Unified framework for measuring segregation resolves how social and geographical space jointly shape connections

• The approach addresses limitations of region-based geographical measures, including the MAUP, checkerboard problem, and the conflation of social and geographical effects in aggregate connectivity.Euclidean proximity kernels alone are insufficient because population density affects effective distance, while average interregional interaction rates mix both segregation dimensions.Source: §1 · p. 2
• The framework distinguishes geographical segregation, social segregation, and their combined effect, total segregation, rather than treating observed cross-group connectivity as purely social.Geographical segregation reflects distance-driven connectivity decay, whereas social segregation reflects differential preferences for connecting across sociodemographic groups.Source: §1 · pp. 2–4
• Figure 1 shows that identical traditional geographical segregation can conceal different geographical and social segregation, while stronger geospatial homophily alone can increase nominal network assortativity.Societies B and C have identical marginal group-to-group mixing and assortativity but differ in geographical versus social homophily.Source: Figure 1 · p. 3, §1 · p. 3
• The authors use aggregated relational data because large-scale individual-level geosocial networks are generally unavailable and difficult to analyze, while preserving aggregated connection volumes for inference.Their ARD source is the Social Connectedness Index, based on rescaled aggregated Facebook friendship counts between geographical regions.Source: §1 · p. 4
• The study develops a probabilistic geosocial network model that estimates tie probabilities from sociodemographic distributions and aggregated connection data, combining U.S. Census, Pew, and Meta Marketing API estimates.The model supplies a likelihood for SCI observations conditional on the estimated sociodemographic distribution across regions.Source: §1 · p. 4
• The joint geosocial intervening-opportunities model outperforms simpler factorized geographical-kernel and Stochastic Block Model alternatives, and the fitted results show stronger social than geographical segregation.The analysis also finds strong race, education, and income homophily, racial homophily declining with educational attainment and weakening in the highest-income group, plus interactions between distance and social homophily.Source: §1 · p. 4

2.1 Measuring geographical, social, and total segregation

• The framework isolates total, social, geographical, and traditional geographical segregation by comparing geosocial network models with degree-preserving random-mixing nulls.Its homophily decomposition distinguishes homophilous from heterophilous deviations, while TV and KL versions recover modularity, dissimilarity, and Theil indices as special cases [40].Source: §2.1 · pp. 5, 7

2.2 Inferring geosocial connectivity from aggregated relational data

• Aggregated regional SCI, Census, and estimated Facebook population data are combined with a geosocial network model to infer regional-social connection volumes needed for segregation measurement.The inferred volumes are marginalized over target regions to obtain the conditional geosocial coordinate distribution σM.Source: §2.2 · p. 7, Figure 3 · p. 7

2.3 Joint geosocial intervening opportunities best explain SCI connectivity

• Joint fuzzy intervening-opportunity models consistently fit SCI data better than hybrid and distance-based models across 40 states and multiple specifications.Fuzzy intervening opportunities can halve mean squared error relative to a standard intervening-opportunity baseline at small distances.Source: §2.3 · p. 8

2.4 Inferred connectivity reveals strong group homophily along race, education, and income

• Inference reveals strong homophily by race, education, and income, with White–Black, college-degree–no-degree, and high-income–other group distances corresponding to lower connection probabilities.Homophily is strongest under the social counterfactual, weaker under traditional geographical segregation, and weakest under geographical segregation; ref. [15] reports a similar social-versus-spatial pattern.Source: §2.4 · p. 8, Figure 5 · p. 11

2.5 Social segregation predominates over geographical segregation

• Social segregation exceeds geographical segregation across race, education, and income: it is two to three times traditional geographical segregation, which is about twice geographical segregation.Total segregation is only slightly larger than social segregation, indicating that group-based connectivity preferences account for most inferred segregation.Source: §2.5 · p. 10, Figure 6 · p. 12

2.6 Social connectivity patterns vary with geographical distance

• Geographical connectivity patterns vary across groups: distance decay is generally steeper for low-education and low-income areas, while education effects are corroborated across source ZCTAs.Income patterns are heterogeneous, with some affluent areas substantially less geographically homophilous than lower-income areas; homogeneous-ZCTA comparisons require caution.Source: §2.6 · p. 13
• For race, education, and income, same-group ZCTA pairs have higher connectivity than between-group pairs, and this homophily gap widens with geographical distance.The empirical SCI shows these patterns across ZCTA pairs, while the social segregation model is used to assess distance-dependent assortativity.Source: §2.6 · p. 13, Figure 7 · p. 14
• For most ZCTAs, race, education, and income assortativity increase with geographical distance, whereas urban/rural assortativity changes little.The urban/rural slope distribution is tightly centered around zero, unlike the positive distance trends for the other social attributes.Source: §2.6 · p. 13
• These distance-dependent assortativity patterns are estimated from the social segregation model to control for patterns induced by geographical segregation.Source: Figure 7 · p. 14

Framework

• The framework defines segregation through deviations from degree-preserving random mixing, separates homophilous and heterophilous contributions, and recovers major existing indices as special cases.These include Dissimilarity, Relative Diversity, Theil, network modularity, and normalized nominal network assortativity, with corresponding normalization constants.Source: §3 · p. 15

Inference from aggregated data

• Privacy-preserving aggregated relational data combined with regional sociodemographic counts support geosocial connectivity inference, and the joint intervening-opportunities model fits SCI better than geographically or socially separate models.The inferred connectivity enables analysis of total, social, geographical, and traditional geographical segregation in the US Facebook network.Source: §3 · p. 15

Empirical findings

• Social segregation predominates over geographical and traditional geographical segregation for race, education, and income, although college-education and income-separation patterns persist across measures.Traditional geographical segregation is qualitatively similar but larger in magnitude, while social segregation differs qualitatively from it.Source: §3 · p. 15
• Social homophily increases with geographical distance, while greater geographical race isolation in a ZCTA is associated with higher local social race homophily.This suggests geographically missing cross-group opportunities may not be compensated at larger distances because homophily also increases there.Source: §3 · p. 15

Limitations

• Inferred group-to-group connection probabilities remain latent, so unobserved confounding, estimated composition and usage rates, and Facebook-specific ties limit interpretation despite model comparisons and robustness checks.The authors call for validation with richer aggregated data or individual-level networks; prior qualitative support comes from individual-level data in other countries [15].Source: §3 · pp. 15–16

Future directions

• The f-divergence approach is analytically tractable but best suited to categorical variables; structured Wasserstein, Hamming, or learned social distances could extend it to ordinal, continuous, or multidimensional categories.Future work should replicate the analysis internationally and use longitudinal or quasi-experimental data to identify causal feedback between geography and social homophily.Source: §3 · p. 16

4.1.1 Traditional geographical segregation as an f-divergence

• The geosocial formulation extends f-divergences from population distributions to connected-pair coordinate distributions, using a geo-egocentric space that retains the first individual’s region and group while marginalizing the second individual’s region.This makes segregation depend on which groups individuals in each region connect to rather than where those connections are geographically located.Source: §4.1.1 · p. 17, §4.1.2 · p. 18

4.1.3 Representation of segregation via counterfactual and reference models

• The framework measures total, social, geographical, and traditional geographical segregation by comparing a network model with counterfactual models that selectively remove geographical or sociodemographic mechanisms.Social segregation equalizes regional group composition while preserving connectivity patterns; geographical segregation removes direct group dependence from connectivity; traditional geographical segregation restricts links to within-region pairs.Source: §4.1.3 · pp. 18–19

4.1.4 Divergence-based segregation indices

• Segregation indices measure departures from degree-constrained random mixing and normalize them by degree-constrained social isolation, placing values between zero for random mixing and one for maximal segregation.The generator determines how deviations from random mixing are weighted, while the model choice identifies total, social, geographical, or traditional geographical segregation.Source: §4.1.4 · pp. 19–20

4.1.5 Homophilous and heterophilous contributions

• The framework decomposes segregation into nonnegative homophilous and heterophilous contributions and defines a signed homophily measure whose positive values indicate predominantly homophilous departures and negative values predominantly heterophilous departures.The total index equals the sum of the homophilous and heterophilous contributions, while the signed measure lies between -1 and 1.Source: §4.1.5 · p. 20

4.1.6 Recovery of established segregation measures

• Under regional isolation, the framework recovers traditional geographical measures, including the multigroup Dissimilarity, Theil Information Theory, and conventionally normalized squared coefficient of variation indices, while its signed homophily formulation recovers network modularity.It also recovers spatially weighted geographical measures under proximity-kernel connectivity and extends Theil’s regional and sociodemographic decompositions to geosocial networks.Source: §4.1.6 · pp. 20–21

4.2.1 Data sources

• The empirical analysis reconstructs six sociodemographic variables for US ZCTAs from ACS, Decennial Census, CPS ASEC, Pew, and Meta data, and measures regional connectivity with Facebook’s aggregated Social Connectedness Index.Individuals are restricted to age 13 or older, and each region–group cell is a cross-category of Urban, Race, Gender, Age, Education, and Income.Source: §4.2.1 · pp. 21–22, §4.2.2 · p. 22

4.2.2 Region–group cells and census reconstruction

• Joint Census distributions are reconstructed with iterative proportional fitting, then combined with Urban/Rural proportions under within-ZCTA independence to define region–group population counts.IPF matches overlapping ZCTA-level margins while using state-level CPS ASEC distributions for initialization; most ZCTAs are predominantly Urban or Rural.Source: §4.2.2 · p. 22

4.2.3 Facebook-user estimates

• Facebook-user populations are estimated by combining Pew surveys, Meta reach estimates, and Census cells in a Bayesian extension of multilevel regression and poststratification.The model treats Meta reach as noisy measurements of latent age–gender-specific counts and estimates usage conditional on Urban, Race, Gender, Age, Education, Income, and Census division.Source: §4.2.3 · p. 22

4.2.4 Social Connectedness Index and aggregated relational data

• The regional Social Connectedness Index is rescaled using an average Facebook degree of 350 and reconstructed as noisy regional friendship counts for aggregated relational data.The SCI is interpreted, up to a global constant, as the connection probability between randomly selected Facebook users from two regions.Source: §4.2.4 · p. 23

4.2.5 Geographical distance

• Geographical distance is defined between ZCTA centroids, while within-ZCTA distance uses an equal-area circular approximation capped at 80% of the nearest-other-ZCTA distance.These definitions use US Census Gazetteer coordinates for distinct ZCTAs and expected pairwise distance for within-ZCTA connections.Source: §4.2.5 · p. 23

4.3.1 Distance-based and hybrid geosocial models

• Distance-based models combine geographical distance with symmetric group-pair effects, while hybrid models replace distance with marginal geographical intervening opportunities to account for population density.The distance-based formulation is motivated by an approximately power-law decline in SCI connection probability, whereas marginal IOs remain limited by their failure to capture group-specific local density.Source: §4.3.1 · pp. 23–24

4.3.2 Joint intervening opportunities in geosocial space

• The joint IO model counts intervening opportunities in combined geographical and sociodemographic space, producing group-specific geographical ranges and containing hybrid and distance-based models as special cases.Because individual coordinates are unavailable, regional group counts and centroids are evaluated with interpolation and symmetrized into region–group-cell IOs.Source: §4.3.2 · pp. 24–25

4.3.3 Fuzzy intervening opportunities

• Fuzzy IOs retain full weight within the standard boundary and apply exponentially decreasing weights beyond it, with ν = 4 km and k = 15 selected before fitting.The same fuzzy weights construct marginal IOs for the hybrid model and group-specific opportunity counts for the joint IO model.Source: §4.3.3 · p. 25

4.3.4 ARD likelihood and conditional geosocial connectivity

• Conditional ARD likelihoods fix observed regional friendship totals while distributing them across group pairs using Negative Binomial and Dirichlet–multinomial models.The framework avoids relying on unconditional regional connectivity predictions and can extend to partially observed sociodemographic ARD.Source: §4.3.4 · pp. 25–26

4.5 Construction of connectivity and segregation estimates

• The fitted connectivity model defines total segregation, while marginal models define social, geographical, and traditional geographical segregation.The four models are M, MHB, MAB, and MRI.Source: §4.5 · p. 27
• Geo-egocentric edge distributions enable segregation measures and regional and sociodemographic group decompositions for Race, Education, and Income analyses.The distributions are constructed separately for Race, Education, and Income, and jointly for Race with Age, Education, or Income.Source: §4.5 · p. 27
• The fitted model produces countrywide marginal group-to-group connection probabilities by additionally marginalizing over the focal ZCTA.Source: §4.5 · p. 27

Data availability

• The study used publicly available U.S. Census, Pew Research Center Facebook-usage, Meta Marketing API reach, and Social Connectedness Index data.Census sources included the American Community Survey, 2020 Decennial Census Demographic and Housing Characteristics File, and Current Population Survey Annual Social and Economic Supplement; Pew sources were cited as, Meta sources as, and the SCI dataset as.Source: Data availability · p. 27

5 Extended Data

• Marginal connectivity patterns remain qualitatively similar across alternative model fits, although estimated marginal degrees for age, education, and income vary substantially and should be interpreted cautiously.Without age homophily constraints, inferred connectivity shows a slight increase between the youngest and oldest age groups, suggesting the constrained model may be slightly misspecified if that pattern reflects the ground truth.Source: Figure 1 · p. 31
• Using census rather than estimated Facebook counts preserves connectivity patterns except for age, while permuting census or Facebook counts weakens social structure and reduces explanatory power for SCI across all 40 state-level units.With both sources permuted, only distance decay remains clearly detectable; the unperturbed model fits SCI better than models using any perturbed data.Source: Figure 2 · p. 32
• Toy examples show that omitting a confounder or using biased Facebook-usage weights can distort inferred age connectivity, whereas a latent dimension or true usage weights recover the age-marginal target more accurately.The examples isolate omitted structure and misspecified usage as distinct sources of inference error.Source: Figure 5 · p. 35
• Posterior predictive checks accurately reproduce marginal ARD patterns but show incomplete capture of residual spatial autocorrelation, while Facebook usage peaks at ages 25–34 and 35–44 and varies across demographic groups and states.API reach estimates overstate usage for ages 25–34 and 35–44 and understate usage for older groups; the distance-only, IO, and full IO checks assess residual Moran’s I and local variability.Source: Figure 6 · p. 36, Figure 7 · p. 37
• Fuzzy intervening opportunities substantially reduce SCI prediction error at small geographical distances, with negligible effects at large distances; ν ∈ {3 km, 4 km} and k ∈ {7, 10, 15} appear advantageous.This pattern holds across three predictive models and four distance-based data subsets; standard intervening opportunities are the baseline.Source: Figure 8 · p. 38

S1 Segregation framework

• The framework compares edge distributions under network models with no-segregation and maximum-segregation reference models to distinguish geographical, social, and total segregation.The maximum-segregation model maximizes f-divergence from the relevant null over the specified class of models.Source: S1 Segregation framework · p. 40, S1.2 Traditional and spatially… · p. 43
• The proposed measures recover the Dissimilarity and Theil indices, network modularity, nominal assortativity, and Relative Diversity as exact special cases.Total Variation yields network modularity and nominal assortativity through the signed homophily measure, while Relative Diversity appears as another special case.Source: S1 Segregation framework · p. 40
• The framework extends Theil’s regional and group decompositions to geosocial networks and supplies normalizations consistent with traditional Theil decompositions.It also establishes that the proposed segregation measures cannot increase under deterministic coarsening of the coordinate space.Source: S1 Segregation framework · pp. 40–41

S1.1 Traditional geographical segregation

• Traditional geographical segregation indices are f-divergences between the joint region–group distribution and the product of their marginal distributions.Choosing f(x) = x log x yields the unnormalized Theil index, while f(x) = 1/2|x − 1| yields the unnormalized Dissimilarity index.Source: S1.1 Traditional geographical… · p. 41
• Spatially weighted measures address dependence on predefined regions by replacing region-based group proportions with kernel-weighted geographical proportions.Under suitable symmetric kernels preserving marginal group proportions, the weighted Dissimilarity measure is a Total Variation distance and the weighted Information Theory measure coincides with Kullback–Leibler divergence.Source: S1.1 Traditional geographical… · pp. 41–42

S1.2 Traditional and spatially weighted geographical segregation as connectivity models

• A symmetric spatial kernel defines a connectivity model with constant expected degrees, so spatially weighted segregation measures quantify connectivity differences relative to an Erdős–Rényi null model.The kernel’s connection probabilities depend only on geographical locations, whereas the Erdős–Rényi model uses the marginal connection probability.Source: S1.2 Traditional and spatially… · pp. 42–43
• Traditional and spatially weighted indices compare expected fractions of edges ending in each group under a geographical connectivity model and an Erdős–Rényi reference model.The region-based kernel produces a regional isolation model as a special connectivity model.Source: S1.2 Traditional and spatially… · p. 43

S1.3 Coordinate spaces and geosocial network models

• The framework supports full geosocial, geo-egocentric, and social coordinate spaces, with regional coordinates obtained by coarsening point-level geography.Full space uses geographical and sociodemographic coordinates for both individuals; geo-egocentric space retains geography for one individual, and social space retains only group coordinates.Source: S1.3 Coordinate spaces and geosocial… · p. 43, S1.4 Representation of segregation via… · p. 46
• Geosocial networks are modeled by coordinate distributions and conditional probabilities that individuals with coordinates c1 and c2 are connected.The framework also represents models through the joint distribution of coordinates and the connection indicator A.Source: S1.3 Coordinate spaces and geosocial… · pp. 43–44

S1.4 Representation of segregation via counterfactual and reference models

• Total segregation is defined from the observed model, while social and geographical segregation use counterfactual models and traditional geographical segregation uses a regional reference model.The observed model describes empirical or idealized connectivity patterns and coordinate distributions.Source: S1.4 Representation of segregation via… · p. 44
• The homogeneous sociodemographic-space model removes geographical concentration of groups while retaining group-specific connectivity patterns and their geographical variation.It reallocates individuals so every region has the overall population’s sociodemographic composition without changing conditional connection probabilities.Source: S1.4 Representation of segregation via… · p. 44
• The social ambiphily model removes direct dependence of connectivity on group membership while retaining the observed joint region-group distribution and geographical connectivity pattern.Its resulting connection probability depends on the two regions but not on the individuals’ sociodemographic groups.Source: S1.4 Representation of segregation via… · p. 45
• The regional isolation model preserves the observed population distribution and overall network density but restricts connections to individuals within the same predefined region.It therefore measures segregation from the distribution of groups across chosen regions, independently of actual geographical connectivity patterns.Source: S1.4 Representation of segregation via… · p. 45
• Random-mixing and complete-separation reference models preserve the same geosocial marginal connectivities while representing, respectively, no segregation and complete separation between sociodemographic groups.The degree-configuration model controls for expected-degree differences, whereas the degree-constrained social-isolation model allocates connectivity within groups.Source: S1.4 Representation of segregation via… · pp. 45–46
• The reference models can be marginalized to geo-egocentric or social spaces, where admissibility is less restrictive and connection probabilities typically remain valid for large-scale geosocial networks.At full resolution, preserving every geosocial coordinate’s marginal connectivity can require probabilities above one; in the empirical application, geo-egocentric reference-model probabilities remained below one.Source: S1.4 Representation of segregation via… · pp. 46–47

Divergence-based indices

• Segregation is measured by an f-divergence between a model-induced distribution on geo-egocentric geosocial space and its degree-constrained random-mixing null model.The divergence is zero when the induced and null distributions coincide.Source: S1.5 Divergence-based segregation… · p. 47

Normalization

• Normalizing by the corresponding degree-constrained social-isolation model places indices on a common scale, with zero denoting random mixing and one maximal constrained segregation.For conditional coordinate distributions, the normalized index lies between 0 and 1 under the stated admissibility conditions; analogous normalization applies on social space and to joint coordinate–link distributions.Source: S1.5 Divergence-based segregation… · p. 47

Segregation types

• Choosing M, MHB, MAB, or MRI as the defining model yields total, social, geographical, or traditional geographical segregation, respectively.The generator f controls how departures from random mixing are weighted.Source: S1.5 Divergence-based segregation… · p. 47

Scope and limitations

• The geo-egocentric measures avoid labeling geographically uneven connectivity as segregation when group-to-group connectivity is uniform after marginalizing over regions.On full geosocial space, the proposed MHB–AB reference removes direct and indirect social-coordinate dependence but lacks a useful maximum-segregation normalization, which remains open.Source: S1.5 Divergence-based segregation… · p. 48

Classical special cases

• Under uniform marginal connectivity, the normalization constants recover traditional Theil, Information Theory, Dissimilarity, and Pearson χ2 segregation indices for corresponding generators.The Pearson χ2 case corresponds to the squared coefficient of variation.Source: S1.5 Divergence-based segregation… · p. 48

• The proposed geosocial segregation measures contain traditional geographical segregation measures as exact special cases under the regional isolation model and appropriate null models.On geo-egocentric space, comparing regional isolation with the Erdős–Rényi random graph recovers traditional geographical indices; comparing a spatial kernel model with the same null recovers egocentric or spatially weighted indices.Source: S1.6 Recovery of traditional… · pp. 49–50
• The framework also recovers normalized Theil, Dissimilarity, and χ2 geographical indices, while offering a general normalization method rather than index-specific normalizations.The normalized indices correspond to H, D, and χ2, and the framework’s normalization approach generalizes those used separately in the geographical segregation literature.Source: S1.6 Recovery of traditional… · pp. 49–50
• Using the homogeneous sociodemographic space model as the reference null recovers the traditional geographical divergence only for x log x or −log x, and up to a factor of 2; general f does not recover it.This null model substitutes the product of regional and group marginals for the joint distribution, so it is retained mainly as a counterfactual model of social segregation rather than as a general reference null.Source: S1.6 Recovery of traditional… · p. 50
• On full geosocial space, the homogeneous sociodemographic null controls for geographical homophily and yields the same divergence as geo-egocentric space, whereas the random-mixing null does not.On social space alone, the two null models coincide, yet geographical segregation remains meaningfully measurable.Source: S1.6 Recovery of traditional… · pp. 49–50

Choice of coordinate space

• Geo-egocentric geosocial coordinate space provides principled reference models, preserves regional social-connectivity variation, and recovers traditional geographical segregation measures as exact special cases.The degree-configuration and degree-constrained social isolation models are suitable reference models, while Lemma 1.7 establishes recovery of traditional geographical measures.Source: S1.7 Choice of coordinate space and… · p. 50

Information loss

• Full geosocial coordinate space remains necessary for questions about group-specific geographical connectivity or how own-group friendship preferences vary with distance.Geo-egocentric measures retain regional variation in social connectivity, whereas social-coordinate measures lose it; full coordinates additionally expose distance-dependent patterns.Source: S1.7 Choice of coordinate space and… · pp. 50–51
• Geographical coordinate coarsening cannot increase σ-based segregation values and will typically decrease them, establishing a real information loss.This limitation is formalized in Lemma 1.24.Source: S1.7 Choice of coordinate space and… · p. 50

Distributional foundations

• τ captures complete coordinate and connectivity information, while σ provides the natural normalization and recovers traditional geographical segregation measures.τ-based measures also support a natural homophily decomposition through conditional f-divergences.Source: S1.8 Conditional coordinate and joint… · p. 51

Sparse-network equivalence

• As marginal connectivity ¯P approaches zero, unnormalized and normalized σ- and τ-based f-divergence measures become asymptotically equivalent under smooth generators.For Total Variation distance, normalized measures are exactly equal; sparse large-scale geosocial networks therefore make σ and τ practically interchangeable.Source: S1.8 Conditional coordinate and joint… · p. 51

Divergence decomposition

• With equal marginal connectivity, τ-based divergences decompose into connectivity-weighted σ- and ξ-based divergences, with an exact Total Variation relation.The exact relation is Dg,τ = 2¯P Dg,σ = 2(1−¯P)Dg,ξ.Source: S1.8 Conditional coordinate and joint… · p. 51

Illustration

• Figure 1.1 illustrates the lemma and corollary by comparing σ-, τ-, and ξ-based measures across ¯P ∈ (0,1) for several generators.The figure’s first two rows visualize how the three distribution-based measures relate as marginal connectivity varies.Source: S1.8 Conditional coordinate and joint… · p. 52

S1.9 Homophily, heterophily, and network modularity

• The framework distinguishes homophilous from heterophilous departures from degree-constrained random mixing and defines signed homophily as the homophilous contribution minus the heterophilous contribution.For the τ-based measure, purely homophilous departures yield ϕf,τ(M 1q) = Sf,τ(M 1q), whereas purely heterophilous departures yield ϕf,τ(M 1q) = −Sf,τ(M 1q).Source: S1.9 Homophily, heterophily, and… · pp. 52, 54, 55

S1.9.2 Network modularity

• Under Total Variation distance, the signed homophily measure recovers network modularity, while its normalized version recovers nominal assortativity.For the maximum-likelihood stochastic block model fit, ϕg,σ(M̂) = ρA(M̂) = ρA; the τ- and σ-based approaches are equivalent up to the factor 2 P̄.Source: S1.9 Homophily, heterophily, and… · p. 54, S1.9.2 Network modularity · p. 57

S1.9.1 Positive and unique generators

• For σ-based decompositions, the section selects a unique generator representative with zero derivative or centered subdifferential at 1, ensuring non-negative and uniquely defined coordinate-pair contributions.This resolves the non-uniqueness of f-divergence generators while preserving the same overall divergence.Source: S1.9.1 Positive and unique generators · p. 55

S1.9.2 Network modularity

• The generalized modularity measures signed deviations in within-group edge mass, whereas general f-divergences can weight positive and negative deviations differently and favor large deviations over several smaller ones.For KL, Jeffreys, Le Cam, and squared Hellinger generators, large deviations contribute more than several equal-sum smaller deviations, and deficits contribute more than equally sized surpluses.Source: S1.9.2 Network modularity · p. 59
• For sufficiently small perturbations, strictly decreasing generators yield positive signed homophily, while strictly increasing generators yield negative signed homophily.Strictly convex symmetric generators can also guarantee positivity across the stated perturbation range.Source: S1.9.2 Network modularity · pp. 59–60
• Model-comparison modularity decomposes additively across intermediate models, but applying this to social and geographical segregation would require a different expected-degree-preservation approach.The authors leave that decomposition for future work.Source: S1.9.2 Network modularity · p. 60
• Relative Diversity equals network modularity under Total Variation when the edge distribution comes from the regional isolation model MRI.The result recovers both Relative Diversity and its normalized formulation within the framework.Source: S1.9.2 Network modularity · p. 60
• Traditional geographical segregation is always net homophilous under network modularity, with equality only when regional and marginal sociodemographic compositions match.Thus, this measure cannot produce net heterophily from traditional geographical segregation relative to degree-constrained random mixing.Source: S1.9.2 Network modularity · p. 61
• The Dissimilarity index counts both homophilous and heterophilous deviations toward segregation, whereas Relative Diversity subtracts heterophilous from homophilous contributions.For general generators, α_f,σ therefore measures the magnitude of homophilous departures from random mixing directly.Source: S1.9.2 Network modularity · p. 61

• The KL-based measure D_h,σ(M) decomposes into a between component comparing coarse-region connectivity and within components capturing finer-region and group effects relative to the null model.The between component ignores finer regional and sociodemographic membership, whereas each within component measures their additional connectivity impact.Source: S1.10 Regional decomposability · p. 61
• For the regional isolation model, the proposed decomposition exactly equals the traditional regional decomposition of the unnormalized Theil segregation index H.Source: S1.10 Regional decomposability · p. 62
• The decomposition remains meaningful when each region forms its own super-region: the between term captures regional membership, while the within term captures additional group membership effects.Source: S1.10 Regional decomposability · p. 62
• Alternative decompositions can introduce a non-positive third term or concentrate most connectivity information in the between component, so the preferred decomposition depends on the research question.The proposed lemma distributes information more equally across between and within components than the latter alternative.Source: S1.10 Regional decomposability · p. 62

S1.11 Group decomposability

• D_h,σ(M) separates segregation between sociodemographic groups from within-group segregation.The within component measures the additional impact of finer sociodemographic coordinates conditional on membership in a coarse group.Source: S1.11 Group decomposability · p. 63
• For M = M_RI, the regional isolation model, the decomposition exactly equals the traditional Theil-index decomposition.Here, the associated null model is the Erdős–Rényi random graph model.Source: S1.11 Group decomposability · pp. 63–64

• Group normalizations follow the KL chain rule, mirroring the entropy decomposition into between-group and expected within-group components.This applies the chain rule directly to the global normalization constant DKL rather than normalizing each component independently.Source: S1.12 Normalization of regional and… · p. 64
• Regional between normalization is chosen as an upper bound on mutual information over models preserving the model’s regional and group marginals.The construction uses the regional decomposition of the global normalization and takes the bound HσMDC(B) for the between component.Source: S1.12 Normalization of regional and… · p. 65
• The normalization constants are KL-divergences of edge distributions induced by the connectivity model against degree-constrained isolation or configuration-model distributions.For group decompositions, between components use coarse-coordinate edge distributions, whereas within components use distributions restricted to finer groups within each coarse group.Source: S1.12 Normalization of regional and… · pp. 65–66
• Under the regional isolation model MRI, the proposed group and regional normalizations recover the traditional Theil-index normalization constants.When M = MRI, MDC = MER, and MDSI = MSI, both group and regional decompositions match their corresponding traditional constants.Source: S1.12 Normalization of regional and… · p. 66
• The regional within normalizations are conditional components of the global normalization, not solutions to separate restricted normalization problems within each coarse group.Normalizations from directly restricted models can differ from those in Lemma 1.22.Source: S1.12 Normalization of regional and… · p. 66
• The paper adopts Lemma 1.22’s strategy for comparability because an alternative conditional model class does not recover traditional Theil normalizations under MRI.Although preserving conditional marginals could define a valid maximizing class, the relevant marginal constraints differ and the resulting normalization would fail to reproduce the traditional constants.Source: S1.12 Normalization of regional and… · p. 66

• Coordinate coarsening cannot increase the f-divergence for any generator f and each ρ ∈ {τ, σ, ξ}.Source: S1.13 Additional results for the… · p. 67
• For ρ = σ, normalization preserves the same coarsening inequalities across the listed coordinate spaces.The σ-based reference divergence is invariant under coordinate coarsening, supporting the normalized monotonicity.Source: S1.13 Additional results for the… · p. 67
• For the ξ-based measure, coarsening increased the DSI-scaled divergence from ≈0.0911 to ≈0.0997.The comparison marginalizes the region of the second individual.Source: S1.13 Additional results for the… · pp. 67–68
• For the τ-based measure, coarsening likewise increased the DSI-scaled divergence from ≈0.0809 to ≈0.0821.Source: S1.13 Additional results for the… · p. 68

S2.1 Normalization of measures

• The maximal separation from the random-mixing null occurs when the second coordinate is a deterministic function of the first.The proof uses the principle that association is maximized when conditional distributions differ maximally from the marginal distribution, with convexity establishing the relevant bound and equality for the degree-constrained social isolation model.Source: S2.1 Normalization of measures · pp. 68–69

S2.2 Recovery of traditional geographical segregation measures as special cases

• The framework recovers traditional geographical segregation measures as special cases under constant-degree conditions and region-based or homophily-based connectivity models.In particular, the degree-configuration measure equals MER, while specializing the regional isolation model and related kernels yields the stated traditional measure identities.Source: S2.2 Recovery of traditional… · pp. 71, 73, 74

S2.3 Conditional coordinate and joint coordinate–link distributions

• Under smoothness assumptions, a second-order Taylor expansion with uniformly bounded third derivatives yields asymptotic control of the remainder, while squared deviations satisfy the cited O_p bound.Source: S2.3 Conditional coordinate and joint… · p. 76
• Under equal marginal connectivity, the proofs establish the stated conditional-distribution corollaries through Lemma 1.10 and cancellation of common normalization factors.The arguments apply the relevant identities to marginal connectivities, numerator and denominator terms, and multiplicative factors that cancel upon normalization.Source: S2.3 Conditional coordinate and joint… · p. 77

S2.4 Homophily, heterophily, and network modularity

• Under equal marginals πN = πM and equal average transition probabilities, connectivity comparisons correspond to σ comparisons, while f-divergence arguments use a second-order Taylor expansion around 1 and f''(1) > 0.Source: S2.4 Homophily, heterophily, and… · pp. 78, 80

S2.5 Regional and group decompositions

• The regional and group proofs decompose KL divergence into between and within components, then show that MRI special cases recover the stated coarse-coordinate normalization and isolation or configuration models.The group decomposition identifies group-specific and pair-specific within components; marginalizing to coarser coordinates yields the degree-constrained social isolation and degree configuration models.Source: S2.5 Regional and group decompositions · pp. 81, 82, 84
• The regional and group decomposition proofs establish identities relating the model probabilities and marginal connection probability across decomposed coordinates.Source: S2.5 Regional and group decompositions · p. 85

S2.6 Additional results

• Under deterministic coordinate coarsening, the relevant f-divergence cannot increase, while the denominator is unchanged because its likelihood ratio depends only on sociodemographic coordinates.The proof applies the data-processing inequality for f-divergences and then sums over geographical coordinates with fixed sociodemographic coordinates.Source: S2.6 Additional results · pp. 86–87