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A new measure of spatial clustering treats a city's demographic distribution as probability mass diffusing under a Metropolis-Hastings random walk, defining clustering as slowness to reach uniformity.

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The diffusion distance to uniformity—the minimal number of Metropolis-Hastings steps to get within epsilon of the uniform distribution—is introduced as a global spatial-clustering statistic with permutation p-value bounds.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection The permutation-moment results are the real contribution and they look correct; the city-level claim is undercut by unstandardized cross-graph comparisons. the 4 major comments →

arxiv 2607.14880 v1 pith:NDPDQWO3 submitted 2026-07-16 math.ST cs.LGstat.TH

Measuring Spatial Clustering via Metropolis-Hastings Diffusion Distance

Thomas Weighill , Chidinma Williams This is my paper
classification math.ST cs.LGstat.TH MSC 62H1160J1005C50
keywords spatial clusteringdiffusion distanceMetropolis-HastingsMoran's Ipermutation testgraph spectrasegregationstochastic block model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper proposes a new way to measure spatial clustering in data living on a graph: instead of summarizing local correlations, run a Metropolis-Hastings random walk that targets the uniform distribution and time how many steps the observed distribution needs to get within a tolerance of uniformity. Slow convergence is interpreted as strong clustering. The paper proves spectral and optimal-transport bounds on this diffusion distance, shows that Moran's I is exactly the one-step version of the same quantity, and derives permutation-null expectation and variance formulas that yield a fast p-value bound. Empirically, the diffusion distance outperforms Moran's I on synthetic block-model data and flags cities whose segregation is globally unstable in a way Moran's I misses.

Core claim

The central discovery is that spatial clustering can be quantified by the hitting time of a graph-constrained diffusion: define diff^P_{p,eps}(f→τ) as the least n such that ‖fP^n−τ‖_p < ϵ, with P the Metropolis-Hastings matrix targeting uniform τ and proposals from a graph random walk. This single number interpolates between local and global structure; Moran's I is recovered as the one-step squared-deviation ratio, and multi-step behavior adds geometric information that the one-step statistic cannot see. The paper supports the measure with exact spectral formulas for permutation moments and a concentration bound that makes the resulting hypothesis test computationally feasible.

What carries the argument

The load-bearing object is the Metropolis-Hastings diffusion distance to uniformity: for a graph G, f is a probability distribution on vertices, τ is uniform, and P is the Metropolis-Hastings transition matrix with proposal Q being the simple random walk and target τ. The distance is the smallest integer n such that the ℓ_p distance from fP^n to τ is below a chosen tolerance ϵ. This object converts the question 'are the data clustered?' into 'how many local diffusion steps are needed to erase the data's signature?' The spectral theory of P then supplies both upper bounds and null-model moments, while the Wasserstein lower bound ties the distance to optimal transport.

Load-bearing premise

The whole interpretation rests on the modeling premise that spatial clustering equals slowness of a Metropolis-Hastings random walk to equilibrate from the observed distribution to uniformity; the paper asserts this rather than validating it against an independent ground-truth measure of segregation, and the numerical values depend on the user-chosen pair (p, ϵ).

What would settle it

Find two city-like graphs with identical Moran's I for the same f but opposite segregation ground truth — one with a narrow bottleneck separating the concentrated mass from the rest, one without — and compare diffusion distances. If diffusion distance fails to rank the bottlenecked city as more clustered (or ranks it lower than Moran's I), the claim that it captures global geometry beyond local one-step correlation is falsified.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Moran's I is subsumed as a one-step approximation; any analysis that uses Moran's I can be re-run with multi-step diffusion distance to expose structure that appears only after several diffusion steps.
  • The permutation test for clustering is computable from the graph spectrum alone, so large spatial datasets can be tested without exhaustive permutation sampling.
  • On stochastic block model data, the diffusion-distance test rejects spatial independence more often than Moran's I as the between-block shift grows, implying higher power in locally noisy but globally biased patterns.
  • On Black population shares across 100 U.S. cities, diffusion distance and Moran's I agree broadly (rank correlation 0.728) but diverge for six cities, indicating segregation that is locally sharp yet not a diffusion bottleneck.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If slow mixing is the right model for segregation, then the same framework can be applied to other spatially arranged variables (e.g., disease incidence, income, pollution), turning any graph-supported distribution into a clustering score; this extension is not pursued in the paper.
  • Because the measure is defined through graph geometry, its city rankings are likely sensitive to how census tracts are aggregated (the modifiable areal unit problem); a sensitivity analysis over multiple tract partitions would test how robust the reported city ordering is.
  • The paper's concentration bound uses Cantelli's inequality and is therefore one-sided and potentially loose; tighter bounds or exact null distributions could turn the test into a tool for ranking rather than merely rejecting.
  • The interpretation hinges on the chosen tolerance ϵ; the paper gives heuristics, but a principled data-driven choice of ϵ (e.g., based on the null distribution of ‖f−τ‖_2) would make the measure more portable across graphs of different sizes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper defines a diffusion distance between probability distributions on a graph: the minimal number of Metropolis-Hastings steps (targeting uniformity) needed for a source distribution f to come within an ℓ_p tolerance ε of uniformity. The authors derive spectral upper bounds, a Wasserstein lower bound, a stability result, and a formal connection between Moran's I and the one-step squared deviation. They then derive exact first and second permutation moments for squared deviation and use Cantelli's inequality to obtain a high-probability upper bound on the permutation p-value of an observed diffusion distance. The method is demonstrated on synthetic stochastic block models and on Black population shares across 100 U.S. cities, where the reported rank correlation with Moran's I is 0.728 and six cities are identified as being more locally than globally clustered.

Significance. If the theoretical results are correct, the paper provides a principled graph-geometric extension of Moran's I with closed-form null moments and a potentially efficient permutation test. The permutation moment derivation in Appendix B is detailed and appears correct, and Theorems 3.3–3.5 are mathematically sound. The power analysis on stochastic block models is a useful benchmark. However, the statistical-testing claim rests on a p-value bound that is anti-conservative as stated, and the empirical city comparison uses raw diffusion distances without standardization, so the central empirical claim that the measure 'detects subtle differences in urban segregation patterns that Moran's I does not' is not yet supported. The connection to Moran's I also appears to use a different weight matrix from the one defined as the paper's default, which needs clarification.

major comments (4)
  1. [§4.3, Theorem 4.4] Theorem 4.4 bounds Pπ(diff > n), but the observed statistic computed in Step 1 of §4.3 is n = diff(f→τ). A permutation p-value for this observed value is Pπ(diff ≥ n) = Pπ(diff > n−1). Since {diff > n} is a subset of {diff ≥ n}, the bound is not an upper bound on the required p-value and is anti-conservative whenever Pπ(diff = n) > 0. The theorem should be applied at n−1 (with ρ_{n−1}(ε)>0) or modified to bound Pπ(diff ≥ n). This is load-bearing for the paper's claim of enabling an efficient statistical test.
  2. [§5.2, Figure 4] The empirical comparison uses raw diffusion distance values (p=1, ε=0.25) across 100 city graphs that differ substantially in size and topology (e.g., New York has 1,848 nodes while many cities are much smaller). Diffusion distance is a hitting time of a graph-specific chain, so raw values conflate the spatial distribution f with graph geometry. The paper's own Section 4 shows that the permutation null distribution of diffusion distance is graph-specific. Without standardization (e.g., using null quantiles or spectral normalization), the rank correlation of 0.728 and the identification of six 'more locally than globally clustered' cities may reflect graph size or bottlenecks rather than segregation. The limitation paragraph acknowledges the geometry dependence but does not control for it.
  3. [§3.2 vs §2.1] Theorem 3.6 connects Moran's I to one-step squared deviation only when the spatial weight matrix is M = P P^T. However, Equation (1) and §2.1 define the default Moran weights as row-standardized adjacency weights w_ij = 1/deg_i, which are not equal to P P^T for irregular graphs. If the Moran's I values in Figure 4 are computed using the default row-standardized weights, then the theorem does not apply to the statistic compared in the application. The manuscript must state explicitly which weight matrix was used and, if the row-standardized weights were used, provide the corresponding connection or remove the implication that the empirical comparison is testing the one-step heuristic.
  4. [§5.2, modeling premise] The city-level interpretation that diffusion distance 'detects subtle differences in urban segregation patterns that Moran's I does not' is not validated against any external ground-truth segregation measure or a priori hypothesis. The paper only shows agreement and disagreement with Moran's I on the same data; divergence between two statistics does not establish which one is correct. Given the sensitivity of the measure to the user-chosen parameters (p, ε) and to the graph geometry, the empirical claim is under-supported. A comparison with established segregation indices, or at least a standardization of the measure by each city's permutation null, would substantially strengthen the empirical case.
minor comments (5)
  1. [Definition 3.1] Typo: 'asymetric' should be 'asymmetric'.
  2. [Remark 3.2] The heuristic E||f−τ||_2 ≈ 1/√N for f uniform on the simplex is stated without derivation. Since the permutation null in Section 4 uses fixed values of f, it would be clearer to distinguish this heuristic from the exact permutation moments of Section 4.
  3. [§4.3] The proposal to approximate the required spectral quantities by retaining only the k largest eigenvalues is described without an error bound. A statement about the assumed decay of eigenvalues, or a bound on the resulting approximation error, would be needed to make this recommendation reliable.
  4. [§5.2, Figure 4] The six cities are identified informally by 'ranked significantly higher by Moran's I than by diffusion distance.' No formal criterion (e.g., residual threshold, rank-difference test) is given. A precise definition would make the claim reproducible.
  5. [§5.2] The empirical analysis uses p=1 and ε=0.25, but no sensitivity analysis for these parameters is reported. Since the paper emphasizes that the measure depends on (p, ε), a check that the city-level conclusions are stable under nearby choices would be valuable.

Circularity Check

0 steps flagged

No significant circularity: the diffusion-distance formulas and inference are derived from first principles; the few self-citations are motivational or data-providing, not load-bearing.

full rationale

The paper does not fit parameters and then rename them as predictions. The diffusion distance in Definition 3.1 is a hitting-time quantity defined directly from a Markov chain; the user-chosen parameters p and epsilon are set by stated heuristics (Remark 3.2), not optimized on the data that is later 'predicted.' The spectral upper bound (Theorem 3.3), Wasserstein lower bound (Proposition 3.4), stability result (Proposition 3.5), permutation moments (Theorem 4.1), and Cantelli-based high-probability bound (Theorem 4.4) are all derived from standard Markov chain/spectral/permanent-moment arguments in the paper itself, with proofs included. The Moran's I connection (Theorem 3.6) is proved in the text by direct algebra: for M = P P^T, I_tau(f,M) = dev_tau(fP)/dev_tau(f). Although the paper cites [11] (Duchin-Murphy-Weighill, with an overlapping author) as the origin of the one-step observation, the theorem is not imported by citation alone; it is re-derived self-contained, so the self-citation is not load-bearing. The use of [21] (Kauba-Weighill) is only to obtain preprocessed city dual graphs, which is a data-source citation, not an argument. The modeling premise that slow mixing under Metropolis-Hastings to uniformity corresponds to spatial clustering is an interpretive assumption, not a circular derivation: the paper does not define clustering as the output of its own fitted constants, and it validates the measure against Moran's I and synthetic SBM ground truth. Concerns about cross-city graph-size confounds in Section 5.2 are about statistical validity or robustness, not circularity. Overall the derivation chain is self-contained; at most there are minor non-load-bearing self-citations.

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 0 invented entities

No fitted parameters; the measure's user-chosen p and epsilon are free choices. The mathematical results rely on standard theorems and on domain assumptions about graph ergodicity, normality of P, the permutation null, and the fidelity of graph-MH dynamics to spatial segregation. No new physical or formal entities are introduced.

free parameters (2)
  • p (norm order) = p=2 for statistical tests, p=1 for city analysis
    The l_p norm order is chosen by the user; results vary with p. Remark 3.2 gives heuristics, not data-based selection.
  • epsilon (tolerance) = 0.01 for SBM experiments, 0.25 for city analysis
    Tolerance chosen by hand via the heuristics in Remark 3.2; diffusion distance and the p-value bound depend on it.
axioms (7)
  • standard math Perron-Frobenius theorem for primitive matrices; the MH-to-uniform chain on a connected non-bipartite graph is primitive.
    Used in proof of Theorem 3.3 to guarantee the spectral radius gap lambda < 1.
  • standard math Birkhoff-von Neumann theorem: every doubly stochastic matrix is a convex combination of permutation matrices.
    Used in Proposition 3.5 to show the l_p norm is non-increasing under doubly stochastic P.
  • standard math Cantelli's (one-sided Chebyshev) inequality.
    Used in Theorem 4.4 to convert mean and variance into a tail bound.
  • domain assumption The graph G is connected and non-bipartite so the MH chain targeting uniformity is ergodic.
    Stated after Definition 3.1; needed for finite diffusion distances and spectral convergence.
  • domain assumption For Theorem 4.4, the transition matrix P is normal (symmetric in the default MH-to-uniform case), so P^n(P^n)^T has spectral representation from P P^T.
    Restricts the efficient spectral p-value test to symmetric/normal chains.
  • domain assumption The permutation null model, uniform random relabeling of the observed f values, is a valid null for spatial clustering.
    Used throughout Section 4; all moments are computed under this exchangeability assumption.
  • ad hoc to paper Census tract dual graphs and Black population shares are an appropriate representation of urban segregation, and the MH random walk is a reasonable model of spatial diffusion.
    Underpins the empirical interpretation in Section 5.2; acknowledged as an approximation in Section 1.2 but not independently validated.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Measuring Spatial Clustering via Metropolis-Hastings Diffusion Distance." pith.science (2026). https://pith.science/paper/NDPDQWO3

@misc{pith2026260714880,
  author       = {Pith},
  title        = {Pith review of: Measuring Spatial Clustering via Metropolis-Hastings Diffusion Distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NDPDQWO3}},
  note         = {Machine review of arXiv:2607.14880}
}
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abstract

We propose a novel measure of the discrepancy between two probability distributions $f$ and $g$ on a graph - which we call the diffusion distance - that measures the rate of convergence of $f$ to $g$ under a graph-constrained Markov chain with stationary distribution $g$. As a default choice for this Markov chain, we use the Metropolis-Hastings transition matrix targeting $g$ with proposals given by a random walk on the graph. Our primary case of interest is when the second distribution $g$ is uniform, in which case the diffusion distance becomes a measure of spatial clustering in $f$. Used in this way, (Metropolis-Hastings) diffusion distance to uniformity extends Moran's $I$-type measures of spatial autocorrelation by incorporating global graph geometry rather than just local patterns. Indeed, Moran's $I$, the most well-known measure of spatial autocorrelation, can be viewed as a one-step heuristic for diffusion distance, so long as specific spatial weights are used. We establish theoretical bounds and a stability result for our measure, connecting it to graph spectra and optimal transport. We then turn our attention to outlining a statistical test for spatial clustering using diffusion distance. Under permutation null models, we derive high-probability bounds on diffusion distance underpinned by exact spectral formulas for convergence of distributions, enabling an efficient statistical test for spatial clustering on large datasets. We empirically compare diffusion distance to Moran's $I$ both as a numerical measure and as a statistical test. We show that diffusion distance exhibits higher power on synthetic data using a stochastic block model. Empirical analysis of Black population distributions for 100 U.S. cities shows that diffusion distance detects subtle differences in urban segregation patterns that Moran's $I$ does not.

Figures

Figures reproduced from arXiv: 2607.14880 by Chidinma Williams, Thomas Weighill.

Figure 1
Figure 1. Figure 1: Three graphs with uniform distribtion f supported on the yellow nodes. The graphs have equal diameters and equal Moran’s I for f, but different diffusion distances to uniform with p = 2, ϵ = 0.01 due to the change in geometry. Theorem 3.6 (Moran’s I as normalized one-step squared deviation [11]). Let P be a bistochastic Markov matrix on N states with uniform stationary distribution τ , let f ∈ ∆N be a prob… view at source ↗
Figure 2
Figure 2. Figure 2: Power analysis of diffusion distance and Moran’s [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Chicago tract map and corresponding dual graph, both colored by tract-level Black population [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparing Metropolis-Hastings diffusion distance to uniformity and Moran’s [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dual graphs of six cities with substantial Black populations whose rank by Moran’s [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Rejection rates for ϵ = 0.001 with the same setup as in Section 5.1. Census data obtained from NHGIS is used in accordance with their licence; see https://www.nhgis.org/ citation-and-use-nhgis-data. We use the preprocessed city graphs published by the authors of [21] and used in their analysis in that paper. B Proof of Theorem 4.1 Preliminaries. Since f is a probability vector, PN i=1 fi = 1, and therefore… view at source ↗

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Reference graph

Works this paper leans on

44 extracted references · 1 linked inside Pith

  1. [1]

    Census dual graphs: Properties and random graph models.arXiv preprint arXiv:2604.04960, 2026

    Sara Anderson, Sarah Cannon, Brooke Feinberg, and Anne Friedman. Census dual graphs: Properties and random graph models.arXiv preprint arXiv:2604.04960, 2026

    Pith/arXiv arXiv 2026

  2. [2]

    Springer Science & Business Media, 1988

    Luc Anselin.Spatial econometrics: methods and models. Springer Science & Business Media, 1988

    1988

  3. [3]

    Local indicators of spatial association—lisa.Geographical analysis, 27(2):93–115, 1995

    Luc Anselin. Local indicators of spatial association—lisa.Geographical analysis, 27(2):93–115, 1995

    1995

  4. [4]

    A geometric interpretation of the metropolis-hastings algorithm

    Louis J Billera and Persi Diaconis. A geometric interpretation of the metropolis-hastings algorithm. Statistical Science, pages 335–339, 2001

    2001

  5. [5]

    Tres observaciones sobre el álgebra lineal.Universidad Nacional de Tucumán Revista, Serie A, 5:147–151, 1946

    Garrett Birkhoff. Tres observaciones sobre el álgebra lineal.Universidad Nacional de Tucumán Revista, Serie A, 5:147–151, 1946

    1946

  6. [6]

    500 Cities: City Boundaries.https://chronicdata.cdc

    Centers for Disease Control and Prevention. 500 Cities: City Boundaries.https://chronicdata.cdc. gov/500-Cities-Places/500-Cities-City-Boundaries/n44h-hy2j, 2024. Accessed: 28-May-2024

    2024

  7. [7]

    Spatial statistics: A theory and methods review.International Encyclopedia of Geography: People, the Earth, Environment and Technology, pages 1–14, 2016

    Yongwan Chun and Daniel A Griffith. Spatial statistics: A theory and methods review.International Encyclopedia of Geography: People, the Earth, Environment and Technology, pages 1–14, 2016

    2016

  8. [8]

    Pion London, 1973

    Andrew D Cliff and John K Ord.Spatial autocorrelation. Pion London, 1973

    1973

  9. [9]

    Pion Ltd, 1981

    Andrew D Cliff and John K Ord.Spatial Processes: Models and Applications. Pion Ltd, 1981

    1981

  10. [10]

    Coifman and Stéphane Lafon

    Ronald R. Coifman and Stéphane Lafon. Diffusion maps.Applied and Computational Harmonic Analysis, 21(1):5–30, 2006

    2006

  11. [11]

    Murphy, and Thomas Weighill

    Moon Duchin, James M. Murphy, and Thomas Weighill. Measuring segregation via analysis on graphs. SIAM Journal on Matrix Analysis and Applications, 44(1):80–105, 2023

    2023

  12. [12]

    A. S. Fotheringham, Chris Brunsdon, and Martin Charlton.Geographically Weighted Regression: The Analysis of Spatially Varying Relationships. John Wiley & Sons, Chichester, UK, 2002

    2002

  13. [13]

    A. S. Fotheringham, Wenbai Yang, and Wei Kang. Multiscale geographically weighted regression.Annals of the American Association of Geographers, 107(6):1247–1265, 2017

    2017

  14. [14]

    The contiguity ratio and statistical mapping.The Incorporated Statistician, 5(3):115–146, 1954

    Robert C Geary. The contiguity ratio and statistical mapping.The Incorporated Statistician, 5(3):115–146, 1954

    1954

  15. [15]

    Spatial modeling with spatially varying coefficient processes.Journal of the American Statistical Association, 98(462):387–396, 2003

    Alan E Gelfand, Hyon-Jung Kim, CF Sirmans, and Sudipto Banerjee. Spatial modeling with spatially varying coefficient processes.Journal of the American Statistical Association, 98(462):387–396, 2003

    2003

  16. [16]

    A critique of spatial autocorrelation.Geographical Analysis, 40(3):290–300, 2008

    Arthur Getis. A critique of spatial autocorrelation.Geographical Analysis, 40(3):290–300, 2008

    2008

  17. [17]

    Griffith.Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization

    Daniel A. Griffith.Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization. Springer, Berlin, Heidelberg, 2003

    2003

  18. [18]

    Hammond, Yaniv Gur, and Chris R

    David K. Hammond, Yaniv Gur, and Chris R. Johnson. Graph diffusion distance: A difference measure for weighted graphs based on the graph Laplacian exponential kernel. InProceedings of the IEEE Global Conference on Signal and Information Processing (GlobalSIP), pages 419–422, 2013

    2013

  19. [19]

    Spatio-temporal alignments: Optimal transport through space and time

    Hicham Janati, Marco Cuturi, and Alexandre Gramfort. Spatio-temporal alignments: Optimal transport through space and time. InProceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, pages 1695–1704. PMLR, 2020

    2020

  20. [20]

    Kantorovich

    Leonid V. Kantorovich. Mathematical methods of organizing and planning production.Management Science, 6(4):366–422, 1939

    1939

  21. [21]

    Topological analysis of us city demographics.La Matematica, 3(4):1400–1425, 2024

    Jakini Auset Kauba and Thomas Weighill. Topological analysis of us city demographics.La Matematica, 3(4):1400–1425, 2024. 15

    2024

  22. [22]

    Segregation: A Python package for measuring residential segregation.Journal of Open Source Software, 5(47):1951, 2020

    Eli Knaap, Levi John Wolf, Rey Sergio, Su Han Chan, Taylor Oshan, and Stefanie Lumnitz. Segregation: A Python package for measuring residential segregation.Journal of Open Source Software, 5(47):1951, 2020

    1951

  23. [23]

    The long-run implications of slum clearance: A neighborhood analysis.Journal of Public Economics, 236:105153, 2024

    Jessica LaVoice. The long-run implications of slum clearance: A neighborhood analysis.Journal of Public Economics, 236:105153, 2024

    2024

  24. [24]

    Spatial autocorrelation: trouble or new paradigm?Ecology, 74(6):1659–1673, 1993

    Pierre Legendre. Spatial autocorrelation: trouble or new paradigm?Ecology, 74(6):1659–1673, 1993

    1993

  25. [25]

    LeSage.Spatial econometrics

    J.P. LeSage.Spatial econometrics. Department of Economics, University of Toledo, 1998

    1998

  26. [26]

    Ipums national historical geographic information system: Version 16.0, 2021

    Steven Manson, Jonathan Schroeder, David Van Riper, Tracy Kugler, and Steven Ruggles. Ipums national historical geographic information system: Version 16.0, 2021

    2021

  27. [27]

    Massey and Nancy A

    Douglas S. Massey and Nancy A. Denton. The dimensions of residential segregation.Social Forces, 67(2):281–315, 1988

    1988

  28. [28]

    Meyer.Matrix Analysis and Applied Linear Algebra

    Carl D. Meyer.Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000

    2000

  29. [29]

    Mood, Franklin A

    Alexander M. Mood, Franklin A. Graybill, and Duane C. Boes.Introduction to the Theory of Statistics. McGraw-Hill, 3rd edition, 1974

    1974

  30. [30]

    Journal of the Royal Statistical Society

    Patrick Alfred Pierce Moran.The interpretation of statistical maps. Journal of the Royal Statistical Society. Series B (Methodological), 1948

    1948

  31. [31]

    Notes on continuous stochastic phenomena.Biometrika, 37(1/2):17–23, 1950

    Patrick Alfred Pierce Moran. Notes on continuous stochastic phenomena.Biometrika, 37(1/2):17–23, 1950

    1950

  32. [32]

    Daniel Peng and James M. Murphy. Optimal transport as a metric for measuring segregation. In2025 International Conference on Sampling Theory and Applications (SampTA), pages 1–5, 2025

    2025

  33. [33]

    Gromov-wasserstein averaging of kernel and distance matrices.International Conference on Machine Learning, pages 2664–2672, 2016

    Gabriel Peyré, Marco Cuturi, and Justin Solomon. Gromov-wasserstein averaging of kernel and distance matrices.International Conference on Machine Learning, pages 2664–2672, 2016

    2016

  34. [34]

    Oxford University Press, 2008

    Dirk Pfeiffer, Timothy P Robinson, Mark Stevenson, Kim B Stevens, David J Rogers, and Archie CA Clements.Spatial analysis in epidemiology. Oxford University Press, 2008

    2008

  35. [35]

    Reardon and Glenn Firebaugh

    Sean F. Reardon and Glenn Firebaugh. Measures of multigroup segregation.Sociological Methodology, 32(1):33–67, 2002

    2002

  36. [36]

    Quantifying ethnic segregation in cities through random walks

    Sandro Sousa and Vincenzo Nicosia. Quantifying ethnic segregation in cities through random walks. Nature Communications, 13:5809, 2022

    2022

  37. [37]

    Jonathan B. Thayn. Eigenvector spatial filtering and spatial autoregression. In Shashi Shekhar, Hui Xiong, and Xun Zhou, editors,Encyclopedia of GIS. Springer, Cham, 2017

    2017

  38. [38]

    Michael Tiefelsdorf and Barry Boots. Modelling spatial processes: The identification and analysis of spatial relationships in regression residuals by means of moran’s i.Journal of Geographical Systems, 5(1):25–45, 1998

    1998

  39. [39]

    Random walk-based segregation measures.Review of Economics and Statistics, 96(3):383–398, 2014

    Wietske Van der Laan Bouma-Doff, Coen Van Eijl, Henk Friesema, et al. Random walk-based segregation measures.Review of Economics and Statistics, 96(3):383–398, 2014

    2014

  40. [40]

    L. N. Vaserstein. Markov processes over denumerable products of spaces, describing large systems of automata.Problemy Peredači Informacii, 5(3):64–72, 1969

    1969

  41. [41]

    Optimal transport for structured data with application on graphs

    Titouan Vayer, Laetitia Chapel, Rémi Flamary, Romain Tavenard, and Nicolas Courty. Optimal transport for structured data with application on graphs. InProceedings of the 36th International Conference on Machine Learning (ICML), volume 97 ofProceedings of Machine Learning Research, pages 6275–6284, 2019. 16

    2019

  42. [42]

    Springer Science & Business Media, 2008

    Cédric Villani.Optimal transport: old and new, volume 338. Springer Science & Business Media, 2008

    2008

  43. [43]

    David W. S. Wong. Implementing spatial segregation measures in GIS.Computers, Environment and Urban Systems, 27(1):53–70, 2003

    2003

  44. [44]

    A decomposition of moran’s i for clustering detection.Computa- tional Statistics & Data Analysis, 51(12):6123–6137, 2007

    Tian Zhang, Ge Lin, and Arthur Getis. A decomposition of moran’s i for clustering detection.Computa- tional Statistics & Data Analysis, 51(12):6123–6137, 2007. A Computational and data details The experiments in Section 5 were run on laptops/MacBooks using Python. The largest city dual graph is New York NY with 1848 nodes. Computing the Moran’sI and diffu...

    2007

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.