Graph Characteristics from the Schrödinger Operator

  • Pablo Suau
  • Edwin R. Hancock
  • Francisco Escolano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7877)

Abstract

In this paper, we show how the Schrödinger operator may be applied to the problem of graph characterization. The motivation is the similarity of the Schrödinger equation to the heat difussion equation, and the fact that the heat kernel has been used in the past for graph characterization. Our hypothesis is that due to the quantum nature of the Schrödinger operator, it may be capable of providing richer sources of information than the heat kernel. Specifically the possibility of complex amplitudes with both negative and positive components, allows quantum interferences which strongly reflect symmetry patterns in graph structure. We propose a graph characterization based on the Fourier analysis of the quantum equivalent of the heat flow trace. Our experiments demonstrate that this new method can be succesfully applied to characterize different types of graph structures.

Keywords

graph characterization heat flow Schrödinger equation quantum walks 

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References

  1. 1.
    Aziz, F., Wilson, R.C., Hancock, E.R.: Graph Characterization via Backtrackless Paths. In: Pelillo, M., Hancock, E.R. (eds.) SIMBAD 2011. LNCS, vol. 7005, pp. 149–162. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Peng, R., Wilson, R., Hancock, E.: Graph Characterization vi Ihara Coefficients. IEEE Transactions on Neural Networks 22(2), 233–245 (2011)CrossRefGoogle Scholar
  3. 3.
    Das, K.C.: Extremal Graph Characterization from the Bounds of the Spectral Radius of Weighted Graphs. Applied Mathematics and Computation 217(18), 7420–7426 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Escolano, F., Hancock, E., Lozano, M.A.: Heat Diffusion: Thermodynamic Depth Complexity of Networks. Physical Review E 85(3), 036206(15) (2012)Google Scholar
  5. 5.
    Xiao, B., Hancock, E., Wilson, R.: Graph Characteristics from the Heat Kernel Trace. Pattern Reognition 42(11), 2589–2606 (2009)MATHCrossRefGoogle Scholar
  6. 6.
    Aubry, M., Schlickewei, U., Cremers, D.: The Wave Kernel Signature: A Quantum Mechanical Approach To Shape Analysis. In: IEEE International Conference on Computer Vision (ICCV), Workshop on Dynamic Shape Capture and Analysis (4DMOD) (2011)Google Scholar
  7. 7.
    Rossi, L., Torsello, A., Hancock, E.R.: Approximate Axial Symmetries from Continuous Time Quantum Walks. In: Gimel’farb, G., Hancock, E., Imiya, A., Kuijper, A., Kudo, M., Omachi, S., Windeatt, T., Yamada, K. (eds.) SSPR & SPR 2012. LNCS, vol. 7626, pp. 144–152. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Emms, D., Wilson, R.C., Hancock, E.R.: Graph Embedding Using Quantum Commute Times. In: Escolano, F., Vento, M. (eds.) GbRPR. LNCS, vol. 4538, pp. 371–382. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Farhi, E., Gutmann, S.: Quantum Computation and Decision Trees. Physical Review A 58, 915–928 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Erdös, P., Rényi, A.: On Random Graphs. I. Publicationes Mathematicae 6, 290–297 (1959)MATHGoogle Scholar
  11. 11.
    Barabási, A.L., Albert, R.: Emergence of Scaling in Random Networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Watts, D.J., Strogatz, S.H.: Collective Dynamics of ’Small-World’ Networks. Nature 393(6684), 440–442 (1998)CrossRefGoogle Scholar
  13. 13.
    Han, L., Escolano, F., Hancock, E., Wilson, R.: Graph Characterizations From Von Neumann Entropy. Pattern Recognition Letters 33(15), 1958–1967 (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Pablo Suau
    • 1
  • Edwin R. Hancock
    • 2
  • Francisco Escolano
    • 1
  1. 1.Mobile Vision Research LabUniversity of AlicanteSpain
  2. 2.Department of Computer ScienceUniversity of YorkUK

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