Abstract
In this paper, we apply the solution of the Schrödinger equation, i.e. the Schrödinger operator, to the graph characterization problem. The motivation behind this approach is two-fold. Firstly, the mathematically similar heat kernel has been used in the past for this same problem. And secondly, due to the quantum nature of the Schrödinger equation, our hypothesis is that it may be capable of providing richer sources of information. The two main features of the Schrödinger operator that we exploit in this paper are its non-ergodicity and the presence of quantum interferences due to the existence of complex amplitudes with both positive and negative components. Our proposed graph characterization approach is based on the Fourier analysis of the quantum equivalent of the heat flow trace, thus relating frequency to structure. Our experiments, performed both on synthetic and real-world data, demonstrate that this new method can be successfully applied to the characterization of different types of graph structures.
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References
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)
Peng, R., Wilson, R.C., Hancock, E.R.: Graph Characterization vi Ihara Coefficients. IEEE Transactions on Neural Networks 22(2), 233–245 (2011)
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)
Escolano, F., Hancock, E.R., Lozano, M.A.: Heat Diffusion: Thermodynamic Depth Complexity of Networks. Physical Review E 85(3), 036206(15) (2012)
Xiao, B., Hancock, E.R., Wilson, R.C.: Graph Characteristics from the Heat Kernel Trace. Pattern Reognition 42(11), 2589–2606 (2009)
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)
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)
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)
Farhi, E., Gutmann, S.: Quantum Computation and Decision Trees. Physical Review A 58, 915–928 (1998)
Erdös, P., Rényi, A.: On Random Graphs. I. Publicationes Mathematicae 6, 290–297 (1959)
Barabási, A.L., Albert, R.: Emergence of Scaling in Random Networks. Science 286(5439), 509–512 (1999)
Watts, D.J., Strogatz, S.H.: Collective Dynamics of ’Small-World’ Networks. Nature 393(6684), 440–442 (1998)
Han, L., Escolano, F., Hancock, E., Wilson, R.: Grapph Characterizations From Von Neumann Entropy. Pattern Recognition Letters 33(15), 1958–1967 (2012)
Estrada, E.: Quantifying Network Heterogeneity. Physical Review E 82(6), 066102 (2010)
Estrada, E.: Network Robustness to Targeted Attacks. The Interplay of Expansibility and Degree Distribution. The European Physical Journal B - Condensed Matter and Complex Systems 52(4), 563–574 (2006)
Antiqueira, L., Rodrigues, F.A., Costa, L.F.: Modeling Connectivity in terms of Network Activity. J. Stat. Mech., L09005 (2009)
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Suau, P., Hancock, E.R., Escolano, F. (2013). Analysis of the Schrödinger Operator in the Context of Graph Characterization. In: Hancock, E., Pelillo, M. (eds) Similarity-Based Pattern Recognition. SIMBAD 2013. Lecture Notes in Computer Science, vol 7953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39140-8_13
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DOI: https://doi.org/10.1007/978-3-642-39140-8_13
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