Discrete harmonic morphisms ensure exact random-walk projection under network coarse-graining, and Laplacian renormalization often produces exact instances of them on real networks.
Smirnov, Discrete complex analysis and probability (2010), arXiv:1009.6077 [math-ph]
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Harmonic morphisms and dynamical invariants in network renormalization
Discrete harmonic morphisms ensure exact random-walk projection under network coarse-graining, and Laplacian renormalization often produces exact instances of them on real networks.