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 ⩽ 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.