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Measuring Diversity and Segregation with Convex Functions

Philip S. Chodrow

arXiv:2610.02446v1physics.soc-phcs.SIphysics.data-an

TL;DR

Segregation measurement quantifies how population traits are separated, particularly by space. The article tours a convex-function framework that expresses segregation through Jensen information and illustrates spatial-analysis approaches. It presents this as a general toolbox while noting that many approaches and more complex joint distributions remain outside its scope.

  • Problem

    Segregation measurement seeks compact quantitative descriptions of how population attributes interact with separating distinctions such as spatial location.

    Source: §1 · p. 1

  • Method

    The article uses convex functions on the probability simplex to represent diversity and formulates segregation as Jensen information, with spatial analysis illustrated through smoothing and other approaches.

    Source: §1 · p. 2, §3 · p. 5, §4 · p. 6

  • Results

    The article formalizes segregation as a comparison of global and average local diversity, with the resulting Jensen-information characterization linked to Bregman divergence.

    Source: §1 · p. 2, §4 · p. 6

  • Takeaways & Limitations

    A choice of strictly convex diversity function can support the spatial methods and can extend the same toolbox to ordinal segregation measures such as wealth, age, or education.

    Source: §6 · p. 11

    Many spatial-segregation approaches lie outside the framework, and systematic comparisons of ordinal diversity functions and more complex joint demographic distributions remain future directions.

    Source: Future Directions · p. 12

Abstract

from arXiv · show

The problem of segregation measurement is to quantitatively describe the interaction of a population's attributes (such as race, ethnicity, education level, or income) with a separating distinction (such as organizational rank, school district, or spatial location). This primarily expository article gives an opinionated tour of one mathematical approach to measuring segregation. We begin with some basic intuitions about diversity and unify these through the framework of convex functions on the probability simplex. Formalizing segregation as a local-to-global comparison of diversity measures, resulting in the general class of Jensen informations as segregation measurements. We describe and computationally illustrate three ways to incorporate spatial structure into segregation measurements: spatial smoothing, aggregation, and local Jensen information. We close with several suggestions for future work.

1 Introduction

The article frames segregation measurement as a quantitative account of how population traits are separated, especially across space. It offers an opinionated mathematical tour, motivated by the challenge of making spatial measures robust to arbitrary geographic boundaries.

1 Introduction

  • Segregation measurement seeks compact quantitative descriptions of how strongly space separates people with different demographic traits.The article focuses on spatial segregation, where individuals are assumed to be located in space, often at their residence.Source: §1 · p. 1
  • Spatial segregation measures must contend with the modifiable areal unit problem: changing geographic boundaries can alter measurements, so measures should be robust to partition choice.The article illustrates the issue with county subdivisions and notes that other reporting units, such as census tracts or block groups, may be used.Source: §1 · p. 1
  • The article develops a mathematical perspective conversant with quantitative sociology, but is an opinionated tour rather than a comprehensive review.It connects intuitions about diversity to convex functions, formulates segregation as Jensen information, and illustrates three spatial-analysis methods.Source: §1 · p. 2

2 Preliminaries

The section motivates spatial segregation measures with synthetic cities and defines a population using its locations, demographic distributions, and population density. The examples distinguish diversity, aspatial segregation, and spatial pattern.

2 Preliminaries

  • A diversity measure distinguishes a single-group city from one whose groups are evenly represented, while aspatial segregation measures distinguish both from homogeneous blocks.The checkerboard city and the city of clustered like blocks have the same block-group counts, but differ in spatial arrangement.Source: §2 · pp. 2–3
  • The checkerboard pattern is described as arguably less segregated than monolithic clustering, motivating spatial measures that distinguish the two and address the MAUP.The passage identifies the distinction as the “checkerboard problem.”Source: §2 · p. 3
  • A population is represented as a triple P = (X, Y, µ), encoding locations, demographic distributions by location, and population density across locations.Rows of Y give category proportions at each location, while µ_i gives the share of the population living there.Source: §2 · p. 3

3 Diversity and Convexity

The section uses convex functions on the probability simplex to formalize diversity, with diversity scores expressed as negative convex functions of the population’s mean demographic vector. It relates this framework to diversity axioms and contrasts symmetric measures with a reference-dependent wealth measure.

3 Diversity and Convexity

  • Pielou’s criteria include uniform maximization and monotonicity under group addition, while the article motivates convexity through the broader intuition that diversity increases on average when populations are mixed.Strictly convex functions provide the monotonicity property; symmetry in addition yields uniform maximization.Source: §3 · pp. 4–5
  • Strict convexity of ϕ makes the population measurement −ϕ(ȳ) strictly concave, matching the proposed intuition that mixing distinct populations increases diversity on average.The framework starts from a diversity measurement defined on the mean demographic vector; the mixing intuition leads to convexity on the simplex.Source: §3 · pp. 3–4
  • The wealth measurement ϕeq is strictly convex and encodes a normative preference for a reference wealth distribution, reaching its maximum when the population mean matches that reference.The example uses cumulative wealth distributions and permits equality to be represented by a reference distribution in which everyone has the same wealth.Source: §3 · p. 5
  • Shannon entropy and the Simpson index satisfy Pielou’s uniform maximization property, whereas the wealth measurement ϕeq does not because it is not symmetric.Strict convexity guarantees monotonicity; strict convexity together with symmetry guarantees uniform maximization.Source: §3 · p. 5

4 Segregation as Information

Segregation is measured as the gap between a population’s global diversity and its average local diversity. Convexity makes this gap a Jensen information, with Bregman divergences providing its local-to-global comparison and conditions under which it satisfies segregation criteria.

4 Segregation as Information

  • Jensen information compares global diversity with average local diversity; it is nonnegative and, for strictly convex ϕ, equals zero exactly when every location matches the population mean.This makes the diversity gap a segregation measure, with a precise zero condition.Source: §4 · p. 5
  • Individual terms in the global-minus-local diversity formula can be negative, so they cannot directly measure each location’s contribution to segregation.A weighted sum of nonnegative Bregman-divergence contributions provides an alternative representation.Source: §4 · p. 6
  • Under mild regularity conditions, the local-to-global diversity gap has a unique Bregman-divergence representation.The divergence is nonnegative and zero when a location’s demographic distribution equals the mean, but it is not a metric.Source: §4 · p. 6
  • Shannon mutual information, Simpson information, and wealth information are examples of Jensen information generated by different convex functions.Wealth information is independent of the reference distribution used to define the wealth-equality function.Source: §4 · pp. 5–6

4.1 The Jensen Information as a Segregation Measure

  • Reardon and Firebaugh’s exchanges property holds for Jensen information generated by separable, strictly convex functions; the nonseparable wealth-equality function has no such guarantee.The property captures the intuition that moving groups from locations where they are overrepresented to locations where they are underrepresented reduces segregation.Source: §4.1 · p. 7

5 Spatial Analysis with the Jensen Information

The section develops three spatially aware ways to analyze segregation—smoothing, aggregation, and local information—using spatial structure to distinguish segregation patterns and examine their scale. Their illustrations show different views of spatial variation, with interpretation bounded by choices such as bandwidth, aggregation, and the demographic model.

5 Spatial Analysis with the Jensen Information

  • Spatialization aims to distinguish checkerboard from monolithic segregation through perturbation, spatial scale, or boundaries; the examples use Shannon mutual information and 2024 ACS race and ethnicity data.The three approaches are presented as spatially aware responses to the Modifiable Areal Unit Problem.Source: §5 · pp. 7–8

5.1 A Perturbative View: Spatial Smoothing

  • At γ = 5 km, Atlanta and Detroit retain segregation while Washington D.C. drops faster, though its smaller size may partly explain the contrast.The authors interpret Washington D.C.’s curve as indicating finer-scale segregation than Atlanta’s or Detroit’s.Source: Figure 5 · p. 8, §5.1 · pp. 8–9
  • Spatial smoothing computes Jensen information after replacing each location’s demographics with a kernel-weighted neighborhood average; increasing bandwidth tends to reduce MAUP sensitivity but also measured segregation.Kernels may be Gaussian radial basis functions or data-driven measures such as road-network distance or commute time.Source: §5.1 · p. 8

5.2 A Scale-Based View: Spatial Information Decomposition

  • Jensen information decomposes into between-unit and within-unit segregation under aggregation, with within-unit information nonnegative and zero only when demographics are uniform within each unit.This decomposition helps describe whether segregation is captured between aggregated units or remains within them.Source: §5.2 · p. 9
  • A greedy hierarchical clustering algorithm forms adjacent aggregated units, assigns information content to their boundaries, and produces profiles of information as the number of units increases.The algorithm has no optimality guarantees; aggregation profiles are relatively insensitive to the geographic size of the analyzed region.Source: §5.2 · pp. 9–10, Figure 6 · p. 10

5.3 A Boundary-Based View: Local Information

  • Local information models a differentiable demographic surface and measures changes in composition; unlike total variation, mean local information depends on the chosen diversity function.The method requires the local-information limit to exist and be computationally tractable to approximate.Source: §5.3 · pp. 10–11
  • In Milwaukee, local information is large along boundaries where demographics change rapidly and small inside relatively homogeneous regions; its mean can complement Jensen information as a two-dimensional description.The illustration applies a 1 km smoothing bandwidth and estimates spatial derivatives using local linear regression.Source: §5.3 · p. 11, Figure 7 · p. 11

6 Discussion

The discussion advocates a unified framework that supports broad families of diversity and segregation measures, including ordinal segregation, while noting that some results depend on properties of the chosen convex function. It identifies methodological and application areas for future work.

6 Discussion

  • A strictly convex function ϕ can encode a relevant sense of diversity, letting the same framework address ordinal segregation for wealth, age, or education.Properties that depend on symmetry or separability of ϕ, such as Theorem 4.2, are exceptions.Source: §6 · pp. 11–12
  • The article generalizes beyond particular measures to emphasize properties and methods that apply to broad classes of measures.Source: §6 · p. 11
  • Related measurements of diversity, segregation, and spatial segregation should complement one another rather than compete for a single “best” measure.At least two measurements are needed to distinguish all four panels of Figure 2.Source: §6 · p. 12

Future Directions

  • A systematic comparison of alternative ordinal diversity functions such as ϕeq is an open direction.Source: Future Directions · p. 12
  • Measuring diversity and segregation for more complex joint distributions, such as race and income, remains a direction for future work.Source: Future Directions · p. 12
  • Many approaches to spatial segregation measurement fall outside the framework developed in the article.Source: Future Directions · p. 12
  • High-level boundaries from agglomerative hierarchical clustering are sensitive to small perturbations of input data.The article proposes exploring alternative hierarchical methods, perhaps divisive clustering, with better macroscopic properties.Source: Future Directions · p. 12
  • Further work could connect empirical measurement with models of how spatial segregation changes over time.The author knows relatively few works linking mechanistic or agent-based models with empirical data, with a notable exception.Source: Future Directions · p. 12

A Pielou’s Third Diversity Axiom

The appendix states Pielou’s additivity axiom for independent demographic attributes and notes that, with two other axioms plus mild assumptions, it characterizes Shannon entropy.

A Pielou’s Third Diversity Axiom

B Supplementary Proofs

The supplementary proofs use convexity and separability to analyze changes in Jensen information under demographic exchanges, and relax nonnegativity by shifting the function by its minimum. Other proof fragments invoke permutation symmetry and direct calculation.

B Supplementary Proofs

  • The exchange argument uses convexity and nonnegativity to establish its inequality.Source: B Supplementary Proofs · p. 13
  • A continuous convex function that is not nonnegative can be shifted by its minimum on the compact simplex so the nonnegative-case calculation applies.Source: B Supplementary Proofs · p. 13
  • A proof fragment uses an inductive strict-concavity inequality and symmetry after considering permutations.Source: B Supplementary Proofs · p. 14
  • The proof of Theorem 4.2 is described as a direct calculation.Source: B Supplementary Proofs · p. 14
  • For an exchange between two demographic vectors, the proof approximates the change in Jensen information using their gradients and the exchange size.Separability expresses the gradients through coordinate derivatives; strict convexity and the assumed coordinate ordering establish the sign for sufficiently small exchanges.Source: B Supplementary Proofs · p. 14
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