Abstract
This paper presents a novel approach to quantifying the information flow on a graph. The proposed approach is based on the solution of a wave equation, which is defined using the edge-based Laplacian of the graph. The initial condition of the wave equation is a Gaussian wave packet on a single edge of the graph. To measure the information flow on the graph, we use the average return time of the Gaussian wave packet, referred to as the wave packet commute time. The advantage of using the edge-based Laplacian of a graph over its vertex-based counterpart is that it translates results from traditional analysis to graph theoretic domain in a more natural way. Therefore it can be useful in applications where distance and speed of propagation are important.
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Grenfell, B.T.: Travelling waves and spatial hierarchies in measles epidemics. Nature, 716-723 (2001)
Abramson, G., Kenkre, V.M., Yates, T.L., Parmenter, R.R.: Traveling Waves of Infection in the Hantavirus Epidemics. Bulletin of Mathematical Biology, 519–534 (2003)
Passerini, F., Severini, S.: The von neumann entropy of networks. International Journal of Agent Technologies and Systems, 58–67 (2009)
Han, L., Escolano, F., Hancock, E.R., Wilson, R.C.: Graph characterizations from von Neumann entropy. Pattern Recognition Letters, 1958–1967 (2102)
Escolano, F., Hancock, E.R., Lozano, M.A.: Heat diffusion: Thermodynamic depth complexity of networks. Physics Review E, 036206 (2012)
Escolano, F., Bonev, B., Hancock, E.R.: Heat Flow-Thermodynamic Depth Complexity in Directed Networks. 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. 190–198. Springer, Heidelberg (2012)
Suau, P., Hancock, E.R., Escolano, F.: Analysis of the Schrödinger Operator in the Context of Graph Characterization. In: Hancock, E., Pelillo, M. (eds.) SIMBAD 2013. LNCS, vol. 7953, pp. 190–203. Springer, Heidelberg (2013)
Aziz, F., Wilson, R.C., Hancock, E.R.: Gaussian Wave Packet on a Graph. In: Kropatsch, W.G., Artner, N.M., Haxhimusa, Y., Jiang, X. (eds.) GbRPR 2013. LNCS, vol. 7877, pp. 224–233. Springer, Heidelberg (2013)
Lee, A.B., Luca, D., Klei, L., Devlin, B., Roeder, K.: Discovering genetic ancestry using spectral graph theory. Genetic Epidemiology, 51-59 (2010)
Bradonjic, M., Molloy, M., Yan, G.: Containing Viral Spread on Sparse Random Graphs: Bounds, Algorithms, and Experiments. Internet Mathematics, 406–433 (2013)
Friedman, J., Tillich, J.P.: Wave equations for graphs and the edge based Laplacian. Pacific Journal of Mathematics, 229–266 (2004)
Wilson, R.C., Aziz, F., Hancock, E.R.: Eigenfunctions of the edge-based Laplacian on a graph. Journal of Linear Algebra and its Applications, 4183–4189 (2013)
Erdõs, P., Rényi, A.: On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 17–61 (1960)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature, 440–442 (1998)
Barabási, A., Albert, R.: Emergence of Scaling in Random Networks. Science 509–512 (1999)
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Aziz, F., Wilson, R.C., Hancock, E.R. (2014). Commute Time for a Gaussian Wave Packet on a Graph. In: Fränti, P., Brown, G., Loog, M., Escolano, F., Pelillo, M. (eds) Structural, Syntactic, and Statistical Pattern Recognition. S+SSPR 2014. Lecture Notes in Computer Science, vol 8621. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44415-3_38
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DOI: https://doi.org/10.1007/978-3-662-44415-3_38
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