Abstract
In this paper, we derive a new bound for commute times estimation. This bound does not rely on the spectral gap but on graph densification (or graph rewiring). Firstly, we motivate the bound by showing that implicitly constraining the spectral gap through graph densification cannot fully explain some estimations in real datasets. Then, we set our working hypothesis: if densification can deal with a small/moderate degradation of the spectral gap, this is due to the fact that inter-cluster commute distances are considerably shrunk. This suggests a more detailed bound which explicitly accounts for the shrinking effect of densification. Finally, we formally develop this bound, thus uncovering the deep implications of graph densification in commute times estimation.
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Conversely, if this is not the case, we are forced to fuse more nodes, thus reducing the number of slices from S to, say, \(S'\) with more edges each. This leads to contracting the bound for \(R^{H}(s,t)\) even more.
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Acknowledgements
M. Curado, F. Escolano and M.A. Lozano are funded by the project TIN2015-69077-P of the Spanish Government.
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Curado, M., Escolano, F., Lozano, M.A., Hancock, E.R. (2018). Dirichlet Densifiers: Beyond Constraining the Spectral Gap. In: Bai, X., Hancock, E., Ho, T., Wilson, R., Biggio, B., Robles-Kelly, A. (eds) Structural, Syntactic, and Statistical Pattern Recognition. S+SSPR 2018. Lecture Notes in Computer Science(), vol 11004. Springer, Cham. https://doi.org/10.1007/978-3-319-97785-0_49
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