Urban Expansion Factor in Urban Growth Studies

• Urban Expansion Factor is a set of quantitative constructs that characterize how cities grow, spread, and change through land conversion, thermal dynamics, and scaling processes.
• Studies operationalize it using methods such as radial population scaling, thermal center-of-mass tracking, and conversion probability models, each linking to different aspects of urban structure.
• Diverse applications illustrate its role in assessing density redistribution, morphological anisotropy, and infrastructure influence, highlighting both its potential and definitional challenges.

Searching arXiv for the supplied papers to ground the article in current preprints. Urban Expansion Factor denotes a family of quantitative constructs used to characterize how urban systems grow, spread, reorganize, or intensify. The term is not standardized across the literature. In some studies it refers to a specific scaling parameter, such as the factor $ij$ governing the stretching of a city’s radial population distribution (Peraza-Mues et al., 19 Sep 2025). In others it denotes a shift coefficient in a distributional model of urban land areas (Hu et al., 2023), the movement of a land-surface-temperature center of mass as a proxy for directional growth (Saha et al., 2024), or the combined set of drivers controlling cell-level urban conversion probability in a cellular automaton (Yao et al., 2017). Still other papers do not define a single scalar at all, but operationalize urban expansion through spatial autoregressive choice models, interface roughening exponents, or coupled density–transport equations (Krisztin et al., 2020, Marquis et al., 12 Jun 2025, Barthelemy et al., 9 Mar 2026). A consistent theme is that urban expansion is treated not merely as total land increase, but as a spatio-temporal process involving morphology, anisotropy, conversion mechanisms, density redistribution, and network feedbacks.

1. Conceptual scope and principal meanings

The literature uses the same phrase for different analytical objects because the object of explanation changes across domains. Thermal remote sensing studies seek proxies for expansion in surface warming and directional drift (Saha et al., 2024). Land-use simulation studies define expansion through transition probabilities and their drivers (Yao et al., 2017). Scaling studies define it through transformations of population or land-area distributions (Peraza-Mues et al., 19 Sep 2025, Hu et al., 2023). Spatial econometric and transport studies frame expansion as a choice or allocation process under spillovers and infrastructure constraints (Krisztin et al., 2020, Su et al., 2024). Physics-inspired studies treat it as a roughening interface with anisotropic growth and coalescence (Marquis et al., 12 Jun 2025), while PDE reviews formalize it as the growth of a local urban density field $\rho(x,t)$ (Barthelemy et al., 9 Mar 2026).

Context	Operational meaning	Representative quantity
Summer LST dynamics	Proxy for directional urban development	LST center of mass
RFA-based CA simulation	Learned conversion drivers of non-urban to urban land	$P(k,t)$
Radial population scaling	Stretching or contraction of population footprint	$ij$
Urban land-size distribution	Evolution away from power law	$b$
Surface-growth physics	Land consumption, anisotropy, roughness, coalescence	slope of $A$ vs. $P$; $\alpha_{\text{loc}}, \beta, z$
Spatial multinomial logit	Class-specific spatial spillovers in land take	$\rho_j$
Metro network expansion	Decision criterion for where to expand infrastructure	$C_{od}$

A common misconception is that Urban Expansion Factor is a single scalar index. The record reviewed here indicates the opposite: the phrase is domain-dependent, and in several papers the closest operational quantity is explicitly described as a proxy rather than a formal named index (Saha et al., 2024), or the paper states that no standalone factor is defined (Su et al., 2024, Barthelemy et al., 9 Mar 2026).

2. Thermal and remotely sensed interpretations

A remote-sensing formulation appears in the study of summer land surface temperature (LST) dynamics in Kolkata, Sao Paulo, and Munich (Saha et al., 2024). There, each summer is treated as one time point in a multi-year tensor $\rho(x,t)$0, with each pixel represented by a length-$\rho(x,t)$1 trajectory,

$\rho(x,t)$2

The study uses $\rho(x,t)$3, forms yearly median composites over city-specific warm months, and applies $\rho(x,t)$4-means clustering with $\rho(x,t)$5 to separate two built-up LST groups and two non-built-up or less-urbanized groups. Temporal instability is summarized by the pixelwise standard deviation

$\rho(x,t)$6

followed by Otsu’s thresholding to isolate high-variation zones. Built-up correspondence is evaluated by Intersection-over-Union,

$\rho(x,t)$7

where $\rho(x,t)$8 is the selected LST cluster region and $\rho(x,t)$9 is the built-up map derived from Sentinel-2 NDBI, Otsu thresholding, mean filtering, and re-thresholding (Saha et al., 2024).

Within that framework, the most direct urban-expansion proxy is the movement of the LST center of mass. The latest year’s LST image is thresholded with Otsu, the same threshold is applied to every yearly image, and the center of mass of the resulting warm-pixel mask is tracked over time. The interpretation is geometric: larger shifts indicate more directional, asymmetric expansion; negligible shifts indicate uniform growth or limited expansion. Empirically, Kolkata exhibits a clear increasing LST trend, strong warming in peripheral areas—especially center-east, corresponding to New Town and surrounding growth areas—and an upward and slightly eastward center-of-mass shift, with IoU $P(k,t)$0. Sao Paulo shows peripheral LST increase and outskirts high-variation zones, but little center-of-mass movement, with IoU $P(k,t)$1. Munich shows relatively stable long-term behavior, negligible center-of-mass shift, roughly concentric thermal rings, and IoU $P(k,t)$2 (Saha et al., 2024).

A second remote-sensing line addresses prediction rather than retrospective interpretation. A two-step semantic segmentation plus ConvLSTM pipeline is used to predict urban expansion in Riyadh, Jeddah, and Dammam from multi-date SPOT imagery (Boulila et al., 2021). That work states explicitly that it does not define a formal urban expansion factor. Instead, it produces urban masks through unsupervised segmentation and cleanup, then learns 2015 $P(k,t)$3 2017 and validates 2017 $P(k,t)$4 2019 with a ConvLSTM model using 4 ConvLSTM layers, 3 batch normalization layers, and 1 Conv3D output layer. Its reported outputs are predictive metrics—maximum validation accuracy $P(k,t)$5, Kappa $P(k,t)$6, MSE $P(k,t)$7, RMSE $P(k,t)$8, PSNR $P(k,t)$9, and SSIM $ij$0—plus descriptive indicators such as “about 40% of land cover” changing to urban areas between 2006 and 2013, Riyadh expanding more than three times in size, Jeddah growing over twice, and Dammam expanding by 400% (Boulila et al., 2021). This suggests that, in image-forecasting contexts, the term is often replaced by direct prediction of future urban masks and by descriptive growth indicators rather than by a single latent parameter.

3. Conversion-probability and spillover formulations

A distinct meaning appears in large-scale land-use simulation for China, where urban expansion is modeled as the probability that a non-urban 30 m cell converts to urban land (Yao et al., 2017). In that setting, the “urban expansion factor” is the set of driving variables used to estimate conversion probability: population, distance to provincial capital cities, distance to prefecture-level cities, distance to counties, distance to main railways, distance to main roads, distance to other roads, and DEM slope. Random forest supplies class probabilities through

$ij$1

and the cellular automaton embeds these probabilities as

$ij$2

with neighborhood effect

$ij$3

random disturbance

$ij$4

and the conversion rule $ij$5 (Yao et al., 2017).

This formulation is explicitly heterogeneous across space. China is divided into 19 homogeneous economic development regions using socio-economic indicators and PCA plus clustering. Nationally, the top three drivers are population density, road density, and terrain fluctuation or slope. Regionally, population density is strongest in Northern China, Northeast, Bohai, Inner Mongolia, Xinjiang, and Tibet; terrain becomes almost as important as population in Central China and Southwest China; and road density dominates in economically developed regions such as Beijing, Shanghai, Tianjin, Jiangsu, and Zhejiang (Yao et al., 2017). The model reports OOB error $ij$6, national FoM $ij$7 versus $ij$8 for ANN-based CA, a 70.75% improvement in the paper’s comparison framing, and farmland simulation agreement with standard deviation $ij$9 and $b$0. It concludes that rapid urbanization is the primary factor of farmland loss in China, with total farmland projected to decrease by 5.72% from 2000 to 2030 and annual average loss about $b$1 per year (Yao et al., 2017).

A spatial econometric analogue appears in the Bayesian spatial multinomial logit model for European NUTS-3 regions, which treats urban expansion as a class-specific land-use choice problem (Krisztin et al., 2020). Let $b$2 denote the share of urban expansion in region $b$3 originating from land-use class $b$4. The multinomial probability is

$b$5

and the class-specific log-odds obey a spatial autoregressive structure,

$b$6

Here the operative expansion factor is not one number but the combination of $b$7, $b$8, and the implied direct, indirect, and total effects. The paper reports $b$9 for cropland, $A$0 for forest, $A$1 for grassland, and $A$2 for other natural land, and interprets the significant cropland and grassland coefficients as evidence that land sealing from productive land is spatially clustered (Krisztin et al., 2020).

An infrastructure-planning variant arises in metro network expansion. The reinforcement-learning formulation uses state $A$3, node-selection actions, and reward

$A$4

where satisfied transportation demand is

$A$5

That paper states that it does not define a single scalar urban expansion factor; the effective expansion criterion is increase in $A$6 under constraints from OD flows, population, existing metro structure, and urban geography. In Beijing and Changsha, the proposed framework reports improvements over DRL-CNN reaching $A$7 and $A$8 at budget $A$9 (Su et al., 2024).

4. Scaling parameters and distributional indicators

One of the most explicit uses of the term appears in the analysis of 69 Mexican metropolitan areas from 1990 to 2020 (Peraza-Mues et al., 19 Sep 2025). The city is represented by its radial probability density of population location, $P$0, where $P$1 is distance from the city centre. The later distribution is modeled as a stretched version of the earlier one,

$P$2

and the scaling factor $P$3 is called the urban expansion factor. If $P$4, the distribution stretches outward; if $P$5, it contracts. Because $P$6, the scaling also implies

$P$7

The parameter is estimated by quantile–quantile scaling, using Theil–Sen regression through the origin on quantiles up to the portion where scaling is clearly observed near the centre. The supplementary material reports mean $P$8 (Peraza-Mues et al., 19 Sep 2025).

That framework becomes comparative when $P$9 is decomposed as

$\alpha_{\text{loc}}, \beta, z$0

equivalently

$\alpha_{\text{loc}}, \beta, z$1

The fitted values are $\alpha_{\text{loc}}, \beta, z$2, $\alpha_{\text{loc}}, \beta, z$3, and $\alpha_{\text{loc}}, \beta, z$4. The density-preserving boundary is

$\alpha_{\text{loc}}, \beta, z$5

for density decrease after scaling. Most Mexican cities fall in the “density loss + urban expansion + population growth” region. Numerically, central urban areas lost 2.5 million residents from 1990 to 2020; against a density-preserving counterfactual, the displacement rises to 4.7 million people; the central share drops from more than 40% to 22%; and distances from the city centre increase by 28% on average, corresponding to about 6 meters per year per kilometre from the centre (Peraza-Mues et al., 19 Sep 2025).

A system-level distributional analogue is the shift coefficient $\alpha_{\text{loc}}, \beta, z$6 in the shifted power-law distribution of urban land areas (Hu et al., 2023). The core form is

$\alpha_{\text{loc}}, \beta, z$7

with $\alpha_{\text{loc}}, \beta, z$8 functioning as the central urban expansion indicator. Its temporal evolution is

$\alpha_{\text{loc}}, \beta, z$9

and the coefficients satisfy

$\rho_j$0

In that interpretation, $\rho_j$1 corresponds to a pure power law, large $\rho_j$2 approaches an exponential distribution, and an extreme limit approaches uniformity. The study analyzes 14 regions and countries from 1992 to 2020 and argues that increasing $\rho_j$3 marks a transition from an ordered, heterogeneous system toward a more homogeneous and disordered one, with lower stability and resilience, higher entropy, and higher exposure to extreme heat and PM2.5 pollution (Hu et al., 2023).

The same paper links the dynamics of $\rho_j$4 to globalization and external economies of scale, reports a positive relation between KOF Globalization Index and $\rho_j$5, and forecasts that the power-law phase will shrink drastically, with most regions entering the exponential regime by about 2100 (Hu et al., 2023). This suggests a broad distinction between two scaling traditions: one centered on redistribution of population within cities through $\rho_j$6, and the other centered on transformation of the inter-city land-area distribution through $\rho_j$7.

5. Interface roughness, anisotropy, and PDE-based dynamics

In a surface-growth-physics framework, urban expansion is quantified through the geometry and dynamics of the built-up boundary rather than through a single area or density indicator (Marquis et al., 12 Jun 2025). Using yearly binary built-up maps from the World Settlement Footprint Evolution for 19 cities between 1985 and 2015, cities are defined by the City Clustering Algorithm and the largest connected component (LCC). The relation between built-up area and population is empirically piecewise linear,

$\rho_j$8

The slope is interpreted as inverse density, so larger slope means more land consumption per person and therefore more sprawl, while smaller slope indicates densification. Three empirical patterns are distinguished: linear growth at constant density, piecewise linear growth with densification, and saturation (Marquis et al., 12 Jun 2025).

That framework also measures directional growth. Under isotropic expansion one expects $\rho_j$9 and hence $C_{od}$0, but the measured directional radii satisfy

$C_{od}$1

The observed exponents range from sectors with $C_{od}$2, indicating pinned or stagnant boundaries, to sectors with $C_{od}$3, even up to about $C_{od}$4, indicating fast branch-like extensions. Growth may occur by local interface advance or by coalescence, with aggregates defined as

$C_{od}$5

As demographic pressure

$C_{od}$6

increases, the relative size of aggregates grows roughly as a power law with exponent $C_{od}$7 (Marquis et al., 12 Jun 2025).

Interface roughness is measured by sector-based widths,

$C_{od}$8

with radial scaling ansatz

$C_{od}$9

For small argument, $\rho(x,t)$00, and the central empirical result is

$\rho(x,t)$01

The local roughness exponent is nearly universal across all 19 cities, whereas $\rho(x,t)$02 and $\rho(x,t)$03 vary substantially. The paper also reports anomalous scaling through $\rho(x,t)$04 and

$\rho(x,t)$05

Here Urban Expansion Factor is best understood as a bundle of quantities: the slope of area versus population, anisotropy in directional growth, roughness exponents, and coalescence intensity (Marquis et al., 12 Jun 2025).

A broader mathematical synthesis frames urban sprawl as a non-equilibrium growth process of a local density field $\rho(x,t)$06 (Barthelemy et al., 9 Mar 2026). The generic objective is

$\rho(x,t)$07

with complementary observables $\rho(x,t)$08, $\rho(x,t)$09, and $\rho(x,t)$10. The review treats surface-growth equations such as Edwards–Wilkinson and Kardar–Parisi–Zhang as conceptual templates, notes the empirical $\rho(x,t)$11, and then surveys city-growth PDEs. Ishikawa’s model uses flux

$\rho(x,t)$12

with conservation

$\rho(x,t)$13

Bracken and Tuckwell’s congestion model writes

$\rho(x,t)$14

In these models, diffusion $\rho(x,t)$15, growth rate $\rho(x,t)$16 or $\rho(x,t)$17, carrying capacity $\rho(x,t)$18 or $\rho(x,t)$19, congestion strength $\rho(x,t)$20, and accessibility parameters such as $\rho(x,t)$21 become effective expansion factors because they determine whether growth is compact, sprawling, anisotropic, or transport-led (Barthelemy et al., 9 Mar 2026).

6. Comparative interpretation, limitations, and recurrent issues

Across these formulations, the phrase Urban Expansion Factor organizes three analytically different questions. The first is where urban growth occurs, addressed through pixelwise thermal trajectories, high-variation peripheries, and center-of-mass drift (Saha et al., 2024). The second is why land converts, addressed through conversion drivers, class-specific land rents, population density, and spatial dependence (Yao et al., 2017, Krisztin et al., 2020). The third is how urban form changes, addressed through radial stretching $\rho(x,t)$22, shift coefficient $\rho(x,t)$23, area–population slopes, anisotropy, roughness, and coupled density–transport dynamics (Peraza-Mues et al., 19 Sep 2025, Hu et al., 2023, Marquis et al., 12 Jun 2025, Barthelemy et al., 9 Mar 2026).

Several recurrent findings follow. Built-up growth and thermal intensification are linked, and high temporal variation tends to concentrate in suburban or peripheral regions rather than stable urban cores (Saha et al., 2024). Population growth alone is not sufficient to characterize expansion, because the same population increase may correspond to densification, density-preserving outward growth, or density loss with decentralization (Peraza-Mues et al., 19 Sep 2025). Urban systems also show strong spatial heterogeneity: the dominant drivers of expansion vary by region in China, spatial spillovers differ by land-use class in Europe, and boundary dynamics vary across cities even when local roughness is nearly universal (Yao et al., 2017, Krisztin et al., 2020, Marquis et al., 12 Jun 2025).

The literature also identifies persistent limitations. One is the tendency of many models to rely on monocentric assumptions even though real cities are polycentric and anisotropic (Barthelemy et al., 9 Mar 2026). Another is equilibrium bias: static or equilibrium models miss path dependence, inertia, historical contingency, transitions, and thresholds (Barthelemy et al., 9 Mar 2026). A third is definitional inconsistency. Some papers explicitly define an Urban Expansion Factor, such as $\rho(x,t)$24 (Peraza-Mues et al., 19 Sep 2025); others use the phrase informally for a set of drivers (Yao et al., 2017); others provide only the closest analogue, such as $\rho(x,t)$25 (Hu et al., 2023); and others state that no single scalar is defined (Su et al., 2024, Barthelemy et al., 9 Mar 2026). This suggests that cross-paper comparison requires identifying the object being measured—density redistribution, land conversion, morphology, infrastructure expansion, or system-level land-size structure—before interpreting numerical values.

In encyclopedic terms, Urban Expansion Factor is therefore best understood not as one canonical metric but as a category of formal devices for quantifying urban growth. Depending on the modeling tradition, it may be a scaling coefficient, a probabilistic driver set, a roughness parameter, a spillover coefficient, a drift proxy, or an objective function. The unifying idea is that urban expansion is a spatio-temporal process whose measurement depends on whether the analytical emphasis falls on land conversion, thermal footprint, spatial redistribution of residents, boundary roughening, or infrastructure-enabled accessibility.