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Layered Hodge Decomposition for Urban Transit Networks

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Complex Networks & Their Applications X (COMPLEX NETWORKS 2021)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 1073))

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Abstract

Modeling the amount of passenger flow along any given line segment of an urban transit network is a challenging task due to the complexity of the system. In this paper, we embark on a characterization of these flows on the basis of a combination of (1) a layered decomposition of the origin-destination matrix, and (2) the Hodge decomposition, a discrete algebraic topology technique that partitions flows into gradient, solenoidal, and harmonic components. We apply our method to data from the London Underground. We find that the layered decomposition estimates the contribution of each origin-destination pair to the flow on each network link, and that the solenoidal and harmonic flows can be described by simple equations, thereby reducing much of the solution to determining gradient flows. Our exploratory analysis suggests it may be feasible to develop solution methods for the transit flow problem with a complexity equivalent to the solution of a hydraulic or electric circuit.

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References

  1. Ortúzar, J. de D., Willumsen, L.G.: Modelling Transport. Wiley, New Jersey (2011)

    Google Scholar 

  2. Krylatov, A., Zakharov, V., Tuovinen, T.: Optimization Models and Methods for Equilibrium Traffic Assignment. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-34102-2

  3. Barthélémy, M.: The Structure and Dynamics of Cities: Urban Data Analysis and Theoretical Modeling. Cambridge (2016)

    Google Scholar 

  4. Louail, T., et al.: Uncovering the spatial structure of mobility networks. Nat. Commun. 6, 6007 (2015)

    Article  Google Scholar 

  5. Cajueiro, D.O.: Optimal navigation in complex networks. Phys. Rev. E 79, 046103 (2009)

    Article  Google Scholar 

  6. Gallotti, R., Porter, M.A., Barthélémy, M.: Lost in transportation: information measures and cognitive limits in multilayer navigation. Sci. Adv. 2(2), e1500445 (2016)

    Google Scholar 

  7. Ligthill, M.J., Whitham F.R.S.: On kinetic waves II. A theory of traffic flow on crowded roads. Proc. Royal Soc. Ser A. 229(1178), 317–345 (1995)

    Google Scholar 

  8. Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Ins. Civil Eng. 2, 325–378 (1952)

    Google Scholar 

  9. Beckmann, M.J., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)

    Google Scholar 

  10. Spiess, H., Florian, M.: Optimal strategies: a new assignment model for transit networks. Transp. Res. B 23B(2), 83–102 (1989)

    Article  Google Scholar 

  11. De Cea, J., Fernández, J.E.: Transit assignment to minimal routes: an efficient new algorithm. Traffic Eng. Control 30, 491–494 (1989)

    Google Scholar 

  12. Legara, E.F., Monterola, C., Lee, K.K., Hung, G.G.: Critical capacity, travel time delays and travel time distribution of rapid mass transit systems. Phys. A 406, 100–106 (2014)

    Article  Google Scholar 

  13. Liu, Y., Bunker, J., Ferreira, L.: Transit: users’ route-choice modelling in transit assignment: a review. Transp. Rev. 30(6), 753–769 (2010)

    Article  Google Scholar 

  14. Janson, B.N., Zozaya-Gorostiza, C.: The problem of cyclic flows in traffic assignment. Transp. Res. B 21B(4), 299–310 (1987)

    Article  Google Scholar 

  15. Raveau, S., Muñoz, J.C., de Grange, L.: A topological route choice model for metro. Transp. Res. A 45(2), 138–147 (2011)

    Google Scholar 

  16. Jiang, X., Lim, L.-H., Yao, Y., Ye, Y.: Statistical ranking and combinatorial Hodge theory. Math. Program. 127(1), 203–244 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Persistent Cohomology and Circular Coordinates. Disc. Comput. Geom. 45, 737–759 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Haruna, T., Fujiki, Y.: Hodge decomposition of information flow on small-world networks. Front. Neural Circuit. 10, 77 (2016)

    Article  Google Scholar 

  19. Kichikawa, Y., Iyetomi, H., Iino, T., Inoue, H.: Community structure based on circular flow in a large-scale transaction network. Appl. Netw. Sci. 4(1), 1–23 (2019). https://doi.org/10.1007/s41109-019-0202-8

    Article  Google Scholar 

  20. Grady, L.J., Polimeni, J.R.: Discrete Calculus: Applied Analysis on Graphs for Computational Science. Springer, Cham (2010)

    Book  MATH  Google Scholar 

  21. Lim, L.-H.: Hodge Laplacians on Graphs. SIAM Rev. 62(3), 685–715 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Johnson, J.L., Goldring, T.: Discrete Hodge theory on graphs: a tutorial. Comput. Sci. Eng. 15(5), 42–55 (2013)

    Article  Google Scholar 

  23. Brandes, U.T.: On variants of shortest-path betweenness centrality and their generic computation. Soc. Networks. 30(2), 136–145 (2008)

    Article  Google Scholar 

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Authors and Affiliations

  1. George Mason University, Fairfax, VA, 22030, USA

    Unchitta Kan & Eduardo López

Authors
  1. Unchitta Kan
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  2. Eduardo López
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Correspondence to Unchitta Kan .

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Editors and Affiliations

  1. Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas Universidad Politécnica de Madrid, Universidad Politécnica de Madrid, Madrid, Madrid, Spain

    Rosa Maria Benito

  2. IUT Lumière, University of Lyon, Bron Cedex, France

    Chantal Cherifi

  3. LE2I, UFR Sciences et Techniques, University of Burgundy, Dijon, France

    Hocine Cherifi

  4. Grupo Interdisciplinar de Sistemas Compl, Universidad Carlos III de Madrid, Leganés, Madrid, Spain

    Esteban Moro

  5. Center for Complex Networks and Systems Research School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, USA

    Luis M. Rocha

  6. Department of Chemical Engineering, Universitat Rovira i Virgili, Tarragona, Tarragona, Spain

    Marta Sales-Pardo

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Kan, U., López, E. (2022). Layered Hodge Decomposition for Urban Transit Networks. In: Benito, R.M., Cherifi, C., Cherifi, H., Moro, E., Rocha, L.M., Sales-Pardo, M. (eds) Complex Networks & Their Applications X. COMPLEX NETWORKS 2021. Studies in Computational Intelligence, vol 1073. Springer, Cham. https://doi.org/10.1007/978-3-030-93413-2_66

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  • DOI: https://doi.org/10.1007/978-3-030-93413-2_66

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-93412-5

  • Online ISBN: 978-3-030-93413-2

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