Using Persistent Homology to Analyze Access to Heterogeneous-Quality Resources and Heterogeneous-Severity Nuisances

1.1. Our Contributions.

• The framework studies heterogeneous-quality resources and heterogeneous-severity nuisances by varying distance and quality or severity together, using a computationally efficient approximation of two-parameter PH.It is designed for any resource or nuisance with a scalar quality or severity measure.Source: §1.1 · p. 2, §1 · p. 2

1.1.1. Mathematical Contributions.

• At each parameter pair, the bifiltration represents areas with insufficient total resource quality or at least a threshold level of nuisance exposure; aggregated 0D components yield geographic regions.For resources the total quality within radius r is below q; for nuisances exposure within radius r is at least q.Source: §1.1.1 · p. 2

1.1.2. Case Studies.

• The Chicago case studies examine poor access to quality parks, landfill overexposure, and pubs and bars as both resources and nuisances.The park analysis assumes proximity is desirable, citing reported physical-health and mental-health benefits; the landfill analysis assumes proximity is undesirable, citing adverse health effects.Source: §1.1.2 · pp. 2–3

1.2.1. Topological Data Analysis for Resource Coverage.

• Prior PH studies of resource coverage used different distance measures, including travel time, waiting time, and Euclidean distance, [29], [53].This paper instead uses L1 distance as a proxy for walking time for resources and Euclidean distance for nuisances.Source: §1.2.1 · p. 3

1.2.2. Topological Data Analysis of Other Geospatial and Social Sys-

• Other PH and TDA work has examined voting, street networks, redistricting, mortality, socioeconomic patterns, and urban phenomena including segregation and COVID-19,, [23], [28],.The pubs example draws inspiration from PH work on the distribution of pubs across UK cities.Source: §1.2.2 · pp. 3–4

1.2.3. Multiparameter Persistent Homology.

• Multiparameter persistence lacks a barcode analogue and complete discrete invariant, so existing summaries trade completeness for computability, stability, or interpretability [8], [42].Because such summaries do not locate a feature in the data, this paper instead identifies geographic regions corresponding to bifiltration features.Source: §1.2.3 · p. 4

1.2.4. Geography and Urban Planning; 1.2.5. Approaches from Network Analysis and Machine Learning.

• Accessibility approaches include 15-minute-city measures, fixed-distance and travel-time thresholds, 2SFCA, and estimates of travel distance or population within thresholds,,.Network and optimization methods have also studied green-space, service, grocery, and polling-site access, [7],,, while mobility embeddings have inferred functional distances [74].Source: §1.2.4 · pp. 4–5, §1.2.5 · p. 5

1.3. Organization of the Paper; 1.4. Data and Code.

• The paper is organized around PH background, bifiltration construction and geographic feature identification, Chicago case studies, and discussion of implications and limitations.Data come from the Chicago Data Portal, EPA LMOP, OpenStreetMap, and Google Reviews; code is available in the cited repository.Source: §1.3 · p. 5, §1.4 · p. 6

2.1. One-Parameter PH.

• A one-parameter filtration is a nested sequence of spaces; PH tracks homology classes through their births and deaths, with 0D classes representing connected components.The section reviews mathematical background on TDA and PH.Source: §2.1 · p. 6, §2 · p. 6

2.2. Naive Example of a One-Parameter Filtration for Resource and Nuisance Applications.

• The case-study data comprise a geographic domain, census-unit discretization, and resource or nuisance sites.Census units are blocks or tracts, and sites are subsets of the geographic domain.Source: §2.2 · p. 6
• A naive superlevel-set filtration marks census units farthest from parks first, but they quickly merge into one component, making conclusions difficult.The example uses public parks in Chicago.Source: §2.2 · p. 7, Figure 1 · p. 7
• One-parameter filtrations use distance to the nearest site, but omit site-specific resource quality or nuisance severity.The proposed bifiltration is intended to address this limitation.Source: §2.2 · pp. 6–7

2.3. Multiparameter Persistent Homology.

• A bifiltration indexes nested spaces by two parameters, and inclusions between spaces induce maps between their homology groups.The paper considers bifiltrations indexed by R^op × R and R × R^op.Source: §2.3 · p. 7

3. Our Distance–Quality Bifiltration.

• Site distances and quality or severity scores define cumulative accessibility or exposure for each census unit, which increases as the radius grows.Resource distances estimate travel along city streets; nuisance distances are Euclidean. Higher site scores represent higher resource quality or greater nuisance severity.Source: §3 · p. 8

3.1. Bifiltration for Resource Sites.

• The resource bifiltration uses sublevel sets: its spaces contain census units whose cumulative quality within radius r is at most q, identifying areas with inadequate resource access.The bifiltration property is that Xr′,q ⊆ Xr,q′ when r ≤ r′ and q ≤ q′.Source: §3.1 · p. 8

3.2. Bifiltration for Nuisance Sites.

• The nuisance bifiltration uses superlevel sets and increasing r, decreasing q; its spaces contain census units with cumulative exposure within radius r of at least q.The bifiltration property is that Xr,q′ ⊆ Xr′,q when r ≤ r′ and q ≤ q′.Source: §3.1 · p. 9, §3.2 · p. 9

3.3. Interpretation of 0D Persistent Homology.

• In both bifiltrations, 0D classes track connected regions of disadvantage: earliest resource components have the poorest access, while earliest nuisance components have the greatest exposure.Resource components appear earliest at largest r and smallest q; nuisance components appear earliest at smallest r and largest q.Source: §3.3 · pp. 10–11

3.4. Computing our Bifiltration Using an Equivalent Graph Bifiltra-

• An equivalent one-dimensional graph bifiltration preserves exactly the same 0D persistent homology as the original spaces, while providing a combinatorially simpler representation.The graph vertices correspond to census units and its edges connect adjacent units in the associated space.Source: §3.4 · p. 11

3.5. Biparameter Component Aggregation.

• A 0D persistent homology class is a parameter-indexed collection of connected components that remains consistent under inclusions between bifiltration spaces.Source: §3.5 · p. 11

3.5.1. Truncated PH Classes.

• Truncated classes exclude components larger than a fixed census-unit threshold because late-stage components can cover nearly the entire region and become uninformative.The threshold can alternatively be based on geographic area rather than census-unit count.Source: §3.5.1 · p. 12

3.5.2. Algorithm to Aggregate Truncated PH Classes.

• The algorithm aggregates truncated classes that are nearly alike by comparing terminal components, then merges provisional classes when 90% of one maximal component is contained in another.This addresses the common case in which many early small components merge over only a few parameter steps, leaving similar classes at later stages.Source: §3.5.2 · pp. 14–16
• Merging all aggregated classes in sets whose maximal components meet the 90% overlap condition makes the result independent of merge order, though merged regions can exceed the truncation threshold and are localized in practice.The merging step yields the final aggregated PH classes.Source: §3.5.2 · p. 16

3.5.3. The Lifetime of an Aggregated PH Class.

• An aggregated PH class’s lifetime is the proportion of bifiltration-parameter values where it is nonempty and not too large, calculated over a finite parameter grid.The lifetime measures the area occupied by the class in the distance–quality plane, subject to the size condition.Source: §3.5.3 · p. 16, §3.5.2 · p. 16

4. Our Case Studies.

• The framework uses maximal components to represent coverage holes or high-exposure regions, treating longer lifetimes as more problematic.The case studies compute aggregated PH classes; the parks example merges classes when one maximal component is contained in another, even with disjoint terminal parameter pairs.Source: §4 · p. 16, Figure 7 · p. 17

4.1.3. Results.

• The five longest-lifetime Chicago park components are in the city’s southern half, where residents must travel farthest for reasonable-quality parks.The six longest-lifetime components span various parts of the city.Source: §4.1.3 · p. 18, Figure 9 · p. 19

4.1.4. Comparison to Our One-Parameter PH Approach.

• For parks, the study uses acreage as quality and L1 distance as a street-distance approximation; the one-parameter filtration merges quickly, while the bifiltration gives a more nuanced coverage picture.The one-parameter components mostly identify areas far from parks, and most Chicago components have short lifetimes because parks are roughly uniformly distributed.Source: §4.1.1 · p. 18, §4.1.2 · p. 18, §4.1.4 · p. 19

4.2.3. Results.

• The most-exposed landfill region is in southwest Illinois, despite not having the highest landfill density or the single largest-waste landfill.The Chicago area and the area slightly southwest of Chicago were not the most exposed when location and waste amount were analyzed simultaneously.Source: §4.2.3 · p. 20, Figure 12 · p. 21

4.2.4. Comparison to Our One-Parameter PH Approach.

• For Illinois landfills, census-tract centroids and landfill waste mass define the analysis; distance-only components are numerous and cover most of the state.Census tracts are used instead of blocks because block-level analysis is computationally demanding statewide; the distance measure is Euclidean.Source: §4.2 · pp. 19–20, §4.2.4 · pp. 20–21, §4.2.1 · p. 20

4.3.2. Distances.

• For pubs, the authors analyze both resource access and nuisance exposure, using total review stars versus review counts and L1 versus Euclidean distance, respectively.The two perspectives use different scores and distance measures to represent pubs’ impacts.Source: §4.3.1 · p. 22, §4.3.2 · p. 22, §4.3 · p. 22

4.3.3. Results.

• Some longest-lifetime poor-coverage components for pubs lie near the northeast’s high-density pub region; for nuisances, smaller components outlast the high-density component.The longest-lifetime nuisance component includes O’Hare, where many travelers pass through and nearby pubs have many reviews.Source: §4.3.3 · pp. 22–23, Figure 14 · p. 23

5.1. Summary.

• The framework identified regions with the poorest resource access or greatest nuisance exposure across public parks, landfills, and bars and pubs.The authors also compared these regions with those found using one-parameter PH.Source: §5.1 · p. 23
• One-parameter PH predominantly captures distance, while the two-parameter framework also accounts for nearby resource quality and nuisance severity, yielding a more complete assessment.Nearby low-quality resources and low-severity nuisances can make distance alone misleading.Source: §5.1 · p. 25

5.2. Discussion.

• The framework requires choices that influence qualitative and quantitative results, although its flexibility allows researchers to explore alternatives using the repository code.Source: §5.2 · p. 25
• Researchers can choose a resource-quality or nuisance-severity measure, but qualitatively different results raise the question of how to compare or choose among them.The paper used total park area and two Google Reviews measures for bars and pubs; it also mentions park reviews and attraction-based scores as alternatives.Source: §5.2 · pp. 25–26
• Size thresholds can use census-block counts or geographic area; the count-based option is more computationally efficient, while area accounts for unequal census-unit sizes.Source: §5.2 · p. 26
• With area thresholds of 10% and 20% of Chicago’s total area, most maximal components were the same, but the 20% threshold included a large southern component absent at 10%.The figure compares maximal components under the two thresholds.Source: §5.2 · p. 26, Figure 18 · p. 26

Appendix A. Data Collection.

• The appendix begins a section providing details about data collection.Source: Appendix A · p. 26

A.1.1. Site Data.

• Chicago park locations and list data came from the Chicago Data Portal, whose dataset includes polygons encoding park geometries.Source: §A.1.1 · p. 26

A.1.2. Quality Score.

• The Chicago Data Portal dataset includes the acreage of each park.Source: §A.1.2 · p. 26

• After duplicate entries were removed, the Illinois data yielded 90 landfills.The source data came from the U.S. EPA’s Landfill Methane Outreach Program; duplicates included separate entries for an original landfill and its expansion.Source: §A.2.1 · pp. 26–27

A.2.2. Quality Score.

• Landfills missing waste-in-place data, measured in short tons, were removed.Source: §A.2.2 · p. 27

A.3.1. Site Data.

• OpenStreetMap amenity tags for bars and pubs produced an initial Chicago set of 949 locations.Source: §A.3.1 · p. 27

A.3.2. Quality Score.

• Google Places checks and manual review narrowed the 949 OpenStreetMap listings to 734 pubs, with 19 records manually updated.Matching used location and name-similarity thresholds selected through trial and error; the authors say the reduction likely reflects outdated OpenStreetMap data.Source: §A.3.2 · p. 27