# Spectral properties of hyperbolic nano-networks with tunable aggregation of simplexes

## TL;DR

This paper investigates how the spectral and topological properties of hyperbolic nano-networks formed by simplexes depend on assembly parameters, revealing tunable spectral dimensions and hierarchical structures relevant for complex dynamics.

## Contribution

It introduces a model linking simplexes aggregation with hyperbolic geometry and spectral features, showing how higher-order connectivity controls spectral dimension and hierarchical architecture.

## Key findings

- Spectral dimension varies with clique size and affinity, exceeding 4 for certain parameters.
- Hyperbolicity remains constant at δ_max=1 across assemblies.
- Spectral distribution exhibits characteristic peaks and minima indicating hierarchical structure.

## Abstract

Cooperative self-assembly can result in complex nano-networks with new hyperbolic geometry. However, the relation between the hyperbolicity and spectral and dynamical features of these structures remains unclear. Using the model of aggregation of simplexes introduced in I [Sci. Rep., 8:1987, 2018], here we study topological and spectral properties of a large class of self-assembled structures consisting of monodisperse building blocks (cliques of size $n=3,4,5,6$) which self-assemble via sharing the geometrical shapes of a lower order. The size of the shared sub-structure is tunned by varying the chemical affinity $\nu$ such that for significant positive $\nu$ sharing the largest face is the most probable, while for $\nu < 0$, attaching via a single node dominates. Our results reveal that, while the parameter of hyperbolicity remains $\delta_{max}=1$ across the assemblies, their structure and spectral dimension $d_s$ vary with the size of cliques $n$ and the affinity when $\nu \geq 0$. In this range, we findthat $d_s >4$ can be reached for $n\geq 5$ and sufficiently large $\nu$. For the aggregates of triangles and tetrahedra, the spectral dimension remains in the range $d_s\in [2,4)$, as well as for the higher cliques at vanishing affinity. On the other end, for $\nu < 0$, we find $d_s\eqsim 1.57$ independently on $n$. Moreover, the spectral distribution of the normalised Laplacian eigenvalues has a characteristic shape with peaks and a pronounced minimum, representing the hierarchical architecture of the simplicial complexes. These findings show how the structures compatible with complex dynamical properties can be assembled by controlling the higher-order connectivity among the building blocks.