Spectral properties of hyperbolic nanonetworks with tunable aggregation of simplexes

Cooperative self-assembly is a ubiquitous phenomenon found in natural systems which is used for designing nanostructured materials with new functional features. Its origin and mechanisms, leading to improved functionality of the assembly, have attracted much attention from researchers in many branches of science and engineering. These complex structures often come with hyperbolic geometry; however, the relation between the hyperbolicity and their spectral and dynamical properties remains unclear. Using the model of aggregation of simplexes introduced by Šuvakov et al. [Sci. Rep. 8, 1987 (2018)], here we study topological and spectral properties of a large class of self-assembled structures or nanonetworks 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 substructure is tuned 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 $\nu \ge 0$. In this range, we find that $d_s>4$ can be reached for $n\ge 5$ 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 normalized 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.

Figure 1

Aggregates of tetrahedra with strong repulsion, a segment is shown in the top panel, and the case with strong attraction resulting in the network with communities indicated by different colors is shown in the bottom panel.

Figure 2

The networks of the aggregated cliques of mixed sizes $n\in [3,6]$ distributed according to $\propto n^{-2}$ for $\nu =5$ (top), and aggregates of triangles for $\nu =9$ (bottom). The community structure is indicated by different colors of nodes.

Figure 3

(a) Ranking distribution of nodes $i=1,2,...,5000$ according to the decreasing degree. The degree $k_i$ of the vertex $i$ is plotted against its rank $r_i$ for different assemblies of cliques of size $n$, indicated in the legend, and $\nu =9$. Stretched exponential curve approximates the data for the random tree ($n=2$), while the asymptotic power-law decay with the exponent $\gamma$ is appropriate for $n\ge 3$. (b) The number of simplexes and faces $f_q$ at different topology level $q$ is plotted against $q+1$ for the same monodisperse assemblies of the size $n$ as in the top panel (the same legend applies). The additional dotted and dashed lines with diamonds are for the mixed sizes $n\in [3,6]$ with the distribution $\sim n^{-\alpha }$ and two values of $\alpha$ given in the legend of panel (b).

Figure 4

For the assemblies of simplexes of the size $n$ given in the legend, and three indicated values of the chemical affinity parameter $\nu$. Top panels show the shortest-path distance distributions $P\left(d\right)$ vs the distance $d$. The corresponding bottom panels display the hyperbolicity parameter ${\delta }_{max}$ (upper curves, full lines) and $\langle \delta \rangle$ (lower curves, dashed lines) against $d_{min}$. Thin dotted line indicates the level ${\delta }_{max}=1$.

Figure 5

The lines with different symbols represent the spectral dimension $d_s$ plotted against chemical affinity $\nu$ for the aggregates of monodisperse cliques of sizes $n=3,4,5$ and a mixture of cliques of different sizes in the range $n\in [3,6]$. The bottom line corresponds to the random tree case $n=2$.

Figure 6

Several examples of the cumulative spectral density $P_c\left(\lambda \right)$ in the range of small $\lambda$ for the Laplacian operator (4) for the aggregates of triangles (top), tetrahedra (middle), and 5-cliques (bottom). In each panel, three lines in the top-to-bottom order correspond to different values of the chemical affinity $\nu =-5$, 0, and $+5$, respectively. The corresponding symbol and color for the online version are indicated in the legend.

Figure 7

Spectral distribution (left column) and the corresponding scatter plots of the eigenvectors $v1,v2,v3$ of the three lowest nonzero eigenvalues of the normalized Laplacian (right column) for the aggregates of tetrahedra $n=4$ for three different values of the affinity parameter $\nu =-9$ (a, b), $\nu =0$ (c), (d), and $\nu =+9$ (e), (f). The bottom panels (g) and (h) are for the random tree structure $n=2$, which is independent of $\nu$. For comparison, we also show the spectra for the cliques of different sizes $n=3$, 5, and 6, and the mixture $n\in [3,6]$ in panel (e); the corresponding lines are explained in the legend. The orientation of each 3D plot in panels (b), (d), (f), and (h) is such that it best depicts the number and size of different branches of nonzero components of the corresponding eigenvectors.