhang, D., Kinsner, W. (eds) Advances in Cognitive Informatics and Cognitive Computing. Studies in Computational Intelligence, vol 323. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16083-7_14

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