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Random walks across dimensions: Exploring simplicial complexes

Diego Febbe1,2,*, Duccio Fanelli1, and Timoteo Carletti2

  • *Contact author: diego.febbe@unifi.it
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Phys. Rev. E 114, 044307 – Published 5 October, 2026

DOI: https://doi.org/10.1103/ppm1-mxhz

Abstract

We introduce an operator to describe a random walk process on a simplicial complex. Walkers are allowed to wander across simplices of various dimensions, bridging nodes to edges, and edges to triangles, via a nested organization that hierarchically extends to higher structures of arbitrarily large, but finite, dimension. The asymptotic distribution of the walkers provides a natural ranking to gauge the relative importance of higher-order simplices. Optimal search strategies in the presence of stochastic teleportation are addressed, and the peculiar interplay of noise with higher-order structures is unraveled.

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Phys. Rev. E 114, 044307– Published 5 October, 2026
  • Received 22 January 2026
  • Revised 9 July 2026
  • Accepted 9 September 2026
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