Congestion and Almost Invariant Sets in Dynamical Systems

  • Michael Dellnitz
  • Robert Preis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)


An almost invariant set of a dynamical system is a subset of state space where typical trajectories stay for a long period of time before they enter other parts of state space. These sets are an important characteristic for analyzing the macroscopic behavior of a given dynamical system. For instance, recently the identification of almost invariant sets has successfully been used in the context of the approximation of so-called chemical conformations for molecules.

In this paper we propose new numerical and algorithmic tools for the identification of the number and the location of almost invariant sets in state space. These techniques are based on the use of set oriented numerical methods by which a graph is created which models the underlying dynamical behavior. In a second step graph theoretic methods are utilized in order to both identify the number of almost invariant sets and for an approximation of these sets. These algorithmic methods make use of the notion of congestion which is a quantity specifying bottlenecks in the graph. We apply these new techniques to the analysis of the dynamics of the molecules Pentane and Hexane. Our computational results are compared to analytical bounds which again are based on the congestion but also on spectral information on the transition matrix for the underlying graph.


Cost Function Short Path Invariant Measure External Cost Graph Partitioning 
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.


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  1. 1.
    N. Alon. On the edge-expansion of graphs. Combinatories, Probability and Computing, 6:145–152, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    S.L. Bezroukov, R. Elsässer, B. Monien, R. Preis, and J.-P. Tillich. New spectral lower bounds on the bisection width of graphs. In Workshop on Graph-Theoretic Concepts in Computer Science (WG), LNCS 1928, pages 23–34, 2000.CrossRefGoogle Scholar
  3. 3.
    N. Bouhmala. Impact of different graph coarsening schemes on the quality of the partitions. Technical Report RT98/05-01, University of Neuchatel, Department of Computer Science, 1998.Google Scholar
  4. 4.
    F.R.K. Chung. Spectral Graph Theory, volume 92 of CBMS Regional conference series in mathematics. American Mathematical Society, 1997.Google Scholar
  5. 5.
    T.H. Cormen, C.E. Leiserson, and R.L. Rivest. Introduction to Algorithms. MIT Press, 1990.Google Scholar
  6. 6.
    M. Dellnitz and A. Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik, 75:293–317, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Dellnitz and O. Junge. On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal., 36(2):491–515, 1999.CrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Deuflhard, M. Dellnitz, O. Junge, and Ch. Schütte. Computation of essential molecular dynamics by subdivision techniques. In Deuflhard et al., editor, Computational Molecular Dynamics: Challenges, Methods, Ideas, LNCSE 4, 1998.Google Scholar
  9. 9.
    P. Deuflhard, W. Huisinga, A. Fischer, and Ch. Schütte. Identification of almost invariant aggregates in reversible nearly uncoupled markov chains. Lin. Alg. Appl., 315:39–59, 2000.zbMATHCrossRefGoogle Scholar
  10. 10.
    R. Diekmann, A. Frommer, and B. Monien. Efficient schemes for nearest neighbor load balancing. Parallel Computing, 25(7):789–812, 1999.CrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Diekmann, B. Monien, and R. Preis. Using helpful sets to improve graph bisections. In D.F. Hsu, A.L. Rosenberg, and D. Sotteau, editors, Interconnection Networks and Mapping and Scheduling Parallel Computations, volume 21 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 57–73. AMS, 1995.Google Scholar
  12. 12.
    R. Elsässer, A. Frommer, B. Monien, and R. Preis. Optimal and alternating-direction loadbalancing schemes. In P. Amestoy et al., editor, Euro Par’99 Parallel Processing, LNCS 1685, pages 280–290, 1999.Google Scholar
  13. 13.
    R. Elsässer, T. Lücking, and B. Monien. New spectral bounds on k-partitioning of graphs. In Proc. of the Symposium on Parallel Algorithms and Architectures (SPAA), pages 255–270, 2001.Google Scholar
  14. 14.
    R. Elsässer, B. Monien, and R. Preis. Diffusive load balancing schemes on heterogeneous networks. In 12th ACM Symp. on Parallel Algorithms and Architectures (SPAA), pages 30–38, 2000.Google Scholar
  15. 15.
    C.M. Fiduccia and R.M. Mattheyses. A linear-time heuristic for improving network partitions. In Proc. IEEE Design Automation Conf., pages 175–181, 1982.Google Scholar
  16. 16.
    M. Fiedler. A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Mathematical J., Praha, 25(100):619–633, 1975.MathSciNetGoogle Scholar
  17. 17.
    A. Fischer, Ch. Schütte, P. Deuflhard, and F. Cordes. Hierarchical uncoupling-coupling of metastable conformations. In Schlick and Gan, editors, Proc. of the 3rd International Workshop on Algorithms for Macromolecular Modeling, LNCSE 24, 2002.Google Scholar
  18. 18.
    G. Froyland and M. Dellnitz. Detecting and locating near-optimal almost-invariant sets and cycles. Technical report, DFG-Priority Program 1095: “Analysis, Modeling and Simulation of Multiscale Problems”, 2001.Google Scholar
  19. 19.
    G. Froyland and M. Dellnitz. μ almost-invariant sets and adaptive boundary refinement. manuscript, 2003.Google Scholar
  20. 20.
    M.R. Garey and D.S. Johnson. Computers and Intractability — A Guide to the Theory of NP-Completeness. Freemann, 1979.Google Scholar
  21. 21.
    A. Gupta. Fast and effective algorithms for graph partitioning and sparse matrix recordering. IBM J. of Research and Development, 41:171–183, 1997.CrossRefGoogle Scholar
  22. 22.
    B. Hendrickson and R. Leland. The chaco user’s guide: Version 2.0. Technical Report SAND94-2692, Sandia National Laboratories, Albuquerque, NM, 1994.Google Scholar
  23. 23.
    B. Hendrickson and R. Leland. A multilevel algorithm for partitioning graphs. In Proc. Supercomputing’ 95. ACM, 1995.Google Scholar
  24. 24.
    J. Hromkovič and B. Monien. The bisection problem for graphs of degree 4 (configuring transputer systems). In Buchmann, Ganzinger, and Paul, editors, Festschriftzum 60. Geburtstag von Günter Hotz, pages 215–234. Teubner, 1992.Google Scholar
  25. 25.
    W. Huisinga. Metastability of Markovian systems. Phd thesis, Freie Universität Berlin, 2001.Google Scholar
  26. 26.
    G. Karypis and V. Kumar. METIS Manual, Version 4.0. University of Minnesota, Department of Computer Science, 1998.Google Scholar
  27. 27.
    G. Karypis and V. Kumar. Multilevel k-way partitioning scheme for irregular graphs. J. of Parallel and Distributed Computing, 48:96–129, 1998.CrossRefGoogle Scholar
  28. 28.
    G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. on Scientific Computing, 20(1), 1999.Google Scholar
  29. 29.
    B.W. Kernighan and S. Lin. An effective heuristic procedure for partitioning graphs. The Bell Systems Technical J., pages 291–307, 1970.Google Scholar
  30. 30.
    F.T. Leighton. Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann Publishers, 1992.Google Scholar
  31. 31.
    B. Mohar. Some applications of laplace eigenvalues of graphs. In Graph Symmetry: Algebraic Methods and Applications, NATO ASI Ser. C 497, pages 225–275, 1997.Google Scholar
  32. 32.
    B. Monien and R. Diekmann. A local graph partitioning heuristic meeting bisection bounds. In 8th SIAM Conf. on Parallel Processing for Scientific Computing, 1997.Google Scholar
  33. 33.
    B. Monien and R. Preis. Bisection width of 3-and 4-regular graphs. In 26th International Symposium on Mathematical Foundations of Computer Science (MFCS), LNCS 2136, pages 524–536, 2001.Google Scholar
  34. 34.
    B. Monien, R. Preis, and R. Diekmann. Quality matching and local improvement for multilevel graph-partitioning. Parallel Computing, 26(12):1609–1634, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    F. Pellegrini. SCOTCH 3.1 user’s guide. Technical Report 1137-96, LaBRI, University of Bordeaux, 1996.Google Scholar
  36. 36.
    R. Ponnusamy, N. Mansour, A. Choudhary, and G.C. Fox. Graph contraction for mapping data on parallel computers: A quality-cost tradeoff. Scientific Programming, 3:73–82, 1994.Google Scholar
  37. 37.
    A. Pothen, H.D. Simon, and K.P. Liu. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. on Matrix Analysis and Applications, 11(3):430–452, 1990.zbMATHCrossRefGoogle Scholar
  38. 38.
    R. Preis. The PARTY Graphpartitioning-Library, User Manual — Version 1.99. Universität Paderborn, Germany, 1998.Google Scholar
  39. 39.
    R. Preis. Analyses and Design of Efficient Graph Partitioning Methods. Heinz Nixdorf Institut Verlagsschriftenreihe, 2000. Dissertation, Universität Paderborn, Germany.Google Scholar
  40. 40.
    Ch. Schütte. Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules. Habilitation thesis, Freie Universität Berlin, 1999.Google Scholar
  41. 41.
    N. Sensen. Lower bounds and exact algorithms for the graph partitioning problem using multicommodity flows. In Proc. European Symposium on Algorithms (ESA), LNCS 2161, pages 391–403, 2001.Google Scholar
  42. 42.
    H.D. Simon and S.-H. Teng. How good is recursive bisection? SIAM J. on Scientific Computing, 18(5): 1436–1445, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    A. Sinclair. Algorithms for Random Generation & Counting: A Markov Chain Approach. Progress in Theoretical Computer Science. Birkhäuser, 1993.Google Scholar
  44. 44.
    C. Walshaw. The Jostle user manual: Version 2.2. University of Greenwich, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Michael Dellnitz
    • 1
  • Robert Preis
    • 1
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversität PaderbornPaderbornGermany
  2. 2.Computer Science Research InstituteSandia National LaboratoriesAlbuquerqueUSA

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