Node Accessibility Characterization of Radially-Grown Structures

TL;DR

This work addresses how radial, axis-aligned growth shapes node accessibility in evolving networks. It adopts a lattice-based radial growth model with orientation preferences defined by $p_p$ and $p_n=1-p_p$, and quantifies accessibility via the exponential entropy $α = e^{ε}$ with $ε = - \sum p_i \log p_i$ across hierarchical distance $h$. Results show that growth biased toward the parallel orientation often increases overall node accessibility and interior connectivity, while strong normal growth enlarges borders and can lower interior accessibility; accessibility rises with growth stage $e$ and hierarchy $h$ and tends toward saturation. The study provides quantitative links between growth directionality and network reachability, offering insights for designing or analyzing axis-aligned deposition and urban expansion scenarios where orientation governs connectivity and resource access.

Abstract

Complex systems have motivated continuing interest from the scientific community, leading to new concepts and methods. Growing systems represent a case of particular interest, as their topological, geometrical, and also dynamical properties change along time, as new elements are incorporated into the existing structure. In the present work, an approach is the case in which systems grown radially around some straight axis of reference, such as particle deposition on electrodes, or urban expansion along avenues, roads, coastline, or rivers, among several other possibilities. More specifically, we aim at characterizing the topological properties of simulated growing structures, which are represented as graphs, in terms of a measurement corresponding to the accessibility of each involved node. The incorporation of new elements (nodes and links) is performed preferentially to the angular orientation respectively to the reference axis. Several interesting results are reported, including the tendency of structures grown preferentially to the orientation normal to the axis to have smaller accessibility.

Figures (8)

• Figure 1: Illustration of the node accessibility in quantifying how effectively the reference node (in cyan) can access the neighbors at the $h-$th neighborhood. Each dashed link represents a path from the reference node up to its $h-$th neighbors. The numbers indicate the respective transition probabilities. The maximum accessibility is equal to the number of neighbors at the considered hierarchy, which takes place when all transition probabilities are identical, as illustrated in (c).
• Figure 2: Examples of synthetic regions obtained along growth stages $e$ = 500, 1000, 1500, 2000, and $2500$ for $p_=0.1, 0.5$ and $0.9$ and $L=60$.
• Figure 3: Diagram illustrating the region where the structure is allowed to grow, which has dimension $L \times H$. In order to avoid border effects, the value $H$ is selected to be large enough, and the two lateral regions with extent $h_{max}$ (the maximum considered neighborhood hierarchy) are excluded from the analysis. Only the node accessibility values of nodes comprised in the region delimited by the blue rectangle are considered for the generation of the node accessibility density.
• Figure 4: Examples of regions obtained for $p_n=0.1, 0.5, 0.9$ for $e=1500$ and $h=3$, with the node accessibility values shown in terms of a respective heatmap (a--c). The respectively obtained border nodes (respectively shown in d--f) correspond to the nodes with accessibility smaller or equal to $6$.
• Figure 5: The distribution (average $\pm$ standard deviation) of the node accessibility respective to synthetic regions obtained at $e=500, 1000, 1500,$ and $2500$ for $p_n=0.1, 0.5, 0.9$ and $h=3$. The average node accessibility tends to increase with $e$.