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REVIEW 4 major objections 5 minor 37 references

Vector fields as a framework for modelling the mobility of commodities

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A vector-field model recovers cattle trade flow directions from sparse origin-destination data.

desk verdict A transparent application of an existing vector-field idea to cattle trade, with a robustness test that is weaker than claimed; the inference claim needs external or clustered-missing validation. read the letter →

arxiv 2506.02047 v1 pith:XKTANHF6 submitted 2025-05-31 physics.soc-ph

Sima Farokhnejad , Angélica S. da Mata , Mariana Macedo , Ronaldo Menezes This is my paper
classification physics.soc-ph
keywords vectorfieldsorigin-destinationdatacommoditymobilitycattletradespatialinterpolationautocorrelationcriticalpointsdiseasesurveillance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that commodity movements, normally stored as origin-destination (OD) records, can be converted into a vector field—an arrow at every location giving the dominant direction and strength of trade. The payoff is that locations absent from the records, which ordinary network models simply ignore, receive interpolated arrows based on nearby known flows. Using cattle trade in Minas Gerais, Brazil, the authors find the field is resilient to sparsity: more than 60% of spatial information must be removed before half of the vectors change, and most angular deviations stay below 15 degrees. If this holds, disease-surveillance and logistics models could estimate likely flow directions in unrepresented areas from incomplete records.

What carries the argument

The load-bearing object is the resultant vector per spatial cell: every outgoing origin-destination trade is drawn as a vector from the cell's centre to the destination's centre, and the vectors are averaged or summed into one arrow representing that cell's dominant flow. Missing arrows are then produced by triangle-based interpolation, which triangulates the known cell centres (with boundary points set to zero vectors to limit edge artefacts) and assigns each new point a barycentric blend of the vectors at the three vertices of the triangle it falls in, $v_p = \frac{h_1 v_1 + h_2 v_2 + h_3 v_3}{h_1 + h_2 + h_3}$. This machinery carries the whole argument because the robustness result—that most interpolated directions stay within 15 degrees until over 60% of sites are removed—is a property of this interpolation scheme on the Minas Gerais data. Supporting analyses use Shannon entropy on binned monthly directions, cosine similarity between consecutive months, Moran's I on vector magnitudes, and eigenvalue classification of critical points to label sinks and sources.

What would settle it

Hold out a contiguous block of municipalities from the Minas Gerais records, rebuild the field from the remaining data, and compare interpolated vectors with the actual resultant vectors at the held-out municipalities; if more than half of the comparisons deviate by more than 15 degrees, the smoothness assumption that carries the method is false for this dataset.

Watch

Extended reading notes

Core claim

The central claim is that an origin-destination network can be re-expressed as a continuous vector field without losing the essential spatial structure of commodity flows, and that this representation supports inference where the original data are silent. Each cell's outgoing trades are summed into one resultant vector; triangle-based interpolation then fills cells with no outgoing edges, producing a field over the whole region. The authors demonstrate on Minas Gerais cattle trade that these inferred fields preserve direction under heavy data removal, reveal regions of stable versus shifting direction via entropy and cosine similarity, cluster municipalities by trade distance using Moran's I, and locate sinks and sources that coincide with slaughterhouses and breeding-season supply hubs. They present the method as a complement to network models, aimed at applications such as foot-and-mouth disease surveillance in data-poor areas.

Load-bearing premise

The load-bearing premise is that neighbouring locations influence one another's trade, so flow directions change smoothly across space; if real trade is sharp-edged, with markets and slaughterhouses pulling traffic in isolated ways, interpolated vectors at unrepresented locations will be unreliable.

Editorial extensions

If this is right

  • Spatially incomplete OD datasets, common for livestock and other commodities, can still yield complete directional fields, so unrepresented municipalities get first-pass flow estimates rather than blanks.
  • Public-health applications could use the interpolated directions and the seasonal sinks and sources to target foot-and-mouth-disease surveillance at places that never appear as origins or destinations.
  • Because robustness is nearly invariant across years, the method can be applied to short windows such as monthly or seasonal fields, which is precisely the resolution needed to track disease-relevant movements.
  • The pipeline transfers directly to other OD-format movement data, including human mobility records, without changing the core method.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural stress test the paper does not run is block removal: deleting contiguous municipalities rather than random ones would probe whether the smoothness assumption holds across real market boundaries, and could overestimate robustness if field gradients are steep there.
  • If the method were applied to multi-commodity or multimodal data, critical points could be compared with infrastructure maps such as slaughterhouses, ports, and warehouses; coincidences would validate the field, while mismatches would reveal where interpolation smears local structure.
  • The angular-deviation statistic could be turned into a surveillance metric: a region whose interpolated direction disagrees strongly with newly collected OD records would flag anomalies such as diversions or unreported trade.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a vector-field framework for commodity mobility, in which origin-destination trade records are aggregated into per-cell resultant vectors and triangle-based interpolation (Eq. 1) fills in cells lacking data. The method is applied to cattle trade in Minas Gerais, Brazil, for 2013-2016 at municipality and micro-region scales. The authors assess robustness by randomly removing spatial cells and measuring angular deviations, then analyze temporal direction diversity via Shannon entropy, temporal regularity via cosine similarity, spatial autocorrelation via Moran's I, and topology via critical points (sinks and sources). The central claims are that the vector-field approach reveals fundamental patterns in commodity mobility and can infer movement information for unrepresented locations.

Significance. If the central claim were firmly established, the framework would provide a useful alternative to network-based models, enabling interpolation, visualization, and topological analysis of commodity flows from incomplete OD data. The paper has clear strengths: the data and code are publicly available, the interpolation pipeline is explicitly specified, the leave-some-out robustness check is a reasonable internal diagnostic, and the Moran's I significance test is correctly framed with simulations. However, the validation is internal and uses a favorable random-missingness mechanism, so the paper does not currently establish the key claim about inferring flow for truly unrepresented locations. The critical-point analysis also rests on a smoothness assumption that is inconsistent with the piecewise-linear interpolation. These issues are fixable within the manuscript's scope, but they require substantive additional analysis.

major comments (4)
  1. [Robustness of vector fields (Fig. 3)] The robustness test only removes randomly chosen spatial cells. The stated use case is inference for locations absent from OD records, which in real data are likely to be peripheral, low-density, and spatially clustered rather than randomly scattered. Under clustered removal, the Delaunay triangles used in Eq. (1) become much larger, and interpolated vectors rely on distant vertices, so angular errors can grow substantially beyond the reported 15 degrees. To support the central claim, the authors should add experiments with spatially clustered removal (e.g., contiguous blocks or peripheral zones) and report error statistics conditioned on distance to the nearest observed cell or on triangle size. This is load-bearing because the abstract and Discussion explicitly claim inference for unrepresented locations.
  2. [Appendix, Interpolation of vector fields] The interpolation assigns zero vectors to selected boundary points 'to mitigate boundary effects', but this modeling choice is neither varied nor tested. Zero-vector boundary conditions pull interpolated directions toward the border and can bias fields precisely in peripheral areas where unrepresented cells are most likely to lie. This also affects the location and classification of critical points. The authors should provide a sensitivity analysis over boundary treatments (e.g., no boundary points, extrapolation, or different boundary values) or otherwise justify that the boundary choice does not drive the reported patterns.
  3. [Appendix, Critical points (Eqs. 4-6)] The critical-point classification uses a Taylor expansion and Jacobian eigenvalues that assume a smooth, differentiable vector field. However, the triangle-based interpolation of Eq. (1) produces a piecewise-linear field that is continuous but not differentiable along triangle edges. Critical points lying on edges have undefined Jacobians, and classifications may be artifacts of the triangulation rather than properties of the flow. The analysis should be restricted to critical points interior to triangles, or the topological analysis should use a smooth interpolation method (e.g., radial basis functions). At minimum, this limitation must be stated explicitly in the critical-points section.
  4. [Discussion, robustness claim] The statement that even when more than 50% of the data is removed the deviation remains below 15 degrees is presented without a precise definition of 'any change' or a confidence interval, and it is derived from the random-removal experiment only. Because the evaluation compares interpolated vectors with the original vectors from the same dataset used to construct the field, it is an internal consistency check, not an external validation. The Discussion should qualify the claim accordingly and report the full error distribution (e.g., median and quantiles) for both random and clustered missingness.
minor comments (5)
  1. [Appendix, Moran's I] The text refers to 'as shown in Fig. A 7A', but the relevant figure is Fig. 7A in the main text; please correct the reference.
  2. [Diversity and regularity of (cattle) commodity flows] The sentence containing 'commodity 1 flow direction' appears to have a stray footnote marker and should be reworded for clarity.
  3. [Fig. 3 caption] The caption does not define how a vector is considered to 'experience any change'; please define the threshold used in the main text.
  4. [Appendix, Shannon entropy (Eq. 3)] For outcomes with zero probability, the term p_i log p_i is undefined unless the convention 0 log 0 = 0 is explicitly stated; please add this convention.
  5. [Appendix, Moran's I (Eq. 7)] The text says z_i is the standardized value and then defines z_i = y_i - ybar, which is a centered value rather than a standardized one; please reconcile the notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the interpolation and robustness tests are internally consistent and do not reduce to fitted inputs or self-citation.

full rationale

The paper's derivation chain is self-contained and does not exhibit a circular step. Origin-destination records are aggregated into per-cell resultant vectors, and missing vectors are obtained by triangle-based interpolation using barycentric weights (Eq. 1), an externally established method cited to [23,24]. No parameter is fitted to the held-out cells during the robustness test: cells are randomly removed, the remaining known vectors are used to interpolate, and the original vectors at removed cells serve only as ground truth. This is standard cross-validation, not a case where a fitted input is renamed as a prediction. The only self-citation, [25], provides detailed cosine similarity values and is not load-bearing for the central claim; it is not used to justify an ansatz, uniqueness theorem, or modeling choice. The Appendix's zero-vector boundary assignment and the smoothness assumption are modeling limitations that may affect accuracy, but they are not definitional equivalences between input and output. Thus no specific reduction of the claimed result to its own inputs can be quoted, and the paper's inference has independent methodological content despite the absence of an external benchmark.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central inference claim rests on a spatial smoothness assumption and on the interpolation choices (triangle-based, zero boundary vectors). No new physical entities are introduced. The k-medoids cluster count and quadrant discretization are free parameters that influence the reported patterns.

free parameters (3)
  • Number of clusters k in k-medoids = 4
    Chosen without a described model-selection criterion; clustering results (Fig 5) depend on this choice.
  • Boundary vector values in interpolation = 0
    Boundary points are assigned zero vectors to mitigate boundary effects (Appendix, Interpolation of vector fields); this choice can distort interpolated directions near the state border.
  • Quadrant discretization for entropy = 4 quadrants
    Monthly vectors are binned into four directional quadrants to compute Shannon entropy (Results, Diversity and regularity); this loses magnitude and within-quadrant angle information.
assumptions (3)
  • domain assumption Neighboring locations exhibit similar commodity flow patterns (spatial smoothness)
    Stated in Introduction: 'neighbouring locations are likely to participate in and influence similar activities'. This underpins the interpolation and is not validated against ground-truth flows at unrepresented locations.
  • domain assumption The interpolated vector field is smooth and differentiable for critical point analysis
    Section Critical points and Appendix: 'by assuming smooth transitions between regions' and Taylor expansion; however triangle-based interpolation produces a piecewise-linear, non-differentiable field, so Jacobian-based classification of sinks/sources may not be well-defined.
  • ad hoc to paper Zero-vector boundary conditions mitigate edge effects
    Appendix: 'selected boundary points assigned zero vectors to mitigate boundary effects'; this is an ad hoc modelling choice with no sensitivity analysis.

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Cite this review

Pith. "Pith review of Vector fields as a framework for modelling the mobility of commodities." pith.science (2026). https://pith.science/paper/XKTANHF6

@misc{pith2026250602047,
  author       = {Pith},
  title        = {Pith review of: Vector fields as a framework for modelling the mobility of commodities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XKTANHF6}},
  note         = {Machine review of arXiv:2506.02047}
}
read the original abstract

Commodities, including livestock, flow through trade networks globally, with trajectories that can be effectively captured using mobility pattern modelling approaches similar to those used in human mobility studies. However, documenting these movements comprehensively presents significant challenges; it can be unrealistic, costly, and may conflict with data protection regulations. As a result, mobility datasets typically contain uncertainties due to sparsity and limitations in data collection. Origin-destination (OD) representations offer a powerful framework for modelling movement patterns and are widely adopted in mobility studies. However, these matrices possess inherent limitations: locations absent from the OD framework lack spatial information on potential mobility directions and intensities. This spatial incompleteness creates analytical gaps across different geographical scales, constraining our ability to characterise movement patterns in underrepresented areas. In this study, we introduce a vector-field-based method to address these data challenges, transforming OD data into vector fields capturing spatial flow patterns comprehensively enabling us to study mobility directions solidly. We use cattle trade data from Minas Gerais, Brazil, as our case study for commodity flows. This region's large livestock trading network makes it an ideal test case. Cattle movements are significant as they affect disease transmission, including foot-and-mouth disease. Accurately modelling these flows allows better surveillance and control strategies. Our vector-field approach reveals fundamental patterns in commodity mobility and can infer movement information for unrepresented locations. Our approach offers an alternative to traditional network-based models, enhancing our capacity to infer mobility patterns from incomplete datasets and advancing our understanding of large-scale commodity trades.

Figures

Figures reproduced from arXiv: 2506.02047 by the authors.

Figure 1
Figure 1. Data sparsity can impact network modelling as it can disregard [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Generating a vector field from origin-destination data. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Robustness of vector field estimations under spatial data removal. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Entropy of monthly vectors for (A) municipalities and (B) micro [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Cosine-based clustering of (A, B) municipalities and (C, D) micro [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Maps of trading distances and spatial lag values. (A) [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Global Moran’s I calculation. (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 I value calculated from the vector field to those from 1000 simulations in whic…
Figure 8
Figure 8. Figure 8: Sinks (A) and sources (B) in the cattle trade vector field. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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Works this paper leans on

37 extracted references · 35 canonical work pages

  1. [1]

    Human mobility: Models and applications

    Barbosa H, Barthelemy M, Ghoshal G, James CR, Lenormand M, Louail T, et al. Human mobility: Models and applications. Physics Reports. 2018;734:1–74

    work page 2018

  2. [2]

    Mobility network models of COVID-19 explain inequities and inform reopening

    Chang S, Pierson E, Koh PW, Gerardin J, Redbird B, Grusky D, et al. Mobility network models of COVID-19 explain inequities and inform reopening. Nature. 2021;589(7840):82–87

    work page 2021

  3. [3]

    Cities and countries in the global scientist mobility network

    Verginer L, Riccaboni M. Cities and countries in the global scientist mobility network. Applied Network Science. 2020;5:1–16

    work page 2020

  4. [4]

    Urban Human Mobility: Data-Driven Modeling and Prediction

    Wang J, Kong X, Xia F, Sun L. Urban Human Mobility: Data-Driven Modeling and Prediction. SIGKDD Explor Newsl. 2019;21(1):1–19. doi:10.1145/3331651.3331653

    work page arXiv 2019

  5. [5]

    Modeling communicable diseases, human mobility, and epidemics: A review

    Soriano-Pa˜ nos D, Cota W, Ferreira SC, Ghoshal G, Arenas A, G´ omez-Garde˜ nes J. Modeling communicable diseases, human mobility, and epidemics: A review. Annalen der Physik. 2022;534(6):2100482

    work page 2022

  6. [6]

    Optimizing surveillance for livestock disease spreading through animal movements

    Bajardi P, Barrat A, Savini L, Colizza V. Optimizing surveillance for livestock disease spreading through animal movements. Journal of the Royal Society Interface. 2012;9(76):2814–2825

    work page 2012

  7. [7]

    Disease spread through animal movements: a static and temporal network analysis of pig trade in Germany

    Lentz HH, Koher A, H¨ ovel P, Gethmann J, Sauter-Louis C, Selhorst T, et al. Disease spread through animal movements: a static and temporal network analysis of pig trade in Germany. PloS one. 2016;11(5):e0155196

    work page 2016

  8. [8]

    A Data-Driven Approach to Cattle Epidemic Modelling Under Uncertainty

    Farokhnejad S, Cardoso D, Rocha C, da Mata AS, Menezes R. A Data-Driven Approach to Cattle Epidemic Modelling Under Uncertainty. In: Pacheco D, Teixeira AS, Barbosa H, Menezes R, Mangioni G, editors. Complex Networks XIII. Cham: Springer International Publishing; 2022. p. 55–67

    work page 2022

Show all 37 references
  1. [9]

    Trajectory data mining: An overview

    Zheng Y. Trajectory data mining: An overview. ACM Transactions on Intelligent Systems and Technology (TIST). 2015;6(3):1–41

    2015

  2. [10]

    Understanding individual human mobility patterns

    Gonzalez MC, Hidalgo CA, Barabasi AL. Understanding individual human mobility patterns. nature. 2008;453(7196):779–782

    2008

  3. [11]

    Human mobility, social ties, and link prediction

    Wang D, Pedreschi D, Song C, Giannotti F, Barabasi AL. Human mobility, social ties, and link prediction. In: Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining; 2011. p. 1100–1108

    2011

  4. [12]

    A survey of results on mobile phone datasets analysis

    Blondel VD, Decuyper A, Krings G. A survey of results on mobile phone datasets analysis. EPJ data science. 2015;4:1–55

    2015

  5. [13]

    Evaluating origin– destination matrices obtained from CDR data

    Mamei M, Bicocchi N, Lippi M, Mariani S, Zambonelli F. Evaluating origin– destination matrices obtained from CDR data. Sensors. 2019;19(20):4470

    2019

  6. [14]

    Patterns of ocean current variability on the West Florida Shelf using the self-organizing map

    Liu Y, Weisberg RH. Patterns of ocean current variability on the West Florida Shelf using the self-organizing map. Journal of Geophysical Research: Oceans. 2005;110(C6)

    2005

  7. [15]

    Glyphs for visualizing uncertainty in vector fields

    Wittenbrink CM, Pang AT, Lodha SK. Glyphs for visualizing uncertainty in vector fields. IEEE transactions on Visualization and Computer Graphics. 1996;2(3):266– 279

    1996

  8. [16]

    Visual analytics of movement: An overview of methods, tools and procedures

    Andrienko N, Andrienko G. Visual analytics of movement: An overview of methods, tools and procedures. Information visualization. 2013;12(1):3–24. June 4, 2025 14/19

    2013

  9. [17]

    Livestock infectious diseases and zoonoses; 2009

    Tomley FM, Shirley MW. Livestock infectious diseases and zoonoses; 2009

    2009

  10. [18]

    Potential for transmission of infections in networks of cattle farms

    Volkova V, Howey R, Savill N, Woolhouse M. Potential for transmission of infections in networks of cattle farms. Epidemics. 2010;2(3):116–122

    2010

  11. [19]

    The foot-and-mouth epidemic in Great Britain: pattern of spread and impact of interventions

    Ferguson NM, Donnelly CA, Anderson RM. The foot-and-mouth epidemic in Great Britain: pattern of spread and impact of interventions. Science. 2001;292(5519):1155–1160

    2001

  12. [20]

    A mathematical theory of communication

    Shannon CE. A mathematical theory of communication. The Bell system technical journal. 1948;27(3):379–423

    1948

  13. [21]

    Being critical of criticality in the brain

    Beggs JM, Timme N. Being critical of criticality in the brain. Frontiers in physiology. 2012;3:163

    2012

  14. [22]

    Vector field interpolation with radial basis functions

    Smolik M, Skala V. Vector field interpolation with radial basis functions. In: Proceedings of SIGRAD 2016, May 23rd and 24th, Visby, Sweden. 127. Link¨ oping University Electronic Press; 2016. p. 15–21

    2016

  15. [23]

    Triangle based interpolation

    Watson D, Philip G. Triangle based interpolation. Journal of the International Association for Mathematical Geology. 1984;16(8):779–795

    1984

  16. [24]

    A refinement of inverse distance weighted interpolation

    Watson DF, Philip G. A refinement of inverse distance weighted interpolation. Geo-processing. 1985;2(4):315–327

    1985

  17. [25]

    Using Vector Fields in the Mod- elling of Movements as Flows

    Farokhnejad S, Oliveira M, Ribeiro E, Menezes R. Using Vector Fields in the Mod- elling of Movements as Flows. In: Teixeira AS, Botta F, Mendes JF, Menezes R, Mangioni G, editors. Complex Networks XIV. Cham: Springer Nature Switzerland

  18. [26]

    Brazilian beef production

    Paulino P, Duarte M, Kahn L, Cottle D. Brazilian beef production. Beef cattle production and trade. 2014;107

    2014

  19. [27]

    Regional expansion of the beef industry in Brazil: from the coast to the Amazon, 1966–2017

    Vale R, Vale P, Gibbs H, Pedr´ on D, Engelmann J, Pereira R, et al. Regional expansion of the beef industry in Brazil: from the coast to the Amazon, 1966–2017. Regional Studies, Regional Science. 2022;9(1):641–664

    1966

  20. [28]

    Introduction to nonextensive statistical mechanics: approaching a complex world

    Tsallis C. Introduction to nonextensive statistical mechanics: approaching a complex world. vol. 1. Springer; 2009

    2009

  21. [29]

    Geographic data science with python

    Rey S, Arribas-Bel D, Wolf LJ. Geographic data science with python. CRC Press; 2023

    2023

  22. [30]

    workshop-python-spatial-stats; 2021

    Kawano Y. workshop-python-spatial-stats; 2021. https://github.com/yohman/ workshop-python-spatial-stats . Accessed 2025-05-10

    2021

  23. [31]

    Local indicators of spatial association—LISA

    Anselin L. Local indicators of spatial association—LISA. Geographical analysis. 1995;27(2):93–115

    1995

  24. [32]

    Spatial analysis for the social sciences

    Darmofal D. Spatial analysis for the social sciences. Cambridge University Press; 2015

    2015

  25. [33]

    Spatial analysis: a reader in statistical geography

    Berry BJL, Marble DF. Spatial analysis: a reader in statistical geography. Englewood Cliffs, NJ, Prentice-Hall. 1968;. June 4, 2025 15/19 Appendix Dataset and code availability This study examines cattle movement in Minas Gerais, Brazil, using data from the Institute of Agricu...

    1968

  26. [34]

    Generating multiple vector fields over predetermined time intervals ( e.g., months over four years)

  27. [35]

    Computing cosine similarity values between consecutive intervals for each region

  28. [36]

    June 4, 2025 17/19

    Constructing feature vectors for each region based on these similarity values. June 4, 2025 17/19

    2025

  29. [37]

    Applying clustering methods to group regions with similar temporal mobility patterns. This approach allows us to categorise regions based on the stability and evolution of their commodity flow directions, offering insights into the persistence and variability of trade movement...

    2025

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Reviewed August 7, 2026 · model on record in the stance chip above.