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arxiv: 2512.10753 · v2 · submitted 2025-12-11 · 💻 cs.CG · cs.SI

Quantifying displacement: an urban expansion consequence via persistent homology

Rita Rodr\'iguez V\'azquez , Manuel Cuerno This is my paper

Pith reviewed 2026-05-16 23:16 UTC · model grok-4.3

classification 💻 cs.CG cs.SI
keywords urban displacementpersistent homologycubical complexestopological data analysisaddress change datapopulation movementMadrid
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The pith

Persistent homology on cubical complexes from address changes quantifies urban displacement and flags affected neighbourhoods and years.

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

The paper builds four cubical complexes from public address change records that embed both location and time, then applies persistent homology to extract topological signatures of gradual population movement. In a twenty-year Madrid study this reveals displacement patterns and pinpoints specific neighbourhoods and years that remain invisible when the same address records are examined directly. A sympathetic reader would care because displacement is a slow process tied to housing pressures, yet cities have lacked replicable ways to measure it across long spans and different places using only existing data.

Core claim

Constructing four cubical complexes that simultaneously encode geographical and temporal information from address changes, then analysing them with persistent homology, captures displacement as an involuntary residential process and identifies the neighbourhoods and years affected in the Madrid case, patterns that raw address data alone do not display.

What carries the argument

Four cubical complexes built from address change records that incorporate spatial and temporal dimensions, analysed by persistent homology to detect topological features associated with displacement.

If this is right

  • Displacement can be tracked as a gradual process over multi-decade periods using only publicly available address records.
  • The method isolates specific neighbourhoods and calendar years experiencing elevated displacement in the Madrid study.
  • Topological features expose movement patterns that direct inspection of address change counts does not reveal.
  • The same construction of cubical complexes and homology analysis can be repeated on address data from other cities and time windows.
  • Quantification becomes possible without requiring new data collection beyond existing public records.

Where Pith is reading between the lines

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

  • The same pipeline could be run on address data from additional cities to compare how different housing regulations affect displacement rates.
  • Detected persistent features might be checked against concurrent rent or property-value data to test links between market pressures and observed movement.
  • Periodic re-application to fresh address updates could serve as an early-warning system for emerging displacement zones.
  • The approach may generalise to other slow social shifts such as community turnover or service-access changes when suitable longitudinal location data exist.

Load-bearing premise

Address change records accurately stand in for involuntary displacement and the persistent topological features specifically signal displacement rather than other kinds of moves.

What would settle it

If the neighbourhoods and years highlighted by the persistent homology analysis do not align with those identified by independent local surveys or census data on forced residential moves.

Figures

Figures reproduced from arXiv: 2512.10753 by Manuel Cuerno, Rita Rodr\'iguez V\'azquez.

Figure 1
Figure 1. Figure 1: Madrid map and their 21 districts as of April 2025. All the data used in this study is publicly available on the official website of the city council [4]. The main data sources are the tables in the ‘C. Demograf´ıa y Poblaci´on’ section, specifically those containing information about address changes with origin inside the city. This includes information on the number of residents who moved, their destinat… view at source ↗
Figure 2
Figure 2. Figure 2: Grayscale image depicting the percentage of people moving in 2023 that stayed in the same neighbourhood (left) and resulting 100 × 100 covering the city map (right). sequences of cubical complexes indexed by a parameter r ranging from 100 to 0: recall that for a fixed value r, a voxel c is selected if V (c) ≥ r. The resulting filtered complexes preserve the neighbourhoods geography. For a more technical ex… view at source ↗
Figure 3
Figure 3. Figure 3: Example of the cubical complex for people moving that stayed in the same neighbourhood obtained by selecting the parameter value r = 30. We then compute persistent homology for each of the four resulting filtered cell complexes using CubicalRipser, an extension of the Ripser Python library designed for cubical complexes [32]. The reason for this choice is three-fold: It supports the T–construction for cubi… view at source ↗
Figure 4
Figure 4. Figure 4: Example of developing persistence for the group of people that moved from Madrid to another city within the same region. The three images at the top row show the cubical complexes for people who moved from Madrid to another city within the same region obtained by selecting the parameter values r = 10, 30 and 50. The bottom row shows the barcode obtained for this filtered complex, displaying the persistence… view at source ↗
Figure 5
Figure 5. Figure 5: Persistence diagram of the filtered complex for the group of people who moved from Madrid to another city within the same region, displaying the H0, H1 and H2 features. Points close to the diagonal present low persistence. supports the idea that the search for more affordable housing is the most likely driver of moves out of the city, allowing us to infer population displacement. We consider two destinatio… view at source ↗
Figure 6
Figure 6. Figure 6: First two connected components born in the group ‘city’. Both cover only the neighbourhood 27 Atocha. In the ‘Comunidad de Madrid’ group, the first connected components are born at the city border in distant neighbourhoods and first expand to further years, i.e., along the vertical axis, resulting in high persistence. They do not necessarily imply substantial displacement, as residents may simply relocate … view at source ↗
Figure 7
Figure 7. Figure 7: Map of central-south Madrid with neighbourhoods 13, 16, 27 and 35 highlighted. District boundaries are outlined in solid lines and neighbourhood limits by dashed lines. during the 2020-2021 COVID pandemic, exemplifies a housing market relaxation and demonstrates our approach’s ability to capture subtle population displacement patterns. We now examine H2 features, which may exhibit complex dynamics. Note th… view at source ↗
Figure 8
Figure 8. Figure 8: Map of Madrid with districts 13, 15 and 18 highlighted. City boundary is outlined in black [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of the birth times of the topological features of each group, split by dimension. displacement, marked by a persistent cavity in ‘stay’ from 2013-2023 containing it, was briefly interrupted during the COVID-19 pandemic, as shown by a ‘city’ cavity. This highlights the neigh￾bourhood’s distinct responsiveness to market forces, which distinguishes it from its surroundings and acts as a focal poi… view at source ↗
Figure 10
Figure 10. Figure 10: Madrid map and their 21 districts as of April 2025. not at the border, and leaving them blank until 2017 would severely interfere with the topology of the 3D cubical complexes we built. As a consequence of the changes undergone in the neighbourhoods of these two districts, we perform the following modifications to the tabular data: • From 2004 to 2016, we consider the whole district instead of individual … view at source ↗
Figure 11
Figure 11. Figure 11: Madrid map and their 131 neighbourhoods as of April 2025, coloured by district. the map in the areas where 193 and 194 are located, as well as that the change in location for 192 is correctly taken into account. • From 2017 on, we keep data as is for the four neighbourhoods in district 19. • We follow a similar approach to correctly deal with the creation of neighbourhood 183 Ensanche de Vallecas. As its … view at source ↗
Figure 12
Figure 12. Figure 12: Figures 12a and 12b depict the changes undergone in the neighbour￾hoods in district ‘19 Vic´alvaro’ occurred in 2017. The district area remained un￾changed, but the subdivision into neighbourhoods changed. On the top image it can be observed that in 2006, a large part of this district had not been developed yet. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Figures 13a and 13b the changes in district ‘18 Villa de Vallecas’ in 2017. The district area remained unchanged, but the subdivision into neighbour￾hoods changed. On the top image it can be observed that in 2006, a large part of this district had not been developed yet. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Violinplots of the distribution of the birth and death times of the topological features of each group. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Cubical complexes for the group ’stay’ at filtration-parameter values 65, 70 and 75. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Cubical complexes for the group ’city’ at filtration-parameter values 30, 40 and 50. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Cubical complexes for the group ’Comunidad de Madrid’ at filtration￾parameter values 60, 70 and 80. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Cubical complexes for the group ’outside’ at filtration-parameter values 60, 70 and 80. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
read the original abstract

Population displacement is a housing-related involuntary residential dislocation. It has become increasingly widespread in many cities, particularly in neighbourhoods undergoing rapid economic and demographic change, and measuring it is essential to assess the social consequences of urban transformation and housing market pressures. Despite its relevance, quantifying displacement presents difficulties due to limited replicability across cities and time periods and the need to analyse long time spans: displacement is a gradual process, impossible to capture in one data snapshot. We introduce a novel tool to overcome these difficulties. Using publicly available address change data, we construct four cubical complexes simultaneously incorporating geographical and temporal information of people moving, and analyse using Topological Data Analysis tools. Finally, we demonstrate this method through a 20-year case study in Madrid, Spain. The results reveal its ability to capture displacement and identify the neighbourhoods and years affected--patterns not observable from raw address change data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a method to quantify population displacement using persistent homology applied to four cubical complexes built from publicly available address-change records that encode both geographic and temporal movement information. It presents a 20-year case study on Madrid data and claims that the resulting 0- and 1-dimensional persistent features identify affected neighbourhoods and years in patterns invisible from raw address counts.

Significance. If the topological features can be shown to isolate involuntary displacement rather than general mobility, the approach would supply a replicable, data-driven framework for tracking gradual urban displacement over long time spans using only public records. The simultaneous incorporation of space and time in cubical complexes and the emphasis on publicly available inputs are positive features for reproducibility.

major comments (3)
  1. [Methods] Methods section on complex construction: the description of how address-change records are mapped into the four cubical complexes and the choice of filtration parameters (grid resolution, time binning, and persistence thresholds) is insufficiently detailed. These choices directly determine the persistent homology output and are therefore load-bearing for the claim that the detected features capture displacement.
  2. [Results] Results and case-study sections: the central interpretation that persistent 0- and 1-dimensional features specifically flag involuntary displacement (rather than voluntary moves or administrative noise) is asserted without any external validation step, such as correlation with Madrid eviction records, known gentrification timelines, or survey-based displacement measures for 2000–2020. This leaves the claim that TDA reveals “patterns not observable from raw address change data” unsupported.
  3. [Results] Results section: no quantitative comparison (e.g., precision-recall against ground-truth displacement events or statistical tests against null models of random mobility) is provided to demonstrate that the topological summary outperforms simple density counts. This weakens the assertion of added value.
minor comments (2)
  1. [Abstract] Abstract: the four cubical complexes are mentioned without stating what distinguishes them (e.g., different filtrations or projections); a brief clarifying clause would improve readability.
  2. [Methods] Notation: the manuscript uses “cubical complexes” and “persistent homology” without referencing standard definitions or software (e.g., Cubical Ripser or Gudhi); adding one or two citations would aid readers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments on our manuscript. We have addressed each of the major comments point by point below, making revisions to the manuscript where appropriate to improve clarity and support for our claims.

read point-by-point responses

  1. Referee: [Methods] Methods section on complex construction: the description of how address-change records are mapped into the four cubical complexes and the choice of filtration parameters (grid resolution, time binning, and persistence thresholds) is insufficiently detailed. These choices directly determine the persistent homology output and are therefore load-bearing for the claim that the detected features capture displacement.

    Authors: We agree with the referee that additional details are necessary for reproducibility. In the revised version, we have substantially expanded the Methods section. We now provide a detailed description of the mapping process from address-change records to the four cubical complexes, including the specific spatial grid resolution of 200m x 200m, temporal binning into yearly intervals, and the filtration parameters used, such as the maximum filtration value and persistence thresholds for considering features as significant. We have also included a figure illustrating the construction pipeline and pseudocode for the complex building algorithm. revision: yes

  2. Referee: [Results] Results and case-study sections: the central interpretation that persistent 0- and 1-dimensional features specifically flag involuntary displacement (rather than voluntary moves or administrative noise) is asserted without any external validation step, such as correlation with Madrid eviction records, known gentrification timelines, or survey-based displacement measures for 2000–2020. This leaves the claim that TDA reveals “patterns not observable from raw address change data” unsupported.

    Authors: We acknowledge that external validation would provide stronger evidence. Unfortunately, comprehensive public records of involuntary displacements (e.g., evictions) for the entire 2000-2020 period in Madrid are not available at the neighborhood level, which motivated our use of an unsupervised topological approach. We have revised the manuscript to include a dedicated limitations subsection discussing this issue and have softened the language around the interpretation to indicate that the features highlight anomalous mobility patterns consistent with displacement, while noting the need for future validation with additional data sources. We have also added comparisons to known gentrification timelines from literature where available. revision: partial

  3. Referee: [Results] Results section: no quantitative comparison (e.g., precision-recall against ground-truth displacement events or statistical tests against null models of random mobility) is provided to demonstrate that the topological summary outperforms simple density counts. This weakens the assertion of added value.

    Authors: We have added quantitative comparisons in the revised Results section. Specifically, we generated null models by randomizing the mobility data while preserving marginal distributions and computed persistence diagrams for these. Statistical tests (e.g., Wasserstein distance between diagrams) show significant differences (p-value < 0.05) from the observed data. Additionally, we provide a side-by-side visual and quantitative comparison showing that persistent features identify specific neighborhoods and years not apparent from raw address change counts alone. While we do not have labeled ground-truth for precision-recall metrics, these additions demonstrate the added value of the topological approach. revision: yes

Circularity Check

0 steps flagged

Standard TDA pipeline on data-derived complexes; no self-referential reduction

full rationale

The paper constructs four cubical complexes directly from address-change records (geographic-temporal data) and applies persistent homology, a standard, parameter-free computational procedure whose output features are not defined in terms of the target displacement interpretation. No equations reduce a claimed prediction to a fitted input by construction, no self-citations bear the central load, and no ansatz or uniqueness theorem is smuggled in. The interpretive step that persistent features indicate involuntary displacement is an external modeling assumption, not a circular derivation within the mathematics itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard assumptions of topological data analysis and the interpretation of address changes as displacement proxies; no free parameters or invented entities are specified in the abstract.

axioms (1)
  • domain assumption Persistent homology on cubical complexes built from spatio-temporal data yields features that correspond to displacement processes.
    Invoked when claiming that detected patterns capture displacement not visible in raw data.

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