MVTree for Hierarchical Network Representation Based on Geometric Algebra Subspace

  • Shuai Zhu
  • Shuai Yuan
  • Dongshuang Li
  • Wen Luo
  • Linwang Yuan
  • Zhaoyuan Yu
Part of the following topical collections:
  1. T.C. : Geometric Algebra for Computing, Graphics and Engineering with Yu Zhaoyuan, Guest E-i-C


Most hierarchical representation methods are designed from engineering perspectives, lacking an appropriate mathematical foundation to integrate different problem definitions. To solve this problem, a hierarchical network representation model based on geometric algebra (GA) subspace is proposed. In this paper, we give a new definition of hierarchical network representation, in which the network nodes are divided into several independent components and multi-level network is constructed based on it. Then these network nodes are coded with basis vectors and the subspace in GA is utilized to represent these nodes of component. Within a subspace or between different subspaces, the topological relation between different basis vectors is defined to connect different subspaces and the complete multi-level network can be formed according to these topological relations in subspaces of different level. A case study is used to show how GA is used to realize the hierarchical network representation and the result indicates that GA can provide an appropriate mathematical foundation for hierarchical network representation.


Hierarchical network Geometric algebra subspace Topological relation Upper network 

Mathematics Subject Classification

Primary 99Z99 Secondary 00A00 


  1. 1.
    Breuils, S., Nozick, V., Fuchs, L.: a geometric algebra implementation using binary tree. Adv. Appl. Clifford Algebras 1–19 (2017)Google Scholar
  2. 2.
    Cruz-Sánchez, H., Staples, G.S., Schott, R., et al.: Operator calculus approach to minimal paths: Precomputed routing in a store and forward satellite constellation. In: Global Communications Conference (GLOBECOM), pp. 3431–3436. IEEE (2013).
  3. 3.
    Geisberger, R., Sanders, P., Schultes, D., Delling, D.: Contraction hierarchies: faster and simpler hierarchical routing in road networks. In: McGeoch, C.C. (ed.) Experimental Algorithms. WEA 2008. Lecture Notes in Computer Science, vol 5038, pp. 319–333. Springer, Berlin, Heidelberg (2008)Google Scholar
  4. 4.
    Geisberger, R., Sanders, P., Schultes, D., et al.: Exact routing in large road networks using contraction hierarchies. Transp. Sci. 46(3), 388–404 (2012)CrossRefGoogle Scholar
  5. 5.
    Holzer, M., Schulz, F., Wagner, D.: Engineering multilevel overlay graphs for shortest-path queries. J. Exp. Algorithmics 13, 156–170 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Huang, P., Pi, Y.: An improved location service scheme in urban environments with the combination of GPS and mobile stations. Wirel. Commun. Mobile Comput. 14(13), 1287–1301 (2015)CrossRefGoogle Scholar
  7. 7.
    Jagadeesh, G.R., Srikanthan, T., Quek, K.H.: Heuristic techniques for accelerating hierarchical routing on road networks. IEEE Trans. Intell. Transp. Syst. 3(4), 301–309 (2002)CrossRefGoogle Scholar
  8. 8.
    Luo, W., Hu, Y., Yu, Z., et al.: A hierarchical representation and computation scheme of arbitrary-dimensional geometrical primitives based on CGA. Adv. Appl. Clifford Algebr 27(3), 1977–1995 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Papachristodoulou, A., Li, L., Doyle, J.C.: Methodological frameworks for large-scale network analysis and design. ACM SIGCOMM Comp. Comm. Rev. 34(3), 7–20 (2004)CrossRefGoogle Scholar
  10. 10.
    Quaglia, A., Sarup, B., Sin, G., et al.: Design of a generic and flexible data structure for efficient formulation of large scale network problems. Comp. Aided Chem. Eng. 32(3), 661–666 (2013)CrossRefGoogle Scholar
  11. 11.
    Sanders, P., Schultes, D.: Highway hierarchies hasten exact shortest path queries. In: Brodal, G.S., Leonardi, S. (eds.) Algorithms–ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol 3669, pp. 568–579. Springer, Berlin, Heidelberg (2005)Google Scholar
  12. 12.
    Schott, R., Staples, S.: Cycles and components in geometric graphs: adjacency operator approach. Probability (2009)Google Scholar
  13. 13.
    Schultes, D., Sanders, P.: Dynamic highway-node routing. In: Demetrescu, C. (ed.) Experimental Algorithms. WEA 2007. Lecture Notes in Computer Science, vol 4525, pp. 66–79. Springer, Berlin, Heidelberg (2007)Google Scholar
  14. 14.
    Schulz, F., Wagner, D., Zaroliagis, C.: Using Multi-level Graphs for Timetable Information in Railway Systems. In: Revised Papers From the, International Workshop on Algorithm Engineering and Experiments. pp. 43–59. Springer-Verlag, Heidelberg (2002)Google Scholar
  15. 15.
    Sims, B.H., Sinitsyn, N., Eidenbenz, S.J.: Visualization and modeling of structural features of a large organizational email network. In: Proceedings of the 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (Proceeding ASONAM ’13), pp. 787–791. IEEE, Niagara, Ontario, Canada (2013)Google Scholar
  16. 16.
    Song, Q., Wang, X.: Efficient Routing on Large Road Networks Using Hierarchical Communities. IEEE Trans. Intell. Transp. Syst. 12(1), 132–140 (2011)CrossRefGoogle Scholar
  17. 17.
    Thorup, M.: Integer priority queues with decrease key in constant time and the single source shortest paths problem. J. Comput. Syst. Sci. 69(3), 330–353 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Willhalm, T.: Engineering Shortest Path and Layout Algorithms for Large Graphs. PhD thesis, Universität Karlsruhe (TH), Fakultät für Informatik (2005)Google Scholar
  19. 19.
    Yu, Z., Luo, W., Yuan, L., et al.: Geometric algebra model for geometry-oriented topological relation computation. Trans. GIS 20(2), 259–279 (2016)CrossRefGoogle Scholar
  20. 20.
    Yu, Z., Wang, J., Luo, W., et al.: A dynamic evacuation simulation framework based on geometric algebra. Comput. Environ. Urban Syst. 59, 208–219 (2016)CrossRefGoogle Scholar
  21. 21.
    Yu, Z., Li, D., Zhu, S., et al.: Multisource multisink optimal evacuation routing with dynamic network changes: a geometric algebra approach. Math. Methods Appl. Sci. 2017(1) (2017).
  22. 22.
    Yuan, L., Zhao, Y., Luo, W., et al.: Geometric algebra for multidimension-unified geographical information system. Adv. Appl. Clifford Algebras 23(2), 497–518 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yuan, L., Yu, Z., Luo, W., et al.: Clifford algebra method for network expression, computation, and algorithm construction. Math. Methods Appl. Sci. 37(10), 1428–1435 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yuan, L., Yu, Z., Luo, W., et al.: Multidimensional-unified topological relations computation: a hierarchical geometric algebra-based approach. Int. J. Geogr. Inf. Sci. 28(12), 2435–2455 (2014)CrossRefGoogle Scholar
  25. 25.
    Zhiyuli, A., Liang, X., Zhou, X.: Learning structural features of nodes in large-scale networks for link prediction. In: Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (AAAI’16), pp. 4286–4287. AAAI Press, Phoenix, Arizona (2016)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Shuai Zhu
    • 1
  • Shuai Yuan
    • 1
  • Dongshuang Li
    • 1
  • Wen Luo
    • 1
  • Linwang Yuan
    • 1
  • Zhaoyuan Yu
    • 1
  1. 1.Geography Department of Nanjing Normal UniversityXianLin School of Nanjing Normal UniversityNanjing CityChina

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