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Equation-free analysis of a dynamically evolving multigraph

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  • Session C: Papers II
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Abstract

In order to illustrate the adaptation of traditional continuum numerical techniques to the study of complex network systems, we use the equation-free framework to analyze a dynamically evolving multigraph. This approach is based on coupling short intervals of direct dynamic network simulation with appropriately-defined lifting and restriction operators, mapping the detailed network description to suitable macroscopic (coarse-grained) variables and back. This enables the acceleration of direct simulations through Coarse Projective Integration (CPI), as well as the identification of coarse stationary states via a Newton-GMRES method. We also demonstrate the use of data-mining, both linear (principal component analysis, PCA) and nonlinear (diffusion maps, DMAPS) to determine good macroscopic variables (observables) through which one can coarse-grain the model. These results suggest methods for decreasing simulation times of dynamic real-world systems such as epidemiological network models. Additionally, the data-mining techniques could be applied to a diverse class of problems to search for a succint, low-dimensional description of the system in a small number of variables.

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References

  1. K.A. Bold, K. Rajendran, B. Rth, I.G. Kevrekidis, J. Comput. Dyn. 1, 111 (2014)

    Article  MathSciNet  Google Scholar 

  2. D. Brown, J. Feng, S. Feerick. Phys. Rev. Lett. 82, 4731 (1999)

    Article  ADS  Google Scholar 

  3. H. Bunke, K. Shearer, Pattern Recogn. Lett. 19, 255 (1998)

    Article  Google Scholar 

  4. R.R. Coifman, S. Lafon, Appl. Comput. Harmonic Anal. 21, 5 (2006)

    Article  MathSciNet  Google Scholar 

  5. R. Durrett, J.P. Gleeson, A.L. Lloyd, P.J. Mucha, F. Shi, D. Sivakoff, J.E.S. Socolar, C. Varghese, Proc. Natl. Acad. Sci. 109, 3682 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  6. L.K. Gallos, N.H. Fefferman, Europhys. Lett. 108, 38001 (2014)

    Article  ADS  Google Scholar 

  7. X. Gao, B. Xiao, D. Tao, X. Li, Pattern Anal. Appl. 13, 113 (2010)

    Article  MathSciNet  Google Scholar 

  8. C.W. Gear, J.M. Hyman, P.G. Kevrekidid, I.G. Kevrekidis, O. Runborg, C. Theodoropoulos, Commun. Math. Sci. 1, 715 (2003)

    Article  MathSciNet  Google Scholar 

  9. S.L. Hakimi, J. Soc. Ind. Appl. Math. 10, 496 (1962)

    Article  MathSciNet  Google Scholar 

  10. V. Havel, Asopis Pro Pstovn Matematiky 80, 477 (1955)

    MathSciNet  Google Scholar 

  11. A.M. Hermundstad, K.S. Brown, D.S. Bassett, J.M. Carlson, PLoS Comput. Biol. 7, e1002063 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  12. A.L. Hodgkin, A.F. Huxley, J. Physiol. 117, 500 (1952)

    Article  Google Scholar 

  13. I. Jolliffe, in Wiley StatsRef: Statistics Reference Online (John Wiley & Sons, Ltd., 2014)

  14. J. Joubert, P. Fourie, K. Axhausen, Trans. Res. Rec. J. Trans. Res. Board 2168, 24 (2010)

    Article  Google Scholar 

  15. A.A. Kattis, A. Holiday, A.-A. Stoica, I.G. Kevrekidis, Virulence 7, 153 (2016)

    Article  Google Scholar 

  16. C.T. Kelley, Solving Nonlinear Equations with Newton’s Method (SIAM, 2003)

  17. I.G. Kevrekidis, C.W. Gear, G. Hummer, AIChE J. 50, 1346 (2004)

    Article  Google Scholar 

  18. B. Nadler, S. Lafon, R.R. Coifman, I.G. Kevrekidis, Appl. Comput. Harmonic Anal. 21, 113 (2006)

    Article  MathSciNet  Google Scholar 

  19. K. Rajendran, I.G. Kevrekidis, Phys. Rev. E 84, 036708 (2011)

    Article  ADS  Google Scholar 

  20. K. Rajendran, I.G. Kevrekidis, Analysis of data in the form of graphs [physics], [arXiv:1306.3524] (2013)

  21. K. Riesen, H. Bunke, Image and Vision Computing 27, 950 (2009)

    Article  Google Scholar 

  22. B. Roche, J.M. Drake, P. Rohani, BMC Bioinformatics 12, 87 (2011)

    Article  Google Scholar 

  23. B. Rth, Random Struct. Algorithms 41, 365 (2012)

    Article  Google Scholar 

  24. B. Rth, L. Szakcs, Acta Math. Hung. 136, 196 (2012)

    Article  Google Scholar 

  25. C.I. Siettos, Appl. Math. Comput. 218, 324 (2011)

    Article  MathSciNet  Google Scholar 

  26. J.M. Swaminathan, S.F. Smith, N.M. Sadeh, Decision Sci. 29, 607 (1998)

    Article  Google Scholar 

  27. S.V.N. Vishwanathan, N.N. Schraudolph, R. Kondor, K.M. Borgwardt, J. Machine Learning Res. 11, 12011242 (2010)

    MathSciNet  Google Scholar 

  28. Y. Xiao, H. Dong, W. Wu, M. Xiong, W. Wang, B. Shi, Pattern Recognit. 41, 3547 (2008)

    Article  Google Scholar 

  29. Z. Zeng, A.K.H. Tung, J. Wang, J. Feng, L. Zhou, Proc. VLDB Endow. 2, 25 (2009)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey, 08544, USA

    A. Holiday & I.G. Kevrekidis

  2. PACM, Princeton University, Princeton, New Jersey, 08544, USA

    I.G. Kevrekidis

Authors
  1. A. Holiday
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  2. I.G. Kevrekidis
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Holiday, A., Kevrekidis, I. Equation-free analysis of a dynamically evolving multigraph. Eur. Phys. J. Spec. Top. 225, 1281–1292 (2016). https://doi.org/10.1140/epjst/e2016-02672-1

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  • DOI: https://doi.org/10.1140/epjst/e2016-02672-1

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