String Edit Distance, Random Walks and Graph Matching

  • Antonio Robles-Kelly
  • Edwin R. Hancock
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2396)


This paper shows how the eigenstructure of the adjacency matrix can be used for the purposes of robust graph-matching. We commence from the observation that the leading eigenvector of a transition probability matrix is the steady state of the associated Markov chain. When the transition matrix is the normalised adjacency matrix of a graph, then the leading eigenvector gives the sequence of nodes of the steady state random walk on the graph. We use this property to convert the nodes in a graph into a string where the node-order is given by the sequence of nodes visited in the random walk. We match graphs represented in this way, by finding the sequence of string edit operations which minimise edit distance.


Random Walk Edit Distance Transition Probability Matrix Graph Match Graph Edit Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    R. C. Wilson A. M. Finch and E. R. Hancock. An energy function and continuous edit process for graph matching. Neural Computation, 10(7):1873–1894, 1998.CrossRefGoogle Scholar
  2. 2.
    K. Siddiqi A. Shokoufandeh, S. J. Dickinson and S. W. Zucker. Indexing using a spectral encoding of topological structure. In Proceedings of the Computer Vision and Pattern Recognition, 1998.Google Scholar
  3. 3.
    Luo Bin and E. R. Hancock. Procrustes alignment with the em algorithm. In 8th International Conference on Computer Analysis of Images and Image Patterns, pages 623–631, 1999.Google Scholar
  4. 4.
    H. Buke. On a relation between graph edit distance and maximum common subgraph. Pattern Recognition Letters, 18, 1997.Google Scholar
  5. 5.
    Fan R. K. Chung. Spectral Graph Theory. American Mathematical Society, 1997.Google Scholar
  6. 6.
    M. A. Eshera and K. S. Fu. A graph distance measure for image analysis. SMC, 14(3):398–408, May 1984.Google Scholar
  7. 7.
    S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. PAMI, 18(4):377–388, April 1996.Google Scholar
  8. 8.
    R. Horaud and H. Sossa. Polyhedral object recognition by indexing. Pattern Recognition, 1995.Google Scholar
  9. 9.
    L. Lovász. Random walks on graphs: a survey. Bolyai Society Mathematical Studies, 2(2):1–46, 1993.Google Scholar
  10. 10.
    V. I. Levenshtein. Binary codes capable of correcting deletions, insertions and reversals. Sov. Phys. Dokl., 6:707–710, 1966.MathSciNetGoogle Scholar
  11. 11.
    Bin Luo and E. R. Hancock. Structural graph matching using the EM algorithm and singular value decomposition. To appear in IEEE Trans. on Pattern Analysis and Machine Intelligence, 2001.Google Scholar
  12. 12.
    B. J. Oommen and K. Zhang. The normalized string editing problem revisited. PAMI, 18(6):669–672, June 1996.Google Scholar
  13. 13.
    A. Sanfeliu and K. S. Fu. A distance measure between attributed relational graphs for pattern recognition. IEEE Transactions on Systems, Man and Cybernetics, 13:353–362, 1983.zbMATHGoogle Scholar
  14. 14.
    G. Scott and H. Longuet-Higgins. An algorithm for associating the features of two images. In Proceedings of the Royal Society of London, number 244 in B, 1991.Google Scholar
  15. 15.
    L. G. Shapiro and R. M. Haralick. Relational models for scene analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 4:595–602, 82.Google Scholar
  16. 16.
    L. S. Shapiro and J. M. Brady. A modal approach to feature-based correspondence. In British Machine Vision Conference, 1991.Google Scholar
  17. 17.
    S. Ullman. Filling in the gaps. Biological Cybernetics, 25:1–6, 76.Google Scholar
  18. 18.
    S. Umeyama. An eigen decomposition approach to weighted graph matching problems. PAMI, 10(5):695–703, September 1988.Google Scholar
  19. 19.
    R. S. Varga. Matrix Iterative Analysis. Springer, second edition, 2000.Google Scholar
  20. 20.
    R. A. Wagner. The string-to-string correction problem. Journal of the ACM, 21(1), 1974.Google Scholar
  21. 21.
    J. T. L. Wang, B. A. Shapiro, D. Shasha, K. Zhang, and K. M. Currey. An algorithm for finding the largest approximatelycommon substructures of two trees. PAMI, 20(8):889–895, August 1998.Google Scholar
  22. 22.
    R. C. Wilson and E. R. Hancock. Structural matching by discrete relaxation. PAMI, 19(6):634–648, June 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Antonio Robles-Kelly
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkUK

Personalised recommendations