December  2016, 21(10): 3391-3405. doi: 10.3934/dcdsb.2016103

Computational methods for asynchronous basins

1. 

Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, 724 SW Harrison Street, Portland, OR 97201, United States

Received  December 2015 Revised  March 2016 Published  November 2016

For a Boolean network we consider asynchronous updates and define the exclusive asynchronous basin of attraction for any steady state or cyclic attractor. An algorithm based on commutative algebra is presented to compute the exclusive basin. Finally its use for targeting desirable attractors by selective intervention on network nodes is illustrated with two examples, one cell signalling network and one sensor network measuring human mobility.
Citation: Ian H. Dinwoodie. Computational methods for asynchronous basins. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3391-3405. doi: 10.3934/dcdsb.2016103
References:
[1]

J. Abbott and A. M. Bigatti, CoCoALib: A C++ library for doing Computations in Commutative Algebra, 2014. Available from: http://cocoa.dima.unige.it/cocoalib.

[2]

D. Austin, R. M. Cross, T. Hayes and J. Kaye, Regularity and Predictability of Human Mobility in Personal Space, PLoS One, 9 (2014), e90256. doi: 10.1371/journal.pone.0090256.

[3]

D. Austin and I. H. Dinwoodie, Monomials and Basin Cylinders for Network Dynamics, SIAM J. Appl. Dyn. Syst., 14 (2015), 25-42. doi: 10.1137/140975929.

[4]

D. E. Bredesen, Reversal of cognitive decline: A novel therapeutic program, Aging, 6 (2014), 707-717. doi: 10.18632/aging.100690.

[5]

T. Buracchio, H. Dodge, D. Howieson, D. Wasserman and J. Kaye, The trajectory of gait speed preceding MCI, Arch. Neurol., 67 (2010), 980-986.

[6]

M. Chaves, R. Albert and E. D. Sontag, Robustness and fragility of Boolean models for genetic regulatory networks, Jour. Theoret. Biol., 235 (2005), 431-449. doi: 10.1016/j.jtbi.2005.01.023.

[7]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, $2^{nd}$ edition, Springer, New York, 1997.

[8]

W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 - A Computer Algebra System for Polynomial Computations, 2015. Available from: http://www.singular.uni-kl.de.

[9]

G. V. De Ferrari and N. C. Inestrosa, Wnt signaling function in Alzheimer's disease, Brain Res. Rev., 33 (2000), 1-12.

[10]

I. H. Dinwoodie, Conditional tests on basins of attraction with finite fields, Methodol. Comput. Appl. Probab., 16 (2014), 161-168. doi: 10.1007/s11009-012-9304-9.

[11]

I. H. Dinwoodie, Vanishing configurations in network dynamics with asynchronous updates, Proc. Amer. Math. Soc., 142 (2014), 2991-3002. doi: 10.1090/S0002-9939-2014-12044-2.

[12]

I. H. Dinwoodie, Polynomials for classification trees and applications, Stat. Methods Appt., 19 (2009), 171-192. doi: 10.1007/s10260-009-0123-2.

[13]

I. H. Dinwoodie and K. Pandya, Exact tests for singular network data, Ann. Inst. Statist. Math. 67 (2015), 687-706. doi: 10.1007/s10463-014-0472-y.

[14]

D. R. Grayson and M. E. Stillman, Macaulay2, A Software System for Research in Algebraic Geometry, 2014. Available from: http://www.math.uiuc.edu/Macaulay2.

[15]

T. Handorf and E. Klipp, Modeling mechanistic biological networks: An advanced Boolean approach, Bioinformatics, 28 (2012), 557-663.

[16]

T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, $2^{nd}$ edition, Springer, New York, 2009. doi: 10.1007/978-0-387-21606-5.

[17]

T. L. Hayes, T. Riley, M. Pavel and J. A. Kaye, Estimation of rest-activity patterns using motion sensors, Conf. Proc. IEEE Eng. Med. Biol. Soc., 2010 (2010), 2147-2150. doi: 10.1109/IEMBS.2010.5628022.

[18]

M. Hermes, G. Eichoff and O. Garaschuk, Intracellular calcium signalling in Alzheimer's disease, J. Cell. Mol. Med., 14 (2009), 30-41. doi: 10.1111/j.1582-4934.2009.00976.x.

[19]

F. Hinkelmann, M. Brandon, B. Guang, R. McNeill, G. Blekherman, A. Veliz-Cuba and R. Laubenbacher, ADAM: Analysis of discrete models of biological systems using computer algebra, BMC Bioinformatics, 12 (2011), p295. doi: 10.1186/1471-2105-12-295.

[20]

F. Hinkelmann, D. Murrugarra, A. S. Jarrah and R. Laubenbacher, A mathematical framework for agent based models of complex biological networks, Bull. Math. Biol. 73 (2011), 1583-1602. doi: 10.1007/s11538-010-9582-8.

[21]

J. A. Kaye, S. A. Maxwell, N. Mattek, T. L. Hayes, H. Dodge, M. Pavel, H. B. Jimison, K. Wild, L. Boise and T. A. Zitzelberger, Intelligent systems for assessing aging changes: Home-based, unobtrusive, and continuous assessment of ageing, J. Gerontol. B: Psychol. Sci. and Soc. Sci., 66B (2011), i180-i190. doi: 10.1093/geronb/gbq095.

[22]

S. Klamt, J. Saez-Rodriquez, J. A. Lindquist, L. Simeoni and E. D. Gilles, A methodology for the structural and functional analysis of signalling and regulatory networks, BMC Bioinformatics, 7 (2006), 1471-2105.

[23]

M. Kreuzer and L. Robbiano, Computational Commutative Algebra I, Springer, New York, 2000. doi: 10.1007/978-3-540-70628-1.

[24]

R. Laubenbacher and B. Sturmfels, Computer Algebra in Systems Biology, Amer. Math. Monthly, 116 (2009), 882-891. doi: 10.4169/000298909X477005.

[25]

R. K. Layek, A. Datta and E. R. Dougherty, From biological pathways to regulatory networks, Molecular BioSystems, 7 (2011), 843-851. doi: 10.1109/CDC.2010.5716936.

[26]

R. K. Layek, A. Datta, M. Bittner and E. R. Dougherty, Cancer therapy design based on pathway logic, Bioinformatics, 27 (2011), 548-555. doi: 10.1093/bioinformatics/btq703.

[27]

A. Liaw and M. Wiener, Classification and Regression by randomForest, R News, 2 (2002), 18-22. Available from: http://CRAN.R-project.org/doc/Rnews/.

[28]

T. Lu, L. Aron, J. Zullo, Y. Pan, H. Kim, Y. Chen, T.-H. Yang, H.-M. Kim, D. Drake, X. S. Liu, D. A. Bennett, M. P. Colaiácovo and B. A. Yankner, REST and stress resistance in ageing and Alzheimer's disease, Nature, 507 (2014), 448-454. doi: 10.1038/nature13163.

[29]

M. K. Morris, J. Saez-Rodriguez, P. K. Sorger and D. A. Lauffenburger, Logic-based models for the analysis of cell signaling networks, Biochemistry, 49 (2010), 3216-3224. doi: 10.1021/bi902202q.

[30]

D. Murrugarra, A. Veliz-Cuba, B. Aguilar, S. Arat and R. Laubenbacher, Modeling stochasticity and variability in gene regulatory networks, EURASIP J. Bioinform. and Syst. Biol., 2012 (2012), p5. doi: 10.1186/1687-4153-2012-5.

[31]

C. Müssel, M. Hopfensitz and H. A. Kestler, BoolNet - an R package for generation, reconstruction and analysis of Boolean networks, Bioinformatics, 26 (2010), 1378-1380.

[32]

J. Petersen, D. Austin, J. Kaye, M. Pavel and T. Hayes, Unobtrusive in-home detection of time spent out-of-home with applications to loneliness and physical activity, IEEE J. Biomed. Health Inform., 18 (2014), 1590-1596. doi: 10.1109/JBHI.2013.2294276.

[33]

R. C. Petersen, Mild cognitive impairment as a diagnostic entity, J. Intern. Med., 256 (2004), 183-194. doi: 10.1111/j.1365-2796.2004.01388.x.

[34]

G. Pistone, E. Riccomagno and H. Wynn, Algebraic Statistics: Computational Commutative Algebra in Statistics, Chapman and Hall, Boca Raton Florida, 2001.

[35]

R. Poltz and M. Naumann, Dynamics of p53 and NF-$\kappa$B regulation in response to DNA damage and identification of target proteins suitable for therapeutic intervention, BMC Syst. Biol., 6 (2012), p125.

[36]

A. Saadatpour, R-S. Wang, A. Liao, X. Liu, T. P. Loughran, I. Albert and R. Albert, Dynamical and structural analysis of a T cell survival network identifies novel candidate therapeutic targets for large granula lymphocyte leukemia, PLoS Comp. Biol. 7 (2011), e1002267.

[37]

A. Saadatpour, I. Albert and R. Albert, Attractor analysis of asynchronous Boolean models of signal transduction networks, J. Theor. Biol., 266 (2010), 641-656. doi: 10.1016/j.jtbi.2010.07.022.

[38]

R. Schlatter, K. Schmich, I. A. Vizcarra, P. Scheurich, T. Sauter, C. Borner, M. Ederer, I. Merfort and O. Sawodny, ON/OFF and Beyond - A Boolean Model of Apoptosis, PLoS Comput. Biol., 5 (2009), e1000595. doi: 10.1371/journal.pcbi.1000595.

[39]

I. Shmulevich, E. R. Dougherty, S. Kim and W. Zhang, Probabilistic Boolean Networks: A rule-based uncertainty model for gene regulatory networks, Bioinformatics, 18 (2002), 261-274. doi: 10.1093/bioinformatics/18.2.261.

[40]

B. Stigler, Polynomial dynamical systems in systems biology, AMS 2006 Proceedings of Symposia in Applied Mathematics, 64 (2007), 53-84. doi: 10.1090/psapm/064/2359649.

[41]

T. Therneau, B. Atkinson and B. Ripley, Rpart: Recursive Partitioning and Regression Trees, 2015. Available from: http://CRAN.R-project.org.

[42]

R. Thomas, Boolean formalization of genetic control circuits, J. Theoret. Biol., 42 (1973), 563-585.

[43]

A. Veliz-Cuba, An algebraic approach to reverse engineering finite dynamical systems arising from biology, SIAM Jour. Appl. Dyn. Systems, 11 (2012), 31-48. doi: 10.1137/110828794.

[44]

A. Wuensche, Complex and Chaotic Dynamics, Basins of Attraction, and Memory in Discrete Networks, Acta Physica Polonica B, 3 (2010), 463-478.

[45]

R. Zhang, M. V. Shah, J. Yang, S. B. Nyland, X. Liu, J. Yun, R. Albert and T. P. Loughran, Network model of survival signaling in large granular lymphocyte leukemia, Proc. Natl. Acad. Sci. USA, 105 (2008), 16308-16313.

show all references

References:
[1]

J. Abbott and A. M. Bigatti, CoCoALib: A C++ library for doing Computations in Commutative Algebra, 2014. Available from: http://cocoa.dima.unige.it/cocoalib.

[2]

D. Austin, R. M. Cross, T. Hayes and J. Kaye, Regularity and Predictability of Human Mobility in Personal Space, PLoS One, 9 (2014), e90256. doi: 10.1371/journal.pone.0090256.

[3]

D. Austin and I. H. Dinwoodie, Monomials and Basin Cylinders for Network Dynamics, SIAM J. Appl. Dyn. Syst., 14 (2015), 25-42. doi: 10.1137/140975929.

[4]

D. E. Bredesen, Reversal of cognitive decline: A novel therapeutic program, Aging, 6 (2014), 707-717. doi: 10.18632/aging.100690.

[5]

T. Buracchio, H. Dodge, D. Howieson, D. Wasserman and J. Kaye, The trajectory of gait speed preceding MCI, Arch. Neurol., 67 (2010), 980-986.

[6]

M. Chaves, R. Albert and E. D. Sontag, Robustness and fragility of Boolean models for genetic regulatory networks, Jour. Theoret. Biol., 235 (2005), 431-449. doi: 10.1016/j.jtbi.2005.01.023.

[7]

D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, $2^{nd}$ edition, Springer, New York, 1997.

[8]

W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-0-2 - A Computer Algebra System for Polynomial Computations, 2015. Available from: http://www.singular.uni-kl.de.

[9]

G. V. De Ferrari and N. C. Inestrosa, Wnt signaling function in Alzheimer's disease, Brain Res. Rev., 33 (2000), 1-12.

[10]

I. H. Dinwoodie, Conditional tests on basins of attraction with finite fields, Methodol. Comput. Appl. Probab., 16 (2014), 161-168. doi: 10.1007/s11009-012-9304-9.

[11]

I. H. Dinwoodie, Vanishing configurations in network dynamics with asynchronous updates, Proc. Amer. Math. Soc., 142 (2014), 2991-3002. doi: 10.1090/S0002-9939-2014-12044-2.

[12]

I. H. Dinwoodie, Polynomials for classification trees and applications, Stat. Methods Appt., 19 (2009), 171-192. doi: 10.1007/s10260-009-0123-2.

[13]

I. H. Dinwoodie and K. Pandya, Exact tests for singular network data, Ann. Inst. Statist. Math. 67 (2015), 687-706. doi: 10.1007/s10463-014-0472-y.

[14]

D. R. Grayson and M. E. Stillman, Macaulay2, A Software System for Research in Algebraic Geometry, 2014. Available from: http://www.math.uiuc.edu/Macaulay2.

[15]

T. Handorf and E. Klipp, Modeling mechanistic biological networks: An advanced Boolean approach, Bioinformatics, 28 (2012), 557-663.

[16]

T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, $2^{nd}$ edition, Springer, New York, 2009. doi: 10.1007/978-0-387-21606-5.

[17]

T. L. Hayes, T. Riley, M. Pavel and J. A. Kaye, Estimation of rest-activity patterns using motion sensors, Conf. Proc. IEEE Eng. Med. Biol. Soc., 2010 (2010), 2147-2150. doi: 10.1109/IEMBS.2010.5628022.

[18]

M. Hermes, G. Eichoff and O. Garaschuk, Intracellular calcium signalling in Alzheimer's disease, J. Cell. Mol. Med., 14 (2009), 30-41. doi: 10.1111/j.1582-4934.2009.00976.x.

[19]

F. Hinkelmann, M. Brandon, B. Guang, R. McNeill, G. Blekherman, A. Veliz-Cuba and R. Laubenbacher, ADAM: Analysis of discrete models of biological systems using computer algebra, BMC Bioinformatics, 12 (2011), p295. doi: 10.1186/1471-2105-12-295.

[20]

F. Hinkelmann, D. Murrugarra, A. S. Jarrah and R. Laubenbacher, A mathematical framework for agent based models of complex biological networks, Bull. Math. Biol. 73 (2011), 1583-1602. doi: 10.1007/s11538-010-9582-8.

[21]

J. A. Kaye, S. A. Maxwell, N. Mattek, T. L. Hayes, H. Dodge, M. Pavel, H. B. Jimison, K. Wild, L. Boise and T. A. Zitzelberger, Intelligent systems for assessing aging changes: Home-based, unobtrusive, and continuous assessment of ageing, J. Gerontol. B: Psychol. Sci. and Soc. Sci., 66B (2011), i180-i190. doi: 10.1093/geronb/gbq095.

[22]

S. Klamt, J. Saez-Rodriquez, J. A. Lindquist, L. Simeoni and E. D. Gilles, A methodology for the structural and functional analysis of signalling and regulatory networks, BMC Bioinformatics, 7 (2006), 1471-2105.

[23]

M. Kreuzer and L. Robbiano, Computational Commutative Algebra I, Springer, New York, 2000. doi: 10.1007/978-3-540-70628-1.

[24]

R. Laubenbacher and B. Sturmfels, Computer Algebra in Systems Biology, Amer. Math. Monthly, 116 (2009), 882-891. doi: 10.4169/000298909X477005.

[25]

R. K. Layek, A. Datta and E. R. Dougherty, From biological pathways to regulatory networks, Molecular BioSystems, 7 (2011), 843-851. doi: 10.1109/CDC.2010.5716936.

[26]

R. K. Layek, A. Datta, M. Bittner and E. R. Dougherty, Cancer therapy design based on pathway logic, Bioinformatics, 27 (2011), 548-555. doi: 10.1093/bioinformatics/btq703.

[27]

A. Liaw and M. Wiener, Classification and Regression by randomForest, R News, 2 (2002), 18-22. Available from: http://CRAN.R-project.org/doc/Rnews/.

[28]

T. Lu, L. Aron, J. Zullo, Y. Pan, H. Kim, Y. Chen, T.-H. Yang, H.-M. Kim, D. Drake, X. S. Liu, D. A. Bennett, M. P. Colaiácovo and B. A. Yankner, REST and stress resistance in ageing and Alzheimer's disease, Nature, 507 (2014), 448-454. doi: 10.1038/nature13163.

[29]

M. K. Morris, J. Saez-Rodriguez, P. K. Sorger and D. A. Lauffenburger, Logic-based models for the analysis of cell signaling networks, Biochemistry, 49 (2010), 3216-3224. doi: 10.1021/bi902202q.

[30]

D. Murrugarra, A. Veliz-Cuba, B. Aguilar, S. Arat and R. Laubenbacher, Modeling stochasticity and variability in gene regulatory networks, EURASIP J. Bioinform. and Syst. Biol., 2012 (2012), p5. doi: 10.1186/1687-4153-2012-5.

[31]

C. Müssel, M. Hopfensitz and H. A. Kestler, BoolNet - an R package for generation, reconstruction and analysis of Boolean networks, Bioinformatics, 26 (2010), 1378-1380.

[32]

J. Petersen, D. Austin, J. Kaye, M. Pavel and T. Hayes, Unobtrusive in-home detection of time spent out-of-home with applications to loneliness and physical activity, IEEE J. Biomed. Health Inform., 18 (2014), 1590-1596. doi: 10.1109/JBHI.2013.2294276.

[33]

R. C. Petersen, Mild cognitive impairment as a diagnostic entity, J. Intern. Med., 256 (2004), 183-194. doi: 10.1111/j.1365-2796.2004.01388.x.

[34]

G. Pistone, E. Riccomagno and H. Wynn, Algebraic Statistics: Computational Commutative Algebra in Statistics, Chapman and Hall, Boca Raton Florida, 2001.

[35]

R. Poltz and M. Naumann, Dynamics of p53 and NF-$\kappa$B regulation in response to DNA damage and identification of target proteins suitable for therapeutic intervention, BMC Syst. Biol., 6 (2012), p125.

[36]

A. Saadatpour, R-S. Wang, A. Liao, X. Liu, T. P. Loughran, I. Albert and R. Albert, Dynamical and structural analysis of a T cell survival network identifies novel candidate therapeutic targets for large granula lymphocyte leukemia, PLoS Comp. Biol. 7 (2011), e1002267.

[37]

A. Saadatpour, I. Albert and R. Albert, Attractor analysis of asynchronous Boolean models of signal transduction networks, J. Theor. Biol., 266 (2010), 641-656. doi: 10.1016/j.jtbi.2010.07.022.

[38]

R. Schlatter, K. Schmich, I. A. Vizcarra, P. Scheurich, T. Sauter, C. Borner, M. Ederer, I. Merfort and O. Sawodny, ON/OFF and Beyond - A Boolean Model of Apoptosis, PLoS Comput. Biol., 5 (2009), e1000595. doi: 10.1371/journal.pcbi.1000595.

[39]

I. Shmulevich, E. R. Dougherty, S. Kim and W. Zhang, Probabilistic Boolean Networks: A rule-based uncertainty model for gene regulatory networks, Bioinformatics, 18 (2002), 261-274. doi: 10.1093/bioinformatics/18.2.261.

[40]

B. Stigler, Polynomial dynamical systems in systems biology, AMS 2006 Proceedings of Symposia in Applied Mathematics, 64 (2007), 53-84. doi: 10.1090/psapm/064/2359649.

[41]

T. Therneau, B. Atkinson and B. Ripley, Rpart: Recursive Partitioning and Regression Trees, 2015. Available from: http://CRAN.R-project.org.

[42]

R. Thomas, Boolean formalization of genetic control circuits, J. Theoret. Biol., 42 (1973), 563-585.

[43]

A. Veliz-Cuba, An algebraic approach to reverse engineering finite dynamical systems arising from biology, SIAM Jour. Appl. Dyn. Systems, 11 (2012), 31-48. doi: 10.1137/110828794.

[44]

A. Wuensche, Complex and Chaotic Dynamics, Basins of Attraction, and Memory in Discrete Networks, Acta Physica Polonica B, 3 (2010), 463-478.

[45]

R. Zhang, M. V. Shah, J. Yang, S. B. Nyland, X. Liu, J. Yun, R. Albert and T. P. Loughran, Network model of survival signaling in large granular lymphocyte leukemia, Proc. Natl. Acad. Sci. USA, 105 (2008), 16308-16313.

[1]

Keisuke Minami, Takahiro Matsuda, Tetsuya Takine, Taku Noguchi. Asynchronous multiple source network coding for wireless broadcasting. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 577-592. doi: 10.3934/naco.2011.1.577

[2]

Junjie Peng, Ning Chen, Jiayang Dai, Weihua Gui. A goethite process modeling method by Asynchronous Fuzzy Cognitive Network based on an improved constrained chicken swarm optimization algorithm. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1269-1287. doi: 10.3934/jimo.2020021

[3]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Stability of the dynamics of an asymmetric neural network. Communications on Pure and Applied Analysis, 2009, 8 (2) : 655-671. doi: 10.3934/cpaa.2009.8.655

[4]

Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283

[5]

Kangkang Deng, Zheng Peng, Jianli Chen. Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1881-1896. doi: 10.3934/jimo.2018127

[6]

Marina Dolfin, Mirosław Lachowicz. Modeling opinion dynamics: How the network enhances consensus. Networks and Heterogeneous Media, 2015, 10 (4) : 877-896. doi: 10.3934/nhm.2015.10.877

[7]

Winfried Just. Approximating network dynamics: Some open problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 1917-1930. doi: 10.3934/dcdsb.2018188

[8]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279

[9]

Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1249-1274. doi: 10.3934/dcds.2009.25.1249

[10]

Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4247-4269. doi: 10.3934/dcdsb.2019080

[11]

Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010

[12]

Yi Ming Zou. Dynamics of boolean networks. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1629-1640. doi: 10.3934/dcdss.2011.4.1629

[13]

Linhe Zhu, Wenshan Liu. Spatial dynamics and optimization method for a network propagation model in a shifting environment. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1843-1874. doi: 10.3934/dcds.2020342

[14]

Jiangtao Mo, Liqun Qi, Zengxin Wei. A network simplex algorithm for simple manufacturing network model. Journal of Industrial and Management Optimization, 2005, 1 (2) : 251-273. doi: 10.3934/jimo.2005.1.251

[15]

Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem. Conference Publications, 2009, 2009 (Special) : 259-268. doi: 10.3934/proc.2009.2009.259

[16]

Peter Giesl. Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 355-373. doi: 10.3934/dcds.2007.18.355

[17]

Konstantin Avrachenkov, Giovanni Neglia, Vikas Vikram Singh. Network formation games with teams. Journal of Dynamics and Games, 2016, 3 (4) : 303-318. doi: 10.3934/jdg.2016016

[18]

Joanna Tyrcha, John Hertz. Network inference with hidden units. Mathematical Biosciences & Engineering, 2014, 11 (1) : 149-165. doi: 10.3934/mbe.2014.11.149

[19]

T. S. Evans, A. D. K. Plato. Network rewiring models. Networks and Heterogeneous Media, 2008, 3 (2) : 221-238. doi: 10.3934/nhm.2008.3.221

[20]

David J. Aldous. A stochastic complex network model. Electronic Research Announcements, 2003, 9: 152-161.

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (177)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]