American Institute of Mathematical Sciences

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December  2011, 4(6): 1533-1541. doi: 10.3934/dcdss.2011.4.1533

Update sequence stability in graph dynamical systems

Matthew Macauley 1, and  Henning S. Mortveit 2,

1. 

Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, United States
2. 

Department of Mathematics, NDSSL, Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA 24061, United States

Received  May 2009 Revised  November 2009 Published  December 2010

In this article, we study finite dynamical systems defined over graphs, where the functions are applied asynchronously. Our goal is to quantify and understand stability of the dynamics with respect to the update sequence, and to relate this to structural properties of the graph. We introduce and analyze three different notions of update sequence stability, each capturing different aspects of the dynamics. When compared to each other, these stability concepts yield different conclusions regarding the relationship between stability and graph structure, painting a more complete picture of update sequence stability.
Keywords: graph dynamical systems, update sequence., Asychronous update schedule, stability.
Mathematics Subject Classification: Primary: 93D99, 37B99; Secondary: 05C9.
Citation: Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533
References:
[1]

C. L. Barrett, H. H. Hunt, M. V. Marathe, S. S. Ravi, D. Rosenkrantz, R. Stearns and P. Tosic, Gardens of Eden and fixed point in sequential dynamical systems,, DM-CCG 2001, (2001), 95. Google Scholar

[2]

C. L. Barrett, H. S. Mortveit and C. M. Reidys, Elements of a theory of simulation III: Equivalence of SDS,, Appl. Math. Comput., 122 (2001), 325. doi: 10.1016/S0096-3003(00)00042-4. Google Scholar

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C. L. Barrett, H. S. Mortveit and C. M. Reidys, Elements of a theory of simulation IV: Fixed points, invertibility and equivalence,, Appl. Math. Comput., 134 (2003), 153. doi: 10.1016/S0096-3003(01)00277-6. Google Scholar

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C. L. Barrett, H. B. Hunt III, M. V. Marathe, S. S. Ravi, D. J. Rosenkrantz and R. E. Stearns, Predecessor and permutation existence problems for sequential dynamical systems,, DMTCS 2003, (2003), 69. Google Scholar

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A. Björner and F. Brenti, "Combinatorics of Coxeter Groups,'', Springer-Verlag, (2005). Google Scholar

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B. Bollobàs, "Random Graphs,'', Cambridge University Press, (2001). Google Scholar

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R. Laubenbacher E. Sontag, A. Veliz-Cuba and A. Salam Jarrah, The effect of negative feedback loops on the dynamics of Boolean networks,, Biophys. J., 95 (2008), 518. doi: 10.1529/biophysj.107.125021. Google Scholar

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D. L. Vertigan F. Jaeger and D. J. A. Welsh, On the computational complexity of the Jones and Tutte polynomials,, Math. Proc. Cambridge Philos. Soc. \textbf{108} (1990), 108 (1990), 35. doi: 10.1017/S0305004100068936. Google Scholar

[9]

A. Å. Hansson, H. S. Mortveit and C. M. Reidys, On asynchronous cellular automata,, Adv. Complex Systems, 8 (2005), 521. Google Scholar

[10]

U. Karaoz, T. M. Murali, S. Letovsky, Y. Zheng, C. Ding, C. R. Cantor and S. Kasif, Whole-genome annotation by using evidence integration in functional-linkage networks,, Proc. Nat. Acad. Sci., 101 (2004), 2888. doi: 10.1073/pnas.0307326101. Google Scholar

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W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. Royal Soc. London A, 115 (1927), 700. doi: 10.1098/rspa.1927.0118. Google Scholar

[12]

V. S. A. Kumar, M. Macauley and H. S. Mortveit, Limit set reachability in asynchronous graph dynamical systems,, RP 2009, (2009), 217. Google Scholar

[13]

M. Macauley, J. McCammond and H. S. Mortveit, Dynamics groups of asynchronous cellular automata,, J. Algebraic Combinat, 33 (2011). doi: 10.1007/s10801-010-0231-y. Google Scholar

[14]

M. Macauley, J. McCammond and H. S. Mortveit, Order independence in asynchronous cellular automata,, J. Cell. Autom., 3 (2008), 37. Google Scholar

[15]

M. Macauley and H. S. Mortveit, On enumeration of conjugacy classes of Coxeter elements,, Proc. Amer. Math. Soc., 136 (2008), 4157. doi: 10.1090/S0002-9939-08-09543-9. Google Scholar

[16]

M. Macauley and H. S. Mortveit, Cycle equivalence of graph dynamical systems,, Nonlinearity, 22 (2009), 421. doi: 10.1088/0951-7715/22/2/010. Google Scholar

[17]

H. S. Mortveit and C. M. Reidys, "An Introduction to Sequential Dynamical Systems,", Universitext, (2007). Google Scholar

[18]

C. M. Reidys, Sequential dynamical systems over words,, Ann. Comb., 10 (2006), 481. doi: 10.1007/s00026-006-0301-y. Google Scholar

[19]

C. M. Reidys, Combinatorics of sequential dynamical systems,, Discrete Math., 308 (2007), 514. doi: 10.1016/j.disc.2007.03.033. Google Scholar

[20]

I. Shmulevich, E. R. Dougherty and W. Zhang, From Boolean to probabilistic Boolean networks as models of genetic regulatory networks,, Proc. IEEE, 90 (2002), 1778. doi: 10.1109/JPROC.2002.804686. Google Scholar

[21]

S. Wolfram, "Theory and Applications of Cellular Automata,", Adv. Ser. Complex Systems, 1 (1986). Google Scholar

show all references

References:
[1]

C. L. Barrett, H. H. Hunt, M. V. Marathe, S. S. Ravi, D. Rosenkrantz, R. Stearns and P. Tosic, Gardens of Eden and fixed point in sequential dynamical systems,, DM-CCG 2001, (2001), 95. Google Scholar

[2]

C. L. Barrett, H. S. Mortveit and C. M. Reidys, Elements of a theory of simulation III: Equivalence of SDS,, Appl. Math. Comput., 122 (2001), 325. doi: 10.1016/S0096-3003(00)00042-4. Google Scholar

[3]

C. L. Barrett, H. S. Mortveit and C. M. Reidys, Elements of a theory of simulation IV: Fixed points, invertibility and equivalence,, Appl. Math. Comput., 134 (2003), 153. doi: 10.1016/S0096-3003(01)00277-6. Google Scholar

[4]

C. L. Barrett, H. B. Hunt III, M. V. Marathe, S. S. Ravi, D. J. Rosenkrantz and R. E. Stearns, Predecessor and permutation existence problems for sequential dynamical systems,, DMTCS 2003, (2003), 69. Google Scholar

[5]

A. Björner and F. Brenti, "Combinatorics of Coxeter Groups,'', Springer-Verlag, (2005). Google Scholar

[6]

B. Bollobàs, "Random Graphs,'', Cambridge University Press, (2001). Google Scholar

[7]

R. Laubenbacher E. Sontag, A. Veliz-Cuba and A. Salam Jarrah, The effect of negative feedback loops on the dynamics of Boolean networks,, Biophys. J., 95 (2008), 518. doi: 10.1529/biophysj.107.125021. Google Scholar

[8]

D. L. Vertigan F. Jaeger and D. J. A. Welsh, On the computational complexity of the Jones and Tutte polynomials,, Math. Proc. Cambridge Philos. Soc. \textbf{108} (1990), 108 (1990), 35. doi: 10.1017/S0305004100068936. Google Scholar

[9]

A. Å. Hansson, H. S. Mortveit and C. M. Reidys, On asynchronous cellular automata,, Adv. Complex Systems, 8 (2005), 521. Google Scholar

[10]

U. Karaoz, T. M. Murali, S. Letovsky, Y. Zheng, C. Ding, C. R. Cantor and S. Kasif, Whole-genome annotation by using evidence integration in functional-linkage networks,, Proc. Nat. Acad. Sci., 101 (2004), 2888. doi: 10.1073/pnas.0307326101. Google Scholar

[11]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. Royal Soc. London A, 115 (1927), 700. doi: 10.1098/rspa.1927.0118. Google Scholar

[12]

V. S. A. Kumar, M. Macauley and H. S. Mortveit, Limit set reachability in asynchronous graph dynamical systems,, RP 2009, (2009), 217. Google Scholar

[13]

M. Macauley, J. McCammond and H. S. Mortveit, Dynamics groups of asynchronous cellular automata,, J. Algebraic Combinat, 33 (2011). doi: 10.1007/s10801-010-0231-y. Google Scholar

[14]

M. Macauley, J. McCammond and H. S. Mortveit, Order independence in asynchronous cellular automata,, J. Cell. Autom., 3 (2008), 37. Google Scholar

[15]

M. Macauley and H. S. Mortveit, On enumeration of conjugacy classes of Coxeter elements,, Proc. Amer. Math. Soc., 136 (2008), 4157. doi: 10.1090/S0002-9939-08-09543-9. Google Scholar

[16]

M. Macauley and H. S. Mortveit, Cycle equivalence of graph dynamical systems,, Nonlinearity, 22 (2009), 421. doi: 10.1088/0951-7715/22/2/010. Google Scholar

[17]

H. S. Mortveit and C. M. Reidys, "An Introduction to Sequential Dynamical Systems,", Universitext, (2007). Google Scholar

[18]

C. M. Reidys, Sequential dynamical systems over words,, Ann. Comb., 10 (2006), 481. doi: 10.1007/s00026-006-0301-y. Google Scholar

[19]

C. M. Reidys, Combinatorics of sequential dynamical systems,, Discrete Math., 308 (2007), 514. doi: 10.1016/j.disc.2007.03.033. Google Scholar

[20]

I. Shmulevich, E. R. Dougherty and W. Zhang, From Boolean to probabilistic Boolean networks as models of genetic regulatory networks,, Proc. IEEE, 90 (2002), 1778. doi: 10.1109/JPROC.2002.804686. Google Scholar

[21]

S. Wolfram, "Theory and Applications of Cellular Automata,", Adv. Ser. Complex Systems, 1 (1986). Google Scholar

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