Urban systems comprise a complex interplay of urban facilities (e.g., grocery stores, healthcare facilities), road networks, and human populations. The efficiency and equity of population-facility interactions significantly impact residents' well-being. However, urban systems face substantial risks from natural and man-made disasters, leading to disruptions in access to facilities and a decline in urban life quality. Improving urban system resilience is crucial for maintaining essential life activities and meeting the demands of urban populations. Resilience is defined as a system's ability to absorb stresses and maintain functionality during disasters. This study focuses on urban system resilience as the ability to retain access to facilities and the extent to which available facilities can meet population demand. While numerous studies have examined resilience in individual urban systems (transportation networks, businesses, communities), they often overlook the complex interactions between population and facilities and the crucial aspect of equitable access. Recent research has highlighted the need to integrate equality considerations into resilience analysis, acknowledging the unequal impacts of disruptions among different urban populations. Existing inequality research in urban studies has concentrated on definitions, metric development, and quantification methods, focusing on quantifying differences in parameters based on social or demographic stratifications. These studies have examined inequality in various aspects, such as neighborhood isolation, mobility segregation, and differential impacts of emergencies, often based on socio-demographic or geographic factors. However, tools and approaches for reducing inequality in urban systems remain limited. A major constraint has been the limited granularity of available data. Previous studies using longitudinal surveys, randomized experiments, physical sensors, and social media data have lacked the spatial and temporal resolution needed to accurately capture human movement and facility locations. The wide availability of smartphone data offers a new opportunity to address these limitations. These data sets provide detailed temporal and spatial information on human activities, allowing researchers to infer lifestyle patterns and facility demand. This study leverages the opportunities presented by smartphone location data and expands the body of work by developing a granular and generalizable model that connects inequality to urban system resilience through an effective distance variable. The study employs an optimization model to minimize total travel distance for urban populations to access facilities, maximizing equality of access. A second model assesses the resilience of systems in optimal and empirical scenarios based on facility distribution changes. The study aims to experimentally validate the measure of effective distance embeddedness following short-term disruptions, to understand the impact of facility disruption and how improved equality of access enhances resilience.

Existing literature on urban resilience has primarily focused on the resilience of individual urban systems, such as transportation networks, business facilities, and communities, employing various methods and offering insights into the impacts of disasters and recovery processes. However, a significant gap exists in considering the complex interplay between population distribution and facility accessibility, and how this interaction influences both equality and resilience. Prior studies on inequality in urban systems have focused on developing metrics and quantification methods to measure differences in various parameters based on social or demographic stratification. However, these studies often lack the granularity of data necessary to effectively address the challenge of spatial inequality, particularly the impact of facility distribution. The use of smartphone data in previous studies has been limited, often lacking the detail needed to adequately study population-facility interactions and their impact on access equality. This study builds upon these previous efforts by employing a novel combination of large-scale datasets and analytical techniques to fill this gap and provide a more comprehensive understanding of the relationship between equality and resilience in urban systems.

This study utilizes four large-scale datasets: US Census data for population estimates, anonymized smartphone location data from Veraset Inc. for population movement patterns, facility information from SafeGraph, and road networks from OpenStreetMap. The data are preprocessed by creating a grid map dividing each county into 0.5 km x 0.5 km grids. Each grid is associated with attributes such as population density, facility count, and travel costs. Population distribution is estimated using both Census data and smartphone location data. The smartphone data, consisting of over 30 million devices, are filtered for ten metropolitan counties and processed using DBSCAN clustering to estimate home locations. The number of devices in each grid, scaled by an amplification factor derived from comparing device counts to census data, estimates the population density. Facility distribution is obtained from SafeGraph, including facility category (NAICS code), location, and capacity. Capacity is estimated by assuming the total capacity of each facility type is equivalent to the total demand in the county and distributed evenly across facilities of the same type within the county. Travel costs are estimated by considering shortest paths between grids using OpenStreetMap road network data, simplifying calculations by summing travel costs between neighboring grids along routes. The Urban Centrality Index (UCI), which ranges from 0 (polycentric) to 1 (monocentric), is used to measure population and facility distribution. The optimization model, formulated as an integer program (p-median problem), aims to minimize total travel distances to facilities by assigning each residential grid cell to its nearest facility and considering both the redistribution of existing facilities and the addition of new facilities. A fast interchange heuristic is employed to solve the optimization problem. The resilience model uses a bipartite network representing dynamic interactions between population and facilities. This model incorporates the rate of change in population demand (λ), facility capacity, and effective distance (w_eff) between facilities. Effective distance represents the average weighted distance between facilities and is used to simplify the high-dimensional network to a one-dimensional model. The model's steady-state solution provides the fraction of satisfied demand as a function of effective distance. Resilience is quantified by comparing effective distance in optimal and empirical scenarios under simulated facility disruptions (random removal of a fraction of facilities), analyzing the change in w_eff and its impact on satisfied demand. Experiments involving random facility distribution are conducted for comparison. The study assesses both the effects of facility disruptions and the potential for improved equality of access to enhance resilience capacity.

Analysis of ten US metropolitan counties reveals that existing facility distributions are inconsistent with population distributions, leading to unequal access. Facilities are concentrated in central areas, while populations are more evenly distributed. This pattern is observed across all ten counties regardless of size, shape, or population. The complementary cumulative distribution function (CCDF) shows sharper slopes for facility distributions compared to population distributions, indicating monocentric facility distributions and polycentric population distributions. The Urban Centrality Index (UCI) confirms this, with facility UCIs significantly higher than population UCIs. Unequal travel costs result, with residents in some areas traveling significantly longer distances to access facilities. Optimization of facility distribution significantly reduces average travel costs and improves equality of access across all counties and facility types. Comparing optimal, existing, and random facility distributions demonstrates that the optimal distribution effectively minimizes travel costs without dramatically altering the spatial structure of cities. Analysis of the resilience of population-facility networks, defined as the ability to maintain a high fraction of satisfied demand during facility disruptions, shows that enhanced equality of access through optimal facility distribution significantly increases resilience. The resilience model shows that the effective distance embeddedness (w_eff), which accounts for spatial distribution and inequality, is strongly associated with resilience. Optimal scenarios consistently exhibit a 10-30% higher w_eff compared to existing scenarios under various levels of facility disruption. This indicates a substantial increase in the network's ability to maintain satisfied demand during crises. This enhanced resilience is directly linked to the improved equality of access achieved through optimization. The study demonstrates a clear correlation between resilience and equality of access to urban facilities.

This study's findings directly address the research question by demonstrating a significant correlation between equality of access and resilience in urban population-facility networks. The results highlight the importance of considering both equality and resilience jointly in urban planning and development. The inconsistency between population and facility distributions, resulting in unequal access, increases vulnerability to disruptions. Optimizing facility distribution improves both equity and resilience, reducing travel costs and increasing the ability of the system to maintain functionality during crises. The study demonstrates the practical implications of considering these factors together in urban planning by providing a data-driven methodology for improving facility distributions to enhance both equality and resilience. While this study focuses on facility shutdowns, future research should examine the effects of road disruptions and transportation mode choices on these relationships. Furthermore, incorporating social factors such as income and demographic considerations into the models will contribute to a more comprehensive understanding of equitable resilience.

This research presents novel models connecting the resilience and inequality of urban systems, demonstrating the correlation between equitable access and resilience to facility disruptions. Optimizing facility distribution minimizes total travel costs, improving equity and enhancing resilience by 10-30%. The findings underscore the need for a joint consideration of equality and resilience in urban planning. Future research should explore the effects of road disruptions, diverse transportation modes, and social equity factors to further refine the models and their application in urban development.

The study makes several assumptions, including the equal capacity of facilities of the same type, stable facility capacity during the study period, and the use of travel distance as the sole measure of accessibility. The model does not explicitly account for social factors such as income and demographic differences, which can impact access to facilities. Additionally, the study focuses on short-term shocks related to facility shutdowns, without considering other types of disruptions like road closures. Future research should address these limitations for a more comprehensive understanding of the complex interplay of equality and resilience in urban systems.