Discrete harmonic morphisms ensure exact random-walk projection under network coarse-graining, and Laplacian renormalization often produces exact instances of them on real networks.
Bianconi,Higher-Order Networks: An Introduction to Simplicial Complexes(Cambridge University Press, 2021)
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Topological Kuramoto model on cell complexes yields phase-locked states via winding numbers, with multistability requiring boundaries of at least five elements and cascades across dimensions.
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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.
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Phase locking and multistability in the topological Kuramoto model on cell complexes
Topological Kuramoto model on cell complexes yields phase-locked states via winding numbers, with multistability requiring boundaries of at least five elements and cascades across dimensions.