The geometry of diffusion on graphs with magnetic field

Estrada, Ernesto
Submitted (2026)

Abstract. We study diffusion processes on mixed graphs generated by the Hermitian magnetic Laplacian and show that they induce a natural geometric representation of the network. The associated magnetic diffusion defines a map of the vertices into a complex Euclidean space, where distances reflect imbalances in the diffusion flow and encode both connectivity and directionality. We prove that this construction gives rise to a well-defined metric structure and characterize its geometric properties. In particular, we show that the induced geometry admits both Euclidean and hyperbolic interpretations through a suitable normalization, thereby linking diffusion dynamics with non-Euclidean geometry in a direct and constructive way. We further introduce angle-based quantities associated with the magnetic diffusion and relate them to the underlying complex structure of the embedding, as well as to projective geometry. The results provide a unified perspective in which geometry emerges intrinsically from diffusion on directed networks, offering a dynamical alternative to embedding methods based on prescribed ambient spaces.

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