December  2011, 4(4): 1081-1096. doi: 10.3934/krm.2011.4.1081

Averaged kinetic models for flows on unstructured networks

1. 

RWTH Aachen, Department of Mathematics, Templergraben 55, 52056 Aachen, Germany

2. 

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804

Received  July 2011 Revised  September 2011 Published  November 2011

We derive a kinetic equation for flows on general, unstructured networks with applications to production, social and transportation networks. This model allows for a homogenization procedure, yielding a macroscopic transport model for large networks on large time scales.
Citation: Michael Herty, Christian Ringhofer. Averaged kinetic models for flows on unstructured networks. Kinetic & Related Models, 2011, 4 (4) : 1081-1096. doi: 10.3934/krm.2011.4.1081
References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains,, SIAM J. Appl. Math., 66 (2006), 896.  doi: 10.1137/040604625.  Google Scholar

[2]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks,, Science, 286 (1999), 509.  doi: 10.1126/science.286.5439.509.  Google Scholar

[3]

S. Battiston, D. Delli Gatti, M. Gallegati, B. Greenwald and J. E. Stiglitz, Credit chains and bankruptcy propagation in production networks,, Journal of Economic Dynamics and Control, 31 (2007), 2061.  doi: 10.1016/j.jedc.2007.01.004.  Google Scholar

[4]

D. Brockmann, Anomalous diffusion and the structure of human transportation networks,, Eur. Phys. J. Special Topics, 157 (2008), 173.  doi: 10.1140/epjst/e2008-00640-0.  Google Scholar

[5]

C. Cercignani, I. Gamba and D. Levermore, A drift-collision balance for a Boltzmann- Poisson system in bounded domains,, SIAM J. Appl. Math., 61 (2001), 1932.  doi: 10.1137/S0036139999360465.  Google Scholar

[6]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Applied Mathematical Sciences, 106 (1994).   Google Scholar

[7]

F. Della Rossa, C. D'Angelo and A. Quarteroni, A distributed model of traffic flows on extended regions,, Networks and Heterogeneous Media \textbf{5} (2010), 5 (2010), 525.   Google Scholar

[8]

P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns,, SIAM J. Applied Mathematics, 68 (2007), 59.  doi: 10.1137/060674302.  Google Scholar

[9]

A. Ern, A. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted aver- ages for advection-diffusion equations with locally small and anisotropic diffusivity,, IMA J. Numer. Anal., 29 (2009), 235.  doi: 10.1093/imanum/drm050.  Google Scholar

[10]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models,", AIMS Series on Applied Mathematics, (2006).   Google Scholar

[11]

S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks,, European Journal of Operational Research, 202 (2010), 456.  doi: 10.1016/j.ejor.2009.05.028.  Google Scholar

[12]

S. Göttlich, M. Herty and C. Ringhofer, Time-dependent order and distribution policies in supply networks,, in, (2008), 521.   Google Scholar

[13]

W. H. S. C. Graves and D. B. Kletter, A dynamic model for requirements planning with application to supply chain optimization,, Operations Research, 46 (1998), 35.  doi: 10.1287/opre.46.3.S35.  Google Scholar

[14]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Mod. Phys., 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[15]

E. W. Larsen, A generalized Boltzmann equation for "non-classical" particle transport,, in, (2007), 15.   Google Scholar

[16]

E. W. Larsen and R. Vasques, A generalized linear Boltzmann equation for non-classical particle transport,, Journal of Quantitative Spectroscopy and Radiative Transfer, (2010).  doi: 10.1016/j.jqsrt.2010.07.003.  Google Scholar

[17]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317.   Google Scholar

[18]

H. Missbauer, Aggregate order release planning for time varying demand,, Int. J. Production Research, 40 (2002), 699.  doi: 10.1080/00207540110090939.  Google Scholar

[19]

K. Nagel, Particle hopping models and traffic flow theory,, Physical Review E, 3 (1996).  doi: 10.1103/PhysRevE.53.4655.  Google Scholar

[20]

F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory,, Z. Angew. Math. Mech., 72 (1992), 359.  doi: 10.1002/zamm.19920720813.  Google Scholar

[21]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440.  doi: 10.1038/30918.  Google Scholar

show all references

References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains,, SIAM J. Appl. Math., 66 (2006), 896.  doi: 10.1137/040604625.  Google Scholar

[2]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks,, Science, 286 (1999), 509.  doi: 10.1126/science.286.5439.509.  Google Scholar

[3]

S. Battiston, D. Delli Gatti, M. Gallegati, B. Greenwald and J. E. Stiglitz, Credit chains and bankruptcy propagation in production networks,, Journal of Economic Dynamics and Control, 31 (2007), 2061.  doi: 10.1016/j.jedc.2007.01.004.  Google Scholar

[4]

D. Brockmann, Anomalous diffusion and the structure of human transportation networks,, Eur. Phys. J. Special Topics, 157 (2008), 173.  doi: 10.1140/epjst/e2008-00640-0.  Google Scholar

[5]

C. Cercignani, I. Gamba and D. Levermore, A drift-collision balance for a Boltzmann- Poisson system in bounded domains,, SIAM J. Appl. Math., 61 (2001), 1932.  doi: 10.1137/S0036139999360465.  Google Scholar

[6]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Applied Mathematical Sciences, 106 (1994).   Google Scholar

[7]

F. Della Rossa, C. D'Angelo and A. Quarteroni, A distributed model of traffic flows on extended regions,, Networks and Heterogeneous Media \textbf{5} (2010), 5 (2010), 525.   Google Scholar

[8]

P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns,, SIAM J. Applied Mathematics, 68 (2007), 59.  doi: 10.1137/060674302.  Google Scholar

[9]

A. Ern, A. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted aver- ages for advection-diffusion equations with locally small and anisotropic diffusivity,, IMA J. Numer. Anal., 29 (2009), 235.  doi: 10.1093/imanum/drm050.  Google Scholar

[10]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models,", AIMS Series on Applied Mathematics, (2006).   Google Scholar

[11]

S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks,, European Journal of Operational Research, 202 (2010), 456.  doi: 10.1016/j.ejor.2009.05.028.  Google Scholar

[12]

S. Göttlich, M. Herty and C. Ringhofer, Time-dependent order and distribution policies in supply networks,, in, (2008), 521.   Google Scholar

[13]

W. H. S. C. Graves and D. B. Kletter, A dynamic model for requirements planning with application to supply chain optimization,, Operations Research, 46 (1998), 35.  doi: 10.1287/opre.46.3.S35.  Google Scholar

[14]

D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Mod. Phys., 73 (2001), 1067.  doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[15]

E. W. Larsen, A generalized Boltzmann equation for "non-classical" particle transport,, in, (2007), 15.   Google Scholar

[16]

E. W. Larsen and R. Vasques, A generalized linear Boltzmann equation for non-classical particle transport,, Journal of Quantitative Spectroscopy and Radiative Transfer, (2010).  doi: 10.1016/j.jqsrt.2010.07.003.  Google Scholar

[17]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317.   Google Scholar

[18]

H. Missbauer, Aggregate order release planning for time varying demand,, Int. J. Production Research, 40 (2002), 699.  doi: 10.1080/00207540110090939.  Google Scholar

[19]

K. Nagel, Particle hopping models and traffic flow theory,, Physical Review E, 3 (1996).  doi: 10.1103/PhysRevE.53.4655.  Google Scholar

[20]

F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory,, Z. Angew. Math. Mech., 72 (1992), 359.  doi: 10.1002/zamm.19920720813.  Google Scholar

[21]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440.  doi: 10.1038/30918.  Google Scholar

[1]

Thomas Hillen, Peter Hinow, Zhi-An Wang. Mathematical analysis of a kinetic model for cell movement in network tissues. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1055-1080. doi: 10.3934/dcdsb.2010.14.1055

[2]

Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049

[3]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[4]

Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429

[5]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621

[6]

Domenica Borra, Tommaso Lorenzi. Asymptotic analysis of continuous opinion dynamics models under bounded confidence. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1487-1499. doi: 10.3934/cpaa.2013.12.1487

[7]

Paolo Barbante, Aldo Frezzotti, Livio Gibelli. A kinetic theory description of liquid menisci at the microscale. Kinetic & Related Models, 2015, 8 (2) : 235-254. doi: 10.3934/krm.2015.8.235

[8]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[9]

Hung-Wen Kuo. Effect of abrupt change of the wall temperature in the kinetic theory. Kinetic & Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030

[10]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic & Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030

[11]

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

[12]

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

[13]

Shyan-Shiou Chen, Chih-Wen Shih. Asymptotic behaviors in a transiently chaotic neural network. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 805-826. doi: 10.3934/dcds.2004.10.805

[14]

Mirosław Lachowicz, Andrea Quartarone, Tatiana V. Ryabukha. Stability of solutions of kinetic equations corresponding to the replicator dynamics. Kinetic & Related Models, 2014, 7 (1) : 109-119. doi: 10.3934/krm.2014.7.109

[15]

José A. Carrillo, M. R. D’Orsogna, V. Panferov. Double milling in self-propelled swarms from kinetic theory. Kinetic & Related Models, 2009, 2 (2) : 363-378. doi: 10.3934/krm.2009.2.363

[16]

Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic & Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004

[17]

Carlos Escudero, Fabricio Macià, Raúl Toral, Juan J. L. Velázquez. Kinetic theory and numerical simulations of two-species coagulation. Kinetic & Related Models, 2014, 7 (2) : 253-290. doi: 10.3934/krm.2014.7.253

[18]

Simone Farinelli. Geometric arbitrage theory and market dynamics. Journal of Geometric Mechanics, 2015, 7 (4) : 431-471. doi: 10.3934/jgm.2015.7.431

[19]

Mark G. Burch, Karly A. Jacobsen, Joseph H. Tien, Grzegorz A. Rempała. Network-based analysis of a small Ebola outbreak. Mathematical Biosciences & Engineering, 2017, 14 (1) : 67-77. doi: 10.3934/mbe.2017005

[20]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]