Advertisement

Scaffolding of Complex Systems Data

  • Philippe BlanchardEmail author
  • Dimitri Volchenkov
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 8)

Abstract

Complex systems, in many different scientific sectors, show coarse-grain properties at different levels of magnification. Discrete data sequences generated by such systems call for the relevant tools for their classification and analysis. We show that discrete time scale-dependent random walks on the graph models of relational databases can be generated by a variety of equivalence relations imposed between walks (e.g., composite functions, inheritance, property relations, ancestor–descendant relations, data queries, address allocation and assignment polices). The Green function of diffusion process induced by the random walks allows to define scale-dependent geometry. Geometric relations on databases can guide the data interpretation. In particular, first passage times in a urban spatial network help to evaluate the tax assessment value of land. We also discuss a classification scheme of growth laws which includes human aging, tumor (and/or tissue) growth, logistic and generalized logistic growth, and the aging of technical devices. The proposed classification permits to evaluate the aging/failure of combined new bio-technical “manufactured products,” where part of the system evolves in time according to biological-mortality laws and part according to technical device behaviors. Moreover it suggests a direct relation between the mortality leveling-off for humans and technical devices and the observed small cure probability for large tumors.

Keywords

Specific Growth Rate Relational Database Technical Device Drazin Inverse Space Syntax 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Financial support by the project MatheMACS (“Mathematics of Multilevel Anticipatory Complex Systems”), grant agreement no. 318723, funded by the EC Seventh Framework Programme FP7-ICT-2011-8 is gratefully acknowledged.

References

  1. 1.
    Agaev RP, Chebotarev PYu (2002) On determining the eigenprojection and components of a matrix. Autom Rem Contr 63(10):1537MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barlow RE, Proschan F (1975) Statistical theory of reliability and testing. Probability models. Wiley, New YorkzbMATHGoogle Scholar
  3. 3.
    Batty M (2004) A new theory of space syntax. Working paper, 75. UCL Centre For Advanced Spatial Analysis Publications, CASA, LondonGoogle Scholar
  4. 4.
    Becker T et al (2011) Flow control by periodic devices: a unifying language for the description of traffic, production and metabolic systems. J Stat Mech 2011: P05004CrossRefGoogle Scholar
  5. 5.
    Bentzen SM, Thomas HD (1996) Tumor volume and local control probability: clinical data and radiobiological interpretations. Int J Radiat Oncol Biol Phys 36:247–251CrossRefGoogle Scholar
  6. 6.
    Ben-Israel A, Greville ThNE (2003) Generalized inverses: theory and applications, 2nd edn. Springer, New YorkGoogle Scholar
  7. 7.
    Bio Th, Fillard P, Ayache N, Pennec X (2004) A Riemannian framework for tensor computing. Int J Comput Vis 66:41Google Scholar
  8. 8.
    Birkhoff G, Mac Lane S (1979) Algebra, 2nd edn. Macmillan, New York, p 35, Th. 6Google Scholar
  9. 9.
    Blanchard Ph, Volchenkov D (2009) Probabilistic embedding of discrete sets as continuous metric spaces. Stochastics: Int J Prob Stoch Proc (formerly: Stochast Stochast Rep) 81(3):259Google Scholar
  10. 10.
    Blanchard Ph, Volchenkov D (2011) Introduction to random walks on graphs and databases. Springer series in synergetics, vol 10. Springer, Berlin/Heidelberg, ISBN 978-3-642-19591-4Google Scholar
  11. 11.
    Bolton RP (1922) Building For Profit. Reginald Pelham Bolton, New YorkGoogle Scholar
  12. 12.
    Campbell SL, Meyer CD, Rose NJ (1976) Applications of the Drazin Inverse to Linear Systems of Differential Equations with Singular Constant Coefficients. SIAM J Appl Math 31(3):411MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Castorina P, Blanchard Ph (2012) Unified approach to growth and aging in biological, technical and biotechnical systems. SpringerPlus 1(7). doi:10.1186/2193-1801-1-7Google Scholar
  14. 14.
    Castorina P, Delsanto PP, Guiot C (2006) Classification scheme for phenomenological universalities in growth problems in physics and other sciences. Phys Rev Lett 96:188701CrossRefGoogle Scholar
  15. 15.
    Castorina P, Deisboeck TS, Gabriele P, Guiot C (2007) Growth laws in cancer: implications for radiotherapy. Rad Res 169:349CrossRefGoogle Scholar
  16. 16.
    Chung FRK (1997) Lecture notes on spectral graph theory. AMS Publications, ProvidenceGoogle Scholar
  17. 17.
    Drazin, MP (1958) Pseudo-inverses in associative rings and semigroups. Am Math Mon 65:506MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Duistermaat JJ (2001) On the boundary behaviour of the Riemannian structure of a self-concordant barrier function. Asymptotic Anal 27(1):9MathSciNetzbMATHGoogle Scholar
  19. 19.
    Economos AC (1979) A non-gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. Age 2:74–76CrossRefGoogle Scholar
  20. 20.
    Erdélyi I (1967) On the matrix equation Ax = λ B x. J Math Anal Appl 17:119MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gavrilov LA, Gavrilova NS (1991) The Biology of life span: a quantitative approach. Harwood Academic Publisher, New YorkGoogle Scholar
  22. 22.
    Gavrilov LA, Gavrilova NS (2001) The reliability theory of aging and longevity. J Theor Biol 213(4):527–545MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gavrilov LA, Gavrilova NS (2011) Mortality measurement and modeling beyond age 100. In: Living to 100 Symposium, Orlando, FLGoogle Scholar
  24. 24.
    Glaeser EL, Gyourko J (2003) Why is Manhattan So Expensive? Manhattan Institute for Policy Research, Civic Report, No. 39Google Scholar
  25. 25.
    Gompertz B (1825) On the nature of the function expressive of the law of human mortality and a new mode of determining life contingencies. Phil Trans R Soc 115:513CrossRefGoogle Scholar
  26. 26.
    Graham A (1987) Nonnegative matrices and applicable topics in linear algebra. Wiley, New YorkzbMATHGoogle Scholar
  27. 27.
    Guiot C, Degiorgis PG, Delsanto PP, Gabriele P, Deisboeck TS (2003) Does tumor growth follow a universal law? J Theor Biol 225:147–151MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hansen WG (1959) How accessibility shapes land use. J Am Inst Planners 25:73CrossRefGoogle Scholar
  29. 29.
    Hartwig RE (1976) More on the Souriau-Frame algorithm and the Drazin inverse. SIAM J Appl Math 31(1):42MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hey J (2001) The mind of the species problem. Trends Ecol Evol 16(7):326CrossRefGoogle Scholar
  31. 31.
    Hillier B, Hanson J (1984) The social logic of space. Cambridge University Press, Cambridge, ISBN 0-521-36784-0CrossRefGoogle Scholar
  32. 32.
    Hillier B (1999) Space is the Machine: A Configurational Theory of Architecture. Cambridge University Press, Cambridge, ISBN 0-521-64528-X (1999)Google Scholar
  33. 33.
    Horiuchi S, Wilmoth JR (1998) Deceleration in tha age pattern of mortality at older ages. Demography 35:391CrossRefGoogle Scholar
  34. 34.
    Huchet A, Candry H, BelkaceniY (2003) L’effet volume en radiotherapie. Premiere parie: effect volume et tumeur. Canc Radiother 7:79–89Google Scholar
  35. 35.
    Kim JJ, Tannock IF (2005) Repopultaion of cancer cells during therapy: an important cause of treatment failure. Nat Rev Canc 5:516–525CrossRefGoogle Scholar
  36. 36.
    Lovász L (1993) Random walks on graphs: a survey. Bolyai Society Mathematical Studies 2: Combinatorics, Paul Erdös is Eighty, 1, Keszthely (Hungary)Google Scholar
  37. 37.
    Meyer CD (1975) The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev 17:443MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Muller Ch, et al (1988) The survival rate of patient with pace-maker is essentially the same of the normal population. Eur Heart J 9:1003Google Scholar
  39. 39.
    Norton LA (1988) Gompertzian model of human breast cancer growth. Canc Res 48:7067–7071Google Scholar
  40. 40.
    Olshansky SJ (1998) On the biodemography of aging: a review essay. Popul Dev Rev 24:381–393CrossRefGoogle Scholar
  41. 41.
    Pennec X (2004) Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. J Math Imag Vis. 25(1):127MathSciNetCrossRefGoogle Scholar
  42. 42.
    Rigdon SE, Basu AP (2000) Statistical methods for the reliability of repairable systems. Wiley, New YorkzbMATHGoogle Scholar
  43. 43.
    Robert P (1968) On the group inverse of a linear transformation. J Math Anal Appl 22:658MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Stanley JA, Shipley WU, Steel GG (1977) Influence of tumor size on the hypoxic fraction and therapeutic sensitivity of Lewis lung tumour. Br J Canc 36:105–13CrossRefGoogle Scholar
  45. 45.
    Steel GG (1977) Growth kinetics of tumours. Clarendon Press, OxfordGoogle Scholar
  46. 46.
    Vaupel JW, Carey JR, Christensen K, Johnson T, Yashin AI, Holm NV, Iachine IA, Kannisto V, Khazaeli AA, Liedo P, Longo VD, Zeng Y, Manton K, Curtsinger JW (1998) Biodemographic trajectories of longevity. Science 280:855–860CrossRefGoogle Scholar
  47. 47.
    Volchenkov D, Blanchard Ph (2007) Random walks along the streets and channels in compact cities: spectral analysis, dynamical modularity, information, and statistical mechanics. Phys Rev E 75:026104CrossRefGoogle Scholar
  48. 48.
    Volchenkov D, Blanchard Ph (2008) Scaling and universality in city space syntax: between Zipf and Matthew. Physica A 387(10):2353CrossRefGoogle Scholar
  49. 49.
    Volchenkov D (2013) Markov chain scaffolding of real world data, Discontinuity, Nonlinearity, and Complexity 2(3):289–299Google Scholar
  50. 50.
    Wheldon TE (1988) Mathematical models in cancer research. Adam Hilger Publisher, BristolzbMATHGoogle Scholar
  51. 51.
    Wilson AG (1970) Entropy in urban and regional modeling. Pion Press, LondonGoogle Scholar
  52. 52.
    Wolfram S (1984) Cellular automata as models of complexity. Nature 311:419CrossRefGoogle Scholar
  53. 53.
    Wolfram S (2002) A new kind of science. Wolfram media, Champaign, ILzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Faculty of PhysicsBielefeld UniversityBielefeldGermany

Personalised recommendations