Deterministic simplicial complexes

S. N. Dorogovt︠s︡ev

doi:10.1088/1742-5468/ae2092

Return

Deterministic simplicial complexes

2025-12-01

PRE

S. N. Dorogovt︠s︡ev

P. L. Krapivsky *

DOI：10.1088/1742-5468/ae2092

Original

Original request for help

Save

Abstract

En 中文

We investigate simplicial complexes deterministically growing from a single vertex. In the first step, a vertex and an edge connecting it to the primordial vertex are added. The resulting simplicial complex has a 1-dimensional simplex and two 0-dimensional faces (the vertices). The process continues recursively: On the n-th step, every existing d-dimensional simplex ( d <= n-1) joins a new vertex forming a (d+1)-dimensional simplex; all 2d+1-2 new faces are also added so that the resulting object remains a simplicial complex. The emerging simplicial complex has intriguing local and global characteristics. The number of simplices grows faster than n!, and the upper-degree distributions follow a power law. Here, the upper degree (or d-degree) of a d-simplex refers to the number of (d+1)-simplices that share it as a face. Interestingly, the d-degree distributions evolve quite differently for different values of d. We compute the Hodge Laplacian spectra of simplicial complexes and show that the spectral and Hausdorff dimensions are infinite. We also explore a constrained version where the dimension of the added simplices is fixed to a finite value m. In the constrained model, the number of simplices grows exponentially. In particular, for m = 1, the spectral dimension is 2. For m = 2, the spectral dimension is finite, and the degree distribution follows a power law, while the 1-degree distribution decays exponentially.

Keywords:

simplicial complexes

deterministic growth

Laplacian spectra

Hodge Laplacian