Deterministic simplicial complexes (2507.07402v1)

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 ($d\leq n-1$) 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.

• The paper presents a novel deterministic model that recursively generates simplicial complexes, providing clear insights into their topology and spectral properties.
• It details both the unconstrained model with factorial growth and the constrained DSC(m) models that yield controlled exponential growth and deterministic trees.
• The research highlights unique spectral findings such as infinite spectral dimensions and Gaussian density tails, offering robust alternatives to stochastic network models.

Deterministic Simplicial Complexes

Introduction

The paper "Deterministic simplicial complexes" introduces a novel class of simplicial complexes that grow deterministically from a single vertex. This deterministic process offers a powerful framework for analyzing the topology and spectral properties of resulting complexes, presenting an alternative to stochastic models. The research delineates the development of these complexes via a recursive addition mechanism, focusing on both unconstrained and constrained growth scenarios.

Unconstrained Growth Model

Constructive Process

In the unconstrained DSC model, the growth process begins with an initial vertex. At each step, new vertices are added, interacting with existing simplices to form new higher dimensional simplices. Specifically, each $d$-dimensional simplex is converted into a $(d+1)$-dimensional simplex by joining with a new vertex—a process that efficiently generates complex structures.

Figure 1: The first four simplicial complexes generated by the DSC model, shown from top to bottom: $\mathcal{K}(0)$, $\mathcal{K}(1)$, $\mathcal{K}(2)$, $\mathcal{K}(3)$.

Combinatorial and Asymptotic Properties

The complexity of DSCs is captured in the growth of simplicial counts $N_d(n)$, which portray factorial growth rates, specifically $N_d(n) \sim n!$. The power-law distribution of upper-degree, or $d$-degree, indicates the diverse connectivity within the complexes, reflecting different evolution patterns for varying $d$.

Spectral Analysis

The spectral properties, including the Hodge Laplacians, reveal that these complexes have infinite spectral and Hausdorff dimensions, a stark contrast to most random networks. This indicates that DSCs maintain complex connectivity patterns across scales.

Figure 2: The log--log plots of the cumulative upper-degree distributions $p^{(d)}_{\text{cum}(k^{(d)},n)$ for simplicial complexes generated by the DSC model at various values of $n$ and $d$.

Constrained Growth Models

DSC$(m)$ Models

In the constrained DSC$(m)$ model, the dimensionality of the simplices is limited. The results show that within these constraints, the exponential growth of $N_d(n)$ is possible, and specific models such as DSC$(1)$ generate deterministic trees.

Figure 3: Estimated values of the exponent $\gamma^{(d)}$ using the expression, shown for generations $n=70$, $200$, $10000$, and in the limit $n \to \infty$.

Structural Characteristics

These models present a fascinating interplay between deterministic control and topological complexity. Notably, the DSC$(1)$ trees exhibit properties analogous to recursive random trees but are deterministic and thus more analyzable.

Spectral Findings

The adjacency matrix spectra of DSC$(1)$ demonstrate a Gaussian tail for the cumulative spectral density, with the largest eigenvalues scaling as $\sqrt{n}$. The Laplacian spectrum illustrates that these trees exhibit a spectral dimension of $2$, implying a unique scaling relation between eigenvalues and network size.

Conclusion

Overall, the deterministic simplicial complexes investigated in this paper offer a compelling alternative to stochastic models, emphasizing the robustness of deterministic rules in generating complex topological structures. Future research could focus on exploring these properties further and assessing their implications in broader complex system contexts. Their potential applications span diverse areas, from network theory to higher-dimensional data representations, suggesting a fertile area for continued paper and discovery.