Fig 1.
Data sparsity can impact network modelling by disregarding important locations.
(A) Typical OD model of commodity movements where locations not included in the recorded movements are excluded entirely from the network. (B) Certain areas that serve as spatial junctions are omitted simply because they are neither origins nor destinations, despite their potential importance in facilitating flow. (C) Data gaps result in missing regions (i.e., blue cells), limiting our ability to assess their significance within the overall movement system. This toy example illustrates how OD models may fail to capture the full spatial complexity of commodity flows.
Fig 2.
Generating a vector field from origin-destination data.
Given locations where we have information (origins/destinations of commodity movements represented by directed arrows) shown in the map on the left, (A) we interpret each origin-destination pair as a vector and calculate the resultant vector of all movements from each particular location. Once this is done for all locations with available data, (B) we interpolate the missing information, illustrated in the rightmost figure as arrows shown in shades of orange. The final result is a complete vector field where all locations (cities, for instance) have a vector with direction and magnitude. In this example, arrows represent direction and shading represents magnitude. The map used in this figure is freely available (not copyrighted) [31].
Fig 3.
Robustness of vector field directions and magnitudes under spatial data removal.
Number of impacted areas as a function of randomly removed areas for 2014 (A, C) and 2015 (B, D). Panels A-B show directional robustness, where lines represent the number of areas with angular deviations below specified degree thresholds (1°,
5°,
15°,
45°,
90°,
180°). Panels C-D show magnitude robustness, where lines represent the number of areas with spatial deviations below specified distance thresholds (
5 km,
10 km,
25 km,
50 km,
100 km). Percentages indicate the proportion of areas impacted in relation to the areas removed. The results demonstrate the robustness of vector field estimations despite spatial data gaps, with high consistency between 2014 and 2015, highlighting the reliability of our approach across different years. The total number of areas is 853, and the maximum diameter of Minas Gerais is approximately 1250 km.
Fig 4.
Entropy of monthly vectors for (A) municipalities and (B) micro-regions.
Monthly vectors are classified into four directional clusters by dividing the Cartesian plane into four quadrants, each assigned a distinct value (1–4). Shannon entropy is then calculated from these 12 cluster values to yield a single entropy measure per municipality or micro-region, which is colour-coded on the map to indicate the diversity of commodity flow directions over the year. For visualisation, entropy values are normalised using Min–Max normalisation. The base maps used in this figure are freely available and not subject to copyright (Panel A: [34]; Panel B: [35]).
Fig 5.
Cosine-based clustering of (A, B) municipalities and (C, D) micro-regions.
Using k-medoids clustering (k = 4) (see Section S8 in S1 File) for further explanation), locations are grouped by their cosine similarity values. Clusters, indicated by distinct colours, reveal that in many peripheral locations the commodity flow directions remain stable over the years. Freely available maps are used in this figure (Panel A: [34]; Panel C: [35]).
Fig 6.
Distribution and maps of trading distances and spatial lag values.
(A) Municipalities are grouped into five predefined distance-based clusters: short (50 km), short–medium (50–100 km), medium (100–150 km), medium–long (150–200 km), and long (>200 km). For each municipality, the trade vector magnitude was computed as the mean of all monthly vectors across four years. The x and y components of the vectors were computed as the Euclidean differences between the latitude and longitude coordinates of the points and then converted into distance values (km). The distributions of vector magnitudes and distances for each cluster are shown. (B) Map of municipalities coloured by distance-behaviour cluster, with trade vector magnitudes and distances represented as in (A). (C) Spatial lag values of the vector magnitudes in (B), calculated using Queen-based spatial weights. These highlight municipalities whose values differ from their neighbours, revealing patterns such as ‘doughnuts’ (low values encircled by high ones) and ‘diamonds’ (high values encircled by low ones). Spatial lag values are grouped into five ranges based on the boxplot distribution of all 853 municipalities. The base maps in this figure are freely available (not copyrighted) [34].
Fig 7.
(A) Scatter plot of vector magnitudes against spatial lag with a fitted line. (B) To evaluate the significance of the observed pattern, we compared the Moran’s value calculated from the vector field to those from 1000 simulations in which vector magnitudes were randomly shuffled among municipalities. The observed Moran’s value of 0.258 was significantly higher than the simulated values, indicating a non-random spatial structure. (C) The heatmap shows Moran’s values for monthly vector fields across different years. Since no blue shades appear in the heatmap, all calculated Moran’s I values are positive, suggesting positive spatial autocorrelation, meaning that municipalities with high vector magnitudes tend to be located near each other.
Fig 8.
Sinks (A) and sources (B) in the cattle trade vector field.
Initial vector fields were generated from cattle trades in each season of 2016. A triangle-based interpolation method is used to generate a continuous vector field from these initial vectors. Critical points within the field are identified, distinguishing attracting points (sinks) from repelling points (sources). Circles indicate the locations of these points, with larger circles used for individual points to enhance visualisation, and darker shading representing higher numbers of points. The boundary maps shown in this figure are freely available (not copyrighted) [37].