Deterministic simplicial complexes

Abstract

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 ($d\le n-1$) joins a new vertex forming a $(d+1)-$dimensional simplex; all $2^{d+1}-2$ 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.

arXiv comment20 pages, 13 figures

References in corpus (21)

• Networks beyond pairwise interactions: structure and dynamics
• The physics of higher-order interactions in complex systems
• Organization of Growing Random Networks
• Pseudofractal Scale-free Web
• Deterministic Scale-Free Networks
• Generic scale of the "scale-free" growing networks
• Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian
• A Geometric Fractal Growth Model for Scale Free Networks
• Emergent Hyperbolic Network Geometry
• Complex Quantum Network Geometries: Evolution and Phase Transitions
• Spectral dimensions of hierarchical scale-free networks with shortcuts
• Topological Percolation on Hyperbolic Simplicial Complexes
• Statistics of Lead Changes in Popularity-Driven Systems
• The spectral dimension of simplicial complexes: a renormalization group theory
• Complex network view of evolving manifolds
• The spectral dimension of random trees
• Degree-dependent intervertex separation in complex networks
• The higher-order spectrum of simplicial complexes: a renormalization group approach
• Hierarchical scale-free network is fragile against random failure
• Structural and spectral properties of a family of deterministic recursive trees: Rigorous solutions
• Combinatorial Properties for a Class of Simplicial Complexes Extended from Pseudo-fractal Scale-free Web