## 2 Materials and Methods
### Grid Generation Procedures  
In the present work, a hexagonal mesh grid (cells) was used to discretize the study area. Currently, rasters in a GIS-based environment often follow a rectangular-shaped grid. Hence, remotely sensed images should be rectified in hexagonal grids when the objective/aim is to return a better representation of the neighborhood relations (Figure 2) [35; 36]. Compared to the rectangular ones, hexagonal grid cells align along three axes, also having a more variable shape [37]. Due to the \"meet points\" set at their edges, hexagonal grid cells are also less ambiguous. Indeed, rectangular grid lines merge their continuity to the orthogonal neighbor cells, and, being sensitive to lines parallel to the x or y axes, human vision becomes distracted by them [38].  
Hence, this work fits into the current of Cellular Automata (CA) model approaches, performed through hexagonal-shaped grids, a type of geometry already used in various studies such as the current one [39; 40; 41].  
First, we created a hexagonal-shape grid in a GIS-based environment (QGIS software, version 3.16.3, Hannover, Germany) [42], with the extension of the layer represented by the UA. The cell size was set according to the peri-urban areas' perception capacity. The spatial and structural definitions of such areas are still an open discussion for spatial sciences; therefore, the area of the single hexagon was set to 2 square km, arbitrarily set based on doubling the mean urban area values detectable in the 2018 Corine Land Cover dataset [31].  
Grid creation in a GIS-based environment is set on the UA extension, so it is not optimized for the best discretization of urban and nonurban areas. To evaluate the best grid position for our analysis, we applied a 3-directional shift, resumable with the following abbreviations (Figure 3):  
Figure 2: Axis alignment for a hexagonal-shaped grid.  
S\({}_{0}\)--Null shift.  
S\({}_{1}\)--Right shift, equal to the radius of the circumscribed circumference of the single hexagonal cell.  
S\({}_{2}\)--Down shift, equal to the apothem of the hexagonal cell.  
S\({}_{3}\)--Diagonal shift, equal to the radius of the circumscribed circumference of a single hexagonal cell.  
The reason that it was chosen to apply to only these 3 shifts is the redundancy of the translation. In fact, additional movements made with the specified distances lead to one of the four initial grid conditions.  
The Urban Atlas was reorganized into two datasets for data processing. First, we considered the division between artificial areas (class 1, 17 urban sub-classes) and non-artificial areas (classes 2-5, 10 non-urban sub-classes). Then, we performed the same analyses with the five categories of the UA (Table 1).  
After the intersection between the UA and every grid (S\({}_{0}\), S\({}_{1}\), S\({}_{2}\), S\({}_{3}\)) (Figure 4), the areas occupied by individual categories in each hexagon were aggregated. It is important to notice that, for the dataset, only cells with a 2 square km area were used to avoid all the mosaic tiles that, due to the intersection between the different grids and the UA, could not be considered like the larger ones. This operation resulted in 606 tiles for the S\({}_{0}\) shift; 605 tiles for the S\({}_{1}\) shift; 598 tiles for the S\({}_{2}\) shift; and 599 tiles for the S\({}_{3}\) shift.  
Finally, the process from which the grids were obtained was standardized using the integrated graphical modeler option in QGIS. No specific plugins were used.