Development and Implementation of a Graph-Based Framework for Socio-Economic Resilience in Urban Systems

Abstract

Urban systems are becoming increasingly complex due to rapid urbanization, socio-economic disparities, and climate change. Addressing these challenges requires innovative approaches that integrate data-driven methodologies with resilience planning. This paper presents a novel extension to the Urban System Abstraction Hierarchy (USAH) framework by integrating socio-economic indicators into a graph-based modeling environment, enabling a more holistic understanding of urban resilience. Our approach advances existing models by operationalizing multi-domain resilience through a graph-based framework that captures complex interdependencies across critical infrastructure, governance, finance, and vulnerable populations. Unlike prior USAH applications, which focused primarily on acute shocks, the proposed model captures interdependencies across infrastructure, environmental conditions, health systems, economic robustness, public finance, and social cohesion. Several graph metrics were analyzed including betweenness centrality, and system-level resilience metrics. Sensitivity testing of the indicator weighting scheme showed that increasing the influence of the network structure from 0.7 to 0.9 betweenness centrality shifts indicator importance toward structurally central nodes while reducing the influence of sub-indicator averages. However, the system-level resilience remained unchanged across scenarios. Beyond traditional centrality measures, we introduce new network metrics that identify system stabilizers, key policy leverage points, cross-domain dependencies, and overall structural fragility. Together, these measures transform the model from a descriptive mapping tool into a practical decision-support framework for resilience planning.

1. Introduction

Urban systems are inherently complex, encompassing interrelated socio-economic, environmental, and infrastructural components. With the growing challenges posed by rapid urbanization, climate change, and socio-economic disparities, fostering resilience in urban systems has become a critical priority. Resilience refers to the capacity of urban systems to adapt, recover, and thrive amidst disruptions while maintaining essential functions and promoting sustainable development. This concept has gained prominence in policy and research, emphasizing the need for innovative approaches to resilience planning.

Graph networks have emerged as a transformative tool in addressing the complexity of urban systems. By employing graph network analysis, planners and policymakers can model urban systems more intuitively, simulate potential disruptions, and develop strategies to mitigate vulnerabilities. Existing studies have demonstrated the utility of graph frameworks in urban applications ranging from digital twins to disaster management.

Despite these advancements, significant gaps remain in current urban resilience frameworks. Most models struggle to integrate multiple resilience indicators across domains or to provide actionable insights into synergies and trade-offs among stakeholders. Furthermore, managing the vast, heterogeneous data generated by urban systems presents an ongoing challenge. Addressing these limitations requires the development of graph-based models that enhance data integration, ensure secure access, and support evidence-based decision-making.

McClymont [1] introduced the USAH model, demonstrating its potential in mapping urban interdependencies and responses to specific events, such as flooding. However, existing models have a limited capacity to address broader chronic stressors and multi-hazard scenarios [2,3]. A more comprehensive, system-wide approach is needed to enhance USAH modeling, incorporating multi-dimensional resilience strategies that enable cities to withstand, adapt to, and evolve in response to various challenges.

While the foundational USAH model provided a valuable framework for examining urban resilience by capturing the interconnections between physical and functional systems, it primarily focused on acute shocks and specific hazards. However, as urban environments increasingly grapple with chronic stressors, such as socio-economic disparities, air pollution, and long-term climate impacts, existing frameworks alone prove insufficient for supporting sustainable urban management. Our approach builds on and significantly extends this earlier work. We propose a data-driven, graph-based framework that moves beyond current practices by structurally integrating multi-domain resilience indicators, spanning not only infrastructure and environment but also health, economic robustness, public finance, and social cohesion. The model leverages graph-based frameworks to visualize and analyze these interconnections, enabling planners to identify critical nodes, simulate system vulnerabilities, and design targeted interventions that reflect the lived complexity of urban life. This expanded approach captures both the depth of chronic stressors and the breadth of socio-technical systems, offering a more complete, adaptive, and actionable resilience planning tool.

This expansion also addresses the rigid segmentation within traditional resilience frameworks, which often treat physical and social systems as separate entities. By leveraging graph framework methodologies, we propose a scalable and actionable framework that synthesizes insights from multi-dimensional resilience indices and urban governance models. The enhanced USAH model employs network analysis to map and quantify cascading effects of disruptions across multiple domains, thereby supporting both immediate response strategies and long-term planning.

The development of this extended framework is part of the multi-phase URSA (Urban Resilience and Sustainability Alliance) project, initiated to bridge the gap between technical resilience modeling and socio-economic realities in Canadian cities. Mosaic I began by identifying key urban resilience factors and their associated indicators. Mosaic II expanded this work through stakeholder engagement, capturing institutional knowledge and social context to validate and refine those indicators. Mosaic III, outlined in this paper, represents the project’s methodological culmination: the implementation of a graph-based framework capable of storing, querying, and visualizing the full spectrum of resilience data, starting with a testbed application in the city of Vancouver.

A key contribution of this study is the introduction of network-based resilience diagnostics (CECS, ECS, MDPS, IDDR, WDD), which move beyond traditional centrality measures. Existing graph-based planning models often rely on node-level metrics, whereas this study demonstrates the importance of edge-level and path-based analysis in identifying systemic leverage points and hidden dependencies. This advances graph-based urban planning from a visualization tool toward a decision-support framework.

This paper explores the development and implementation of a graph-based framework designed to enhance socio-economic resilience in urban systems, with a focus on Vancouver as a case study. It aims to integrate resilience indicators across critical dimensions, including infrastructure, health, the environment, and governance. By mapping stakeholders, normalizing indicators, and leveraging graph theory, the study seeks to provide a scalable and robust framework for urban resilience planning. This approach not only fills critical gaps in existing models but also sets a foundation for more resilient urban futures.

1.1. Review of Graph Theory and Urban Resilience Indicators

Innovative city platforms play a vital role in enabling the creation of innovative city applications by offering high-level services that developers can reuse. Ensuring interoperability and efficiently managing large volumes of diverse data is crucial to accomplish this. Additionally, these platforms must provide secure access to data, which requires implementing strong security measures to safeguard sensitive information from unauthorized access or breaches. Protecting data privacy and confidentiality is equally important, particularly when handling personal or sensitive information [3].

Monteiro et al. [4] present a comparative evaluation of multiple graph data management systems using the Linked Data Benchmark Council’s Social Network Benchmark (LDBC SNB). The analysis examines performance across key dimensions, including query execution time, data loading efficiency, and computational resource utilization (CPU and memory). The results demonstrate significant variation in system performance, particularly in scalability and responsiveness as data volumes increase. Systems with optimized query processing architectures exhibited lower latency and more efficient handling of complex queries, while others showed comparatively higher resource consumption and reduced performance during data ingestion and large-scale query execution. These findings emphasize the critical role of system efficiency, scalability, and resource management in the effective deployment of graph-based analytical frameworks for data-intensive applications.

Regarding urban resilience, graph networks have been instrumental in modeling complex urban systems. Their ability to efficiently store and analyze interconnected data supports decision-making processes aimed at enhancing city resilience. For instance, in Liu’s work, a graph network has been utilized to create digital twins of urban environments, facilitating real-time monitoring and optimization of city operations [5].

Graph theory has been widely applied in urban systems to model complex interrelations among city elements such as infrastructure, stakeholders, and environmental factors. Recent literature highlights the role of graph networks in urban planning by enabling scalable and efficient storage of nodes and relationships [6,7,8]. Furthermore, resilience indicators have been used to measure urban sustainability across domains such as social, economic, infrastructure, and environmental resilience [9].

Existing urban resilience frameworks, such as the City Resilience Index [10] and Urban Systems Abstraction Hierarchy (USAH) [11], have laid the foundation for assessing city resilience in response to economic and environmental disruptions. However, most frameworks lack integration of multiple resilience indicators across domains, and current models do not incorporate network-driven insights like synergies and trade-offs among stakeholders. Graph network analysis addresses this gap by mapping complex urban relationships, enabling more effective urban resilience planning.

Recent studies have highlighted the effectiveness of graph network analysis in enhancing urban resilience planning [12]. For instance, Bixler et al. [13] employed social network analysis and exponential random graph modeling to examine urban governance structures, revealing that collaborative networks among stakeholders significantly bolster resilience implementation.

1.2. Current Gaps and Challenges in Urban Resilience Frameworks

Current urban resilience models suffer from inconsistencies in indicator hierarchies and limited cross-domain integration [14]. While USAH has been applied successfully in isolated cases [11], scaling the model internationally requires a system that ensures data security across jurisdictions while enabling global comparability.

Graph network analysis provides a powerful tool to model these complex interconnections by representing each system as nodes and their interactions as edges. This allows for a more intuitive and structured approach to identifying vulnerabilities in interconnected systems. Using graph theory, planners can simulate the impact of failures in one part of the network and predict how disruptions may cascade through the system, aiding the development of more robust urban resilience strategies.

Another challenge is managing the vast amount of heterogeneous data generated by various urban systems, which include both spatial and temporal components. Graph networks, as demonstrated in the study of Janus Graph, Neo4j, and other systems [4], excel at handling large-scale, complex data with intricate relationships. By leveraging graph network analysis, urban resilience models can more efficiently manage and query this data, facilitating real-time decision-making in response to crises. This approach also supports better resource allocation and risk assessment by revealing critical nodes in the urban network that, if compromised, could have the most significant negative effects on city resilience.

1.3. Incorporation of Socio-Economic Parameters

While graph theory and digital urban platforms have enhanced the modeling of infrastructure and governance systems, the systematic integration of socio-economic parameters remains limited in many resilience frameworks. Existing models often prioritize physical infrastructure, environmental systems, or institutional capacity, while socio-economic and socio-cultural indicators, such as income inequality, human capital, demographic vulnerability, public finance stability, and social cohesion, are treated as secondary or static contextual variables [9,14].

However, urban resilience is fundamentally a socio-technical construct in which infrastructure performance, governance effectiveness, and community adaptability are deeply intertwined. Socio-economic indicators influence exposure, sensitivity, and adaptive capacity, shaping how shocks and chronic stressors propagate across urban systems. Without explicitly modeling these parameters within network structures, resilience assessments may overlook important dependencies and cross-sector interactions.

Graph-based modeling environments provide a structured mechanism to embed socio-economic variables directly as nodes within multi-domain dependency networks. By representing socio-economic indicators alongside infrastructure, environmental, and governance elements, graph models enable the analysis of cascading effects, trade-offs, and leverage points across domains. This integration supports a shift from siloed indicator lists toward a relational understanding of resilience, where socio-economic dynamics are recognized as both drivers and outcomes of systemic stability.

Despite this potential, few resilience frameworks operationalize socio-economic parameters within weighted, path-sensitive graph architectures. Addressing this gap requires not only indicator inclusion but also the quantification of their structural embeddedness and cross-domain influence; an area this study advances through network-based resilience metrics.

The existing urban-resilience frameworks have a general design flaw, i.e., they list indicators (within different domains) but do not illustrate how these are related to each other. The node-level centrality methods used in previous graph-based studies capture multi-domain interdependence using only a single position-based measure; therefore, they conceal relational leverage points and transitive chain dependencies. This research makes three additional theoretical contributions. First, this research expands on USAH by integrating socio-economic indicators and governance indicators into a weighted dependency network as structurally active nodes, whereas previously they were treated as either contextually or secondarily relevant. Second, this research presents five additional network diagnostics (CECS, ECS, MDPS, IDDR, WDD), which together illustrate the structural attributes of embeddedness, leverage, transitivity, domain coupling, and density of Vancouver’s urban resilience system as measurable structural properties. Finally, this research empirically demonstrates that urban resilience is an emergent attribute of network architecture, not of the performance of specific indicators within each domain, and identifies governance coherence and digital connectivity as the structural foundations of Vancouver’s urban resilience system. These contributions advance graph-based urban resilience modeling from descriptive mapping toward a structural diagnostic framework capable of informing targeted, evidence-based planning interventions.

2. Methods

2.1. Study Area

The current work is part of a multi-phase research initiative called URSA project, aiming to address socio-economic urban resiliency in Vancouver. The city of Vancouver, a major metropolitan center on the west coast of Canada, is characterized by a dense urban core, diverse socio-economic structure, and exposure to multiple environmental stressors. Vancouver was selected due to its advanced data availability, established resilience planning initiatives, and its relevance as a representative case for complex urban systems facing challenges such as climate change, population growth, and infrastructure pressure. The city exhibits a wide range of interconnected domains, including governance, infrastructure, environment, and social systems, making it well-suited for testing a graph-based resilience framework. Its coastal location also exposes it to risks such as flooding and climate-related impacts, further supporting its suitability as a case study for multi-domain resilience analysis.

The three phases of the project are illustrated in

Table 1. URSA Project Phases.

Phase	Objective
Mosaic I	Identify the key factors and their corresponding indicators that can be utilized to develop an Urban System Abstraction Hierarchy (USAH) model.
Mosaic II	Stakeholders’ analysis to provide the necessary critical answers to fill the gaps for those factors and indicators that were not quickly and logically identified in Mosaic I
Mosaic III	Implement a graph-based network for managing resilience data, optimize performance, and do a test run for Vancouver socio-economic data

.

2.2. Identification of Dimensions, Indicators and Sub-Indicators

The methodology used in Mosaic III is structured around three key components: the identification of resilience indicators, the development of a graph network, and the implementation of a secure data model for cross-jurisdictional use.

We began by identifying key stakeholders relevant to urban resilience in Vancouver, grouped under categories such as academic institutions, government, NGOs, private sector, and community organizations. Stakeholders were further mapped to seven resilience dimensions, such as Infrastructure, Health and Wellbeing, Environment, and Governance, and linked to specific areas of engagement, such as education, transportation, or capacity building.

Indicator selection was conducted through a multi-phase validation process within the URSA project (see

Table 2. URSA Domain and Indicators.

Domain	Indicators
Critical Infrastructure	Built Infrastructure BI Digital Infrastructure DI Effective Provision of Critical Services CR Energy EN Transportation TN Water WR
Economics	Budget and Subsidy BS Business Environment BE Diverse Livelihood and Employment DL Economic Robustness ER Finance and Savings FS Households Assets HA Human Capital HC Income IN Innovation and Entrepreneurship IE Public Finance PF
Environment	Air Quality AQ Decarbonization DC Environment and Ecology EE Flooding FL Green Infrastructure GI Heat Stress HS Land Use LU Waste Management WM
Health & Well-being	Access to Quality Healthcare QH Emergency Medical Care EM Minimal Human Vulnerability HV Public Health PH
Institutional	Education ED Inclusivity and Involvement II Institutional Collaboration IC Knowledge Dissemination and Management KD Rule of Law RL
Leadership & Strategy	Disaster Management DM Good Governance GG Integrated Development Planning ID
Social & Demographics	Community Engagement and Preparedness CE Comprehensive Social Security CS Population PN Social Cohesion SC

). In Mosaic I, candidate indicators were identified through a comprehensive review of the literature and evaluated using four specific criteria, the first of which was conceptual relevance to the body of literature on urban resilience. Second, measurable or quantifiable/structured qualitative data existed for each indicator selected for inclusion in this study. Third, to provide a broad representation of all seven identified areas of resiliency (environmental, economic, institutional, health, social, infrastructure, and governance) and to prevent any significant area of resiliency from being omitted, indicators were selected to ensure domain representativeness. Fourth, the indicators were evaluated for their contextual application to Vancouver’s urban system. This evaluation occurred through a structured stakeholder engagement process as part of Mosaic II. Any indicator that did not meet one or more of these criteria was either removed or combined with other indicators that met the same functional requirements.

This approach prioritizes capturing cross-domain interdependencies inherent in socio-economic urban systems. However, formal statistical screening methods (e.g., Principal Component Analysis or Delphi-based consensus) and tests of indicator independence and redundancy were not applied at this stage and are acknowledged as a limitation, to be addressed in future research.

Each indicator is represented as a node in the set, with directed edges illustrating dependency relationships. Edges within the same domain reflect functional interdependence (e.g., Transportation ↔ Energy), while edges across domains highlight cross-sector connections (e.g., Public Finance → Public Health). In this modular network, strong intra-domain coupling suggests subsystem cohesion, whereas bridging edges indicate key interdependencies.

Resilience indicators and sub-indicators were then defined and associated with these dimensions. Data for each sub-indicator was collected, classified as quantitative or qualitative, and normalized using various methods, such as dummy encoding, Likert scaling, min-max scaling, and Z-scoring, depending on the nature and availability of data. Particular care was taken to address indicators that could negatively affect resilience. A full breakdown of normalization procedures and indicator metrics is provided in Appendix A.

Once the data structure was established, a graph-based network was implemented to model the interdependencies between resilience indicators. Each node in the graph represents an indicator, and edges capture the directional relationships between them. This network structure allows us to simulate cascading effects, identify systemic bottlenecks, and analyze the overall integrity of urban systems under stress.

This methodological framework provides the foundation for an adaptive, data-driven approach to resilience planning, offering insights into both local and system-wide vulnerabilities in urban environments.

2.3. Graph Network Development

The graph-based model was implemented using Python’s NetworkX library (Python 3.10.12), in which indicators are represented as nodes, and their interactions define relationships between them (See Supplementary Materials). The model was applied to Vancouver using data on urban resilience. Future work will extend deployment to additional jurisdictions for cross-jurisdictional comparisons.

2.4. Mathematical Framework and Formulas

2.4.1. Normalization of Indicator Values

Data used in this study were collected from publicly available sources, including the City of Vancouver Open Data Portal, Statistics Canada, and Government of British Columbia, as well as relevant institutional reports. Indicators were compiled for the most recent available and consistent time period; where multi-year data existed, aggregated or representative values were used. Missing quantitative values were addressed using indicator-specific approaches depending on data availability and context. Where limited gaps existed in longitudinal datasets, values were estimated using temporal averaging or interpolation based on adjacent years.

To ensure methodological consistency, normalization techniques were selected based on data structure: Z-score normalization was applied where sufficient longitudinal data existed, while Min–Max scaling was used for cross-sectional indicators. For indicators with negative contributions to resilience, directionality was standardized post-normalization using inverse transformation following OECD composite indicator guidelines [15].

• Min-Max Normalization

Applied when indicators had fixed upper and lower bounds.

x′ = (x − x_min)/(x_max − x_min)

where

• x = raw value;
• x_min, x_max = observed minimum and maximum values of the indicator

This method rescales the values to the [0, 1] range, preserving proportional relationships.

b.

Z-score Normalization

This method is used when data distribution approximated normality and historical trends were available.

For each sub-indicator x, the normalized value z is given by:

Z = (x − μ)/σ

where

• μ = mean of the indicator across observations;
• σ = standard deviation of the indicator.

This method highlights deviations from average performance and enables relative comparisons across time or jurisdictions.

2.4.2. Interconnectedness Metric

To measure the structural density within and between domains, we used a custom interconnectedness index:

I = E_realized/E_possible

(1)

where

• E_realized = number of actual edges;
• E_possible = n(n − 1) for a directed graph with n nodes.

This value was computed separately for intra-domain and inter-domain subgraphs to assess modular cohesion and systemic coupling.

2.4.3. Edge Weight Assignment Based on Distance Metrics

To identify strongly related sub-indicators, we computed pairwise distances between them and selected those with minimal distances for high-coupling designations (

Table A9. Sub-indicator distances and clusters.

Indicator	Sub-Indicator 1	Sub-Indicator 2	Distance from y = x
Access to Quality Healthcare	Access to Healthcare	Adequate access to quality healthcare	0
Access to Quality Healthcare	Access to Healthcare	Health Insurance penetration and density	0
Access to Quality Healthcare	Access to Healthcare	Proportion of medical insurance coverage	0
Access to Quality Healthcare	Access to Healthcare	Quality of Healthcare	0
Access to Quality Healthcare	Adequate access to quality healthcare	Health Insurance penetration and density	0
Access to Quality Healthcare	Adequate access to quality healthcare	Proportion of medical insurance coverage	0
Access to Quality Healthcare	Adequate access to quality healthcare	Quality of Healthcare	0
Access to Quality Healthcare	Health Insurance penetration and density	Proportion of medical insurance coverage	0
Access to Quality Healthcare	Health Insurance penetration and density	Quality of Healthcare	0
Access to Quality Healthcare	Proportion of medical insurance coverage	Quality of Healthcare	0
Air Quality	Air Quality Health Index (AQHI)	Days of good air quality	0.707106804
Budget and Subsidy	Availability of subsidies/incentives for residents to rebuild houses	Financial Support	0
Budget and Subsidy	Availability of subsidies/incentives for residents to rebuild houses	Funding for Disaster Resilience Mapping	0.707106781
Budget and Subsidy	Availability of subsidies/incentives for residents to rebuild houses	Percentage of Total Budget Spent on Culture	0.675815062
Budget and Subsidy	Availability of subsidies/incentives for residents to rebuild houses	Refinancing Costs	0.186080732
Budget and Subsidy	Availability of subsidies/incentives for residents to rebuild houses	Structure of Budgetary System	0
Budget and Subsidy	Availability of subsidies/incentives for residents to rebuild houses	Sufficient Budget for Disaster Risk Reduction	0
Budget and Subsidy	Availability of subsidies/incentives for residents to rebuild houses	The capital investment in improving the infrastructure and public service facilities in historical cities and historical and cultural blocks in the last three years	0.707106781
Budget and Subsidy	Financial Support	Funding for Disaster Resilience Mapping	0.707106781
Budget and Subsidy	Financial Support	Percentage of Total Budget Spent on Culture	0.675815062
Budget and Subsidy	Financial Support	Refinancing Costs	0.186080732
Budget and Subsidy	Financial Support	Structure of Budgetary System	0
Budget and Subsidy	Financial Support	Sufficient Budget for Disaster Risk Reduction	0
Budget and Subsidy	Financial Support	The capital investment in improving the infrastructure and public service facilities in historical cities and historical and cultural blocks in the last three years	0.707106781
Budget and Subsidy	Funding for Disaster Resilience Mapping	Percentage of Total Budget Spent on Culture	0.031291719
Budget and Subsidy	Funding for Disaster Resilience Mapping	Refinancing Costs	0.521026049
Budget and Subsidy	Funding for Disaster Resilience Mapping	Structure of Budgetary System	0.707106781
Budget and Subsidy	Funding for Disaster Resilience Mapping	Sufficient Budget for Disaster Risk Reduction	0.707106781
Budget and Subsidy	Funding for Disaster Resilience Mapping	The capital investment in improving the infrastructure and public service facilities in historical cities and historical and cultural blocks in the last three years	0
Budget and Subsidy	Percentage of Total Budget Spent on Culture	Refinancing Costs	0.48973433
Budget and Subsidy	Percentage of Total Budget Spent on Culture	Structure of Budgetary System	0.675815062
Budget and Subsidy	Percentage of Total Budget Spent on Culture	Sufficient Budget for Disaster Risk Reduction	0.675815062
Budget and Subsidy	Percentage of Total Budget Spent on Culture	The capital investment in improving the infrastructure and public service facilities in historical cities and historical and cultural blocks in the last three years	0.031291719
Budget and Subsidy	Refinancing Costs	Structure of Budgetary System	0.186080732
Budget and Subsidy	Refinancing Costs	Sufficient Budget for Disaster Risk Reduction	0.186080732
Budget and Subsidy	Refinancing Costs	The capital investment in improving the infrastructure and public service facilities in historical cities and historical and cultural blocks in the last three years	0.521026049
Budget and Subsidy	Structure of Budgetary System	Sufficient Budget for Disaster Risk Reduction	0
Budget and Subsidy	Structure of Budgetary System	The capital investment in improving the infrastructure and public service facilities in historical cities and historical and cultural blocks in the last three years	0.707106781
Budget and Subsidy	Sufficient Budget for Disaster Risk Reduction	The capital investment in improving the infrastructure and public service facilities in historical cities and historical and cultural blocks in the last three years	0.707106781
Built Infrastructure	Building insulation	Adequate continuity for critical assets and services	0.707106781
Built Infrastructure	Building insulation	Alerts and emergency notification systems	0.707106781
Built Infrastructure	Building insulation	Building code	0.707106781
Built Infrastructure	Building insulation	Building layout and orientation	1.63 × 10−11
Built Infrastructure	Building insulation	Buildings above water logging	0.696500179
Built Infrastructure	Building insulation	Cybersecurity preparedness	1.63 × 10−11
Built Infrastructure	Building insulation	Diligent maintenance of critical services	1.63 × 10−11
Built Infrastructure	Building insulation	Electricity quality	0.004824091
Built Infrastructure	Building insulation	Energy monitoring	0.707106781
Built Infrastructure	Building insulation	Flexibility of grid	1.63 × 10−11
Built Infrastructure	Building insulation	Flexible infrastructure services	0.707106781
Built Infrastructure	Building insulation	Future-proofing the built structures	0.707106781
Built Infrastructure	Building insulation	Generation and utilization of information	1.63 × 10−11
Built Infrastructure	Building insulation	Guarantee of continued critical services	0.707106781
Built Infrastructure	Building insulation	Home ownership	1.63 × 10−11
Built Infrastructure	Building insulation	Housing age	0.707106781
Built Infrastructure	Building insulation	Natural ventilation	0.707106781
Built Infrastructure	Building insulation	Permeable pavement and bioswales	0.707106781
Built Infrastructure	Building insulation	Population living in proximity to polluted industries	6.15 × 10−12
Built Infrastructure	Building insulation	Preservation of housing	1.75 × 10−5
Built Infrastructure	Building insulation	Reducing air infiltration and thermal bridging	1.97 × 10−10
Built Infrastructure	Building insulation	Secure technology networks	0.707106781
Built Infrastructure	Building insulation	The infrastructure coverage rate	0.281994184
Built Infrastructure	Building insulation	Urban energy supply systems for increasing shares of renewable energy	0.707106781
Built Infrastructure	Building insulation	Urban form	1.63 × 10−11

in Appendix A).

Let d(i, j) be the computed distance between sub-indicators i and j, then:

d(i, j) = (|x_i − x_j|)/√2

(2)

where x_i, x_j are normalized values of sub-indicators.

Pairs with d(i, j) = 0 were designated as strongly coupled and represented with weighted edges in the sub-indicator network.

(1)

Shortest Path-Based Distance (for sub-indicator coupling)

Once normalized values were assigned to nodes (indicators), the edges between them were constructed to reflect either empirical dependency relationships or semantic proximity based on expert knowledge.

(2)

Edge Weight Assignment

Edge weights in the sub-indicator subnetwork are inversely proportional to deviation from the baseline:

w_(ij) = 1 − |x_i − x_j|

(3)

This approach emphasizes harmonized relationships, assigning higher weights to more aligned sub-indicators: distance from the 45° line” produces weights in [0, 1] where higher = stronger coupling.

(3)

Indicator Value Computation

Indicator values combine the average normalized sub-indicator scores (data-driven performance) and their average betweenness centrality (node-level betweenness within each indicator’s sub-graph, not at the global level network structure):

Iₖ = 0.3(Xk) + 0.7(Bk)

(4)

This 30/70 ratio prioritizes network structure, reflecting dependencies and bottlenecks, while retaining data realism. These ratios are based on sensitivity analysis and network stability tests.

(4)

Indicator Influence Network Construction

With indicator values established, a 40 × 40 indicator network is generated to represent cross-domain influences. Edges are weighted using the same deviation method, ensuring consistency between sub-indicator and indicator levels.

w_(ij) = 1 − |x_i − x_j|

(5)

2.4.4. Adjacency Matrix Construction

From domain-specific dependencies (e.g., Energy → Digital Infrastructure), we formed a binary or weighted adjacency matrix A:

A_uv = 1 if indicator u directly influences v, otherwise 0.

For weighted cases, we applied edge weights w_uv proportional to influence strength (from expert scoring or empirical alignment).

2.4.5. Network-Based Resilience Metrics

Given the high density of the constructed indicator network, traditional node betweenness centrality provided limited discrimination across nodes. Therefore, we implemented a suite of weighted, edge- and path-based metrics to quantify structural embeddedness, dependency propagation, cross-domain coupling, and overall system density. These metrics are computed directly from the weighted adjacency matrix of the graph.

(1)

Cumulative Edge Criticality Score (CECS)

For each node i, the Cumulative Edge Criticality Score (CECS) is defined as the sum of the weights of all edge’s incident to that node:

$${\mathrm{C}\mathrm{E}\mathrm{C}\mathrm{S}}_{\mathrm{i}}={\sum }_{\mathrm{j}}\mathrm{w}\_\mathrm{i}\mathrm{j}$$

(6)

where w_ij represents the weighted dependency between indicators i and j.

CECS measures the structural embeddedness of an indicator within the resilience network. Higher CECS values indicate greater integration across domains and stronger participation in dependency structures.

2.4.6. Edge Criticality Score (ECS)

The Edge Criticality Score (ECS) quantifies the contribution of a given edge to the strongest dependency paths in the network. For each edge, eij, ECS is computed as the cumulative influence of that edge across all maximum-strength paths between indicator pairs.

ECS captures the structural importance of specific inter-indicator relationships within dominant dependency chains.

2.4.7. Maximum Dependency Path Strength (MDPS)

For each ordered pair of indicators (i, j), the Maximum Dependency Path Strength (MDPS) is defined as the maximum product of edge weights along any path connecting i and j:

$${\mathrm{M}\mathrm{D}\mathrm{P}\mathrm{S}}_{\mathrm{i},\mathrm{j}}=\max \left(\mathrm{P}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{s}{\mathrm{P}}_{\mathrm{i}\mathrm{j}}\right){\prod }_{\left(\mathrm{u},\mathrm{v}\right)\in {\mathrm{P}}_{\mathrm{i},\mathrm{j}}}{\mathrm{w}}_{\mathrm{u}\mathrm{v}}$$

(7)

To ensure computational stability, this is implemented by transforming edge weights using −log(w) and applying shortest-path algorithms.

MDPS identifies the strongest transitive dependency relationships within the network.

2.4.8. Inter-Domain Dependency Ratio (IDDR)

For each domain d, the Inter-Domain Dependency Ratio (IDDR) is computed as:

$${\mathrm{I}\mathrm{D}\mathrm{D}\mathrm{R}}_{\mathrm{d}}={\displaystyle \frac{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{C}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}-\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}-\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}}$$

(8)

This metric measures the relative reliance of each domain on external versus internal relationships.

2.4.9. Weighted Dependency Density (WDD)

The WDD captures the overall structural coupling of the network and is defined as:

$$\mathrm{W}\mathrm{D}\mathrm{D}={\displaystyle \frac{2\sum {\mathrm{w}}_{\mathrm{i}\mathrm{j}}}{\mathrm{n}(\mathrm{n}-1)}}$$

(9)

where nis the total number of nodes in the graph.

WDD represents the average weighted connectivity relative to the maximum possible connectivity in a fully connected network.

3. Results

The results of the normalizations by various methods can be seen in Appendix A. In addition, we identified and classified the stakeholders based on the areas of resilience.

3.1. Identifying Stakeholders

The analysis identified six main categories of stakeholders critical to Vancouver’s resilience: Academic Institutions, Community Organizations, Government, Non-Governmental Organizations (NGOs), groups without “voice,” such as marginalized communities, and the private sector. Within these categories, specific subcategories were identified that reflect the particularities of each group. For example, in Academic Institutions, subcategories such as universities, technical colleges and applied research centers are included, each contributing uniquely to areas such as technical education and innovation. In Community Organizations, local groups focused on social equity, environmental sustainability and cultural development are differentiated. Within Government, levels such as local government, provincial government and specialized agencies are subdivided, each with specific roles in infrastructure and planning. NGOs also include subcategories oriented to disaster management, environmental conservation and capacity building. Finally, the Private Sector is broken down into small businesses, technology corporations and financial institutions that drive innovation and economic resources.

Each of these categories represents a distinct role in strengthening urban resilience, contributing uniquely to areas such as education, sustainable development, capacity building and disaster management.

To identify key stakeholders, a structured approach was used that included collecting data from reliable sources such as official reports (e.g., the Vancouver Resilience Report), institutional websites (such as those of local and provincial governments), and academic papers published in journals such as the Journal of Urban Resilience and the Canadian Journal of Sustainable Development. Subsequently, matrices were developed and analyzed that assessed the power, interest, influence and importance of each stakeholder in the context of urban resilience. This in-depth analysis allowed the stakeholders to be categorized according to their capacity to impact resilience and their level of interest in related initiatives.

Academic Institutions were represented by eight key stakeholders, including Vancouver Community College and the British Columbia Institute of Technology (BCIT). These institutions contributed primarily to the areas of “Education” and “Research.” Their efforts focused on fostering institutional collaboration, advancing technical education, and promoting innovative research, which collectively enhances the city’s preparedness and adaptability.

Community Organizations represented the largest group, with 12 stakeholders. This category included organizations involved in environmental sustainability, social equity and leadership in urban strategies. These groups played a key role in integrating diverse community needs into resilience planning, thereby enhancing social cohesion and local engagement.

Government, with nine stakeholders, was key in areas such as infrastructure development, capacity building and environmental conservation. Their contributions focused on strategic leadership, emergency preparedness and community engagement. These stakeholders provided the necessary regulatory frameworks and resources to facilitate resilience efforts in all sectors.

Non-Governmental Organizations (NGOs) included five stakeholders with a strong focus on sustainable development and capacity building. They facilitated regional coordination, disaster management and energy resilience through their initiatives and advocacy.

Private Sectors had a significant impact on innovation and investment in sustainable technologies. Businesses played an essential role in generating innovative solutions to critical problems, such as clean energy and efficiency in urban infrastructure. Evaluating their impacts, significance, and authority.

3.2. Evaluating Stakeholders Impacts, Significance, and Authority

The evaluation of stakeholder contributions revealed varying degrees of impact, significance, and authority across the categories, and also integrated the main findings of the study. This analysis, based on matrices of power, interest, influence and importance, identified the stakeholders most critical to strengthening Vancouver’s resilience. Government and community organizations demonstrated the greatest direct impact due to their ability to mobilize resources and implement large-scale programs. For example, government stakeholders were instrumental in emergency preparedness and resilient infrastructure development, while community-based organizations excelled in addressing local needs and fostering social cohesion.

NGOs played a key role in addressing gaps in traditional resilience frameworks. They brought specialized expertise in disaster management and sustainability, serving as a bridge between government policies and community needs.

Strategic leadership and innovation emerged as critical factors, especially driven by government and academic institutions. While the government provided regulatory frameworks and allocated essential resources, academic institutions contributed innovative research and quality education, which strengthened the city’s long-term resilience. Finally, collaboration among stakeholders was identified as essential, integrating complementary strengths to address Vancouver’s multifaceted challenges and enhance its resilience in the face of adversity.

3.3. Implementation and Network Analysis

Interconnectedness (or Edge values) quantifies the overall connectivity of the network by measuring the density of existing links relative to all possible directed connections. This metric reflects how tightly knit the system is, both within and across domains. High interconnectedness suggests a high potential for rapid transmission of shocks, feedback loops, and cascading effects, as indicators are more deeply embedded in the network. Low interconnectedness, conversely, may point to structural isolation or modular containment, potentially buffering parts of the system from widespread disruptions.

The computed total edge weights represent a form of edge-betweenness centrality, where lower cumulative weights correspond to shorter and more critical pathways between indicators. In this model, shorter paths signify structurally essential “bridges” in the network, relationships whose disruption would disproportionately constrain information, coordination, or resource flows across the broader urban system. Because the network exhibits dense interconnections among indicators, edge-based centrality offers a more precise diagnostic than node-betweenness centrality alone. In such settings, node betweenness can become less discriminating, whereas edge betweenness captures subtle yet strategic relational vulnerabilities by quantifying the network's dependence on specific influence pathways. As shown in

Table 3. Top 10 edges between Nodes (Indicators).

Indicator 1	Indicator 2	NormWeight1	NormWeight2	Edge Weight
Flooding	Institutional Collaboration	1	1	1
Flooding	Public Health	1	1	1
Institutional Collaboration	Public Health	1	1	1
Public Finance	Transportation	0.5101206443	0.5101214285	0.9999992158
Income	Waste management	0.5432103087	0.5430612245	0.9998509157
Households Assets	Population	0.5592108844	0.5594511945	0.9997596899
Community Engagement & Preparedness	Comprehensive Social Security	0.6566122449	0.6563302862	0.9997180413
Green Infrastructure	Integrated Development Planning	0.5291372473	0.5277460317	0.9986087844
Diverse Livelihood and Employment	Waste management	0.5416190476	0.5430612245	0.9985578231
Green Infrastructure	Innovation and Entrepreneurship	0.5291372473	0.5306057384	0.9985315089

, the lowest total edge weights are associated with Rule of Law, Air Quality, and Human Capital, identifying them as the most critical bottlenecks. These indicators serve as foundational bridges (

Table 4. Important Path between Nodes.

Indicator	Cumulative Path Distance (Lower = More Central)
Air Quality	36.733137
Rule of Law	37.266863
Human Capital	47.299812
Flooding	57.357473
Institutional Collaboration	57.357473
Public Health	57.357473
Business Environment	57.924712
Comprehensive Social Security	58.354250
Access to Quality Healthcare	58.923718
Decarbonization	61.150886
Transportation	63.544240
Built Infrastructure	64.357981
Green Infrastructure	64.600280
Minimal Human Vulnerability	64.784086
Public Finance	64.907095
Economic Robustness	64.925844
Environment and Ecology	64.934659
Digital Infrastructure	65.103356
Innovation and Entrepreneurship	65.227872
Education	65.434159
Population	65.560219
Income	65.674401
Community Engagement & Preparedness	65.700611
Good Governance	65.806184
Household Assets	65.837344
Integrated Development Planning	65.903256
Effective Provision of Critical Services	65.971340
Energy	65.993720
Disaster Management	66.061187
Land Use	66.064641
Water	66.091541
Social Cohesion	66.112887
Waste Management	66.223363
Inclusivity and Involvement	66.225914
Budget and Subsidy	66.233508
Diverse Livelihood and Employment	66.244431
Finance and Savings	66.249209

) across institutional, environmental, and socio-economic domains, underscoring that reinforcing these dimensions is essential for reducing systemic fragility and enhancing the city’s overall resilience.

The Vancouver case study demonstrates the capacity of the proposed graph-based resilience model to capture and visualize systemic dependencies among urban indicators. Figure 1 presents the indicator-level network, where each node represents an indicator and weighted edges represent the magnitude of inferred dependency influence derived from the pairwise influence matrix. Centrality analyses, particularly betweenness centrality, reveal several high-impact bottlenecks, including Good Governance, Disaster Management, and Digital Infrastructure. These nodes occupy structurally critical positions within the urban system, meaning that their disruption could constrain multiple dependent indicators across environmental, institutional, and socio-economic domains.

In the Vancouver context, these bottlenecks reveal distinct vulnerabilities. Good Governance now occupies the most pronounced bottleneck position in the network, indicating its extensive mediation role across institutional, economic, infrastructure, and environmental domains. Its structural centrality suggests that governance effectiveness, encompassing transparency, regulatory coherence, coordination capacity, and institutional trust, acts as the primary integrative backbone of the resilience system. Disruptions or weaknesses in governance structures are therefore likely to cascade widely, fragmenting cross-sector coordination and weakening adaptive response capacity.

Disaster Management emerges as a second-order structural bottleneck, highlighting the system’s sensitivity to preparedness, response coordination, and recovery mechanisms. Its prominence reflects the model’s emphasis on multi-hazard and chronic stressor scenarios, in which emergency planning and risk management capabilities shape performance across health, infrastructure, and economic indicators. This indicates that resilience in Vancouver is strongly contingent on the operational effectiveness of crisis management systems.

Digital Infrastructure also appears as a critical bottleneck, underscoring the increasing reliance of urban systems on information flows, communication networks, and digital service delivery. Its structural role suggests that disruptions to digital connectivity or cybersecurity vulnerabilities could propagate across governance, economic activity, public services, and social coordination functions. The prominence of this node reflects the growing centrality of digital systems in enabling adaptive governance and cross-domain integration.

From a policy perspective, these findings suggest that resilience planning in Vancouver—and, by extension, other urban jurisdictions—should prioritize reinforcing the identified bottlenecks rather than treating them as independent sectoral challenges. A graph-based resilience assessment provides decision-makers with a systems-level diagnostic tool: by quantifying interdependencies among governance, infrastructure, and socio-economic functions, it enables targeted interventions that stabilize the entire network rather than marginally improving individual sectors.

3.4. Alignment Between Stakeholder and Data-Driven Matrices

This subsection evaluates the alignment between the empirical data-driven dependency matrix and the stakeholder-derived adjacency matrix. The objective was to identify instances in which stakeholders perceived no relationship between indicators (value = 0), but empirical analysis revealed a strong dependence (weight ≥ 0.75).

The comparison revealed 106 such discrepancies (

Table 5. Summary of Unaligned High-Weight Relationships where stakeholder weights = 0.

Indicator 1	Dimension 1	Indicator 2	Dimension 2	Domain Relationship	Data-Driven Value
QH	Health and Well-Being	QH	Health and Well-Being	Intra-domain	1.000000
QH	Health and Well-Being	RL	Institutional	Inter-domain	0.827343
AQ	Environment	AQ	Environment	Intra-domain	1.000000
BS	Economic	BS	Economic	Intra-domain	1.000000
BS	Economic	BI	Critical Infrastructure	Inter-domain	0.936756
BS	Economic	CE	Social and Demographic	Inter-domain	0.906749
BS	Economic	CR	Critical Infrastructure	Inter-domain	0.956486
BS	Economic	EE	Environment	Inter-domain	0.924624
BS	Economic	LU	Environment	Inter-domain	0.926652
BS	Economic	WM	Environment	Inter-domain	0.979700
BS	Economic	WR	Critical Infrastructure	Inter-domain	0.934672
BI	Critical Infrastructure	BI	Critical Infrastructure	Intra-domain	1.000000
BI	Critical Infrastructure	DI	Critical Infrastructure	Intra-domain	0.788329
BI	Critical Infrastructure	GG	Leadership and Strategy	Inter-domain	0.815900
BI	Critical Infrastructure	HV	Health and Well-Being	Inter-domain	0.988039
BI	Critical Infrastructure	PN	Social and Demographic	Inter-domain	0.940666
BE	Economic	BE	Economic	Intra-domain	1.000000
BE	Economic	CR	Critical Infrastructure	Inter-domain	0.910004
BE	Economic	II	Institutional	Inter-domain	0.850146
CE	Social and Demographic	CE	Social and Demographic	Intra-domain	1.000000
CE	Social and Demographic	DC	Environment	Inter-domain	0.914800
CE	Social and Demographic	ER	Economic	Inter-domain	0.860923
CE	Social and Demographic	CR	Critical Infrastructure	Inter-domain	0.863235
CE	Social and Demographic	WR	Critical Infrastructure	Inter-domain	0.972077
CS	Social and Demographic	CS	Social and Demographic	Intra-domain	1.000000
CS	Social and Demographic	DC	Environment	Inter-domain	0.915082
CS	Social and Demographic	GI	Environment	Inter-domain	0.872807
CS	Social and Demographic	LU	Environment	Inter-domain	0.980378
CS	Social and Demographic	WM	Environment	Inter-domain	0.886731
CS	Social and Demographic	WR	Critical Infrastructure	Inter-domain	0.972359
DC	Environment	DC	Environment	Intra-domain	1.000000
DC	Environment	DM	Leadership and Strategy	Inter-domain	0.995224
DC	Environment	DL	Economic	Inter-domain	0.970206
DC	Environment	FS	Economic	Inter-domain	0.980420
DC	Environment	RL	Institutional	Inter-domain	0.786450
DC	Environment	SC	Social and Demographic	Inter-domain	0.962383
DI	Critical Infrastructure	DI	Critical Infrastructure	Intra-domain	1.000000
DI	Critical Infrastructure	PN	Social and Demographic	Inter-domain	0.847662
DI	Critical Infrastructure	WM	Environment	Inter-domain	0.831273
DM	Leadership and Strategy	DM	Leadership and Strategy	Intra-domain	1.000000
DM	Leadership and Strategy	DL	Economic	Inter-domain	0.974982
DM	Leadership and Strategy	IN	Economic	Inter-domain	0.976573
DM	Leadership and Strategy	WM	Environment	Inter-domain	0.976424
DL	Economic	DL	Economic	Intra-domain	1.000000
DL	Economic	EE	Environment	Inter-domain	0.946366
DL	Economic	GI	Environment	Inter-domain	0.987518
DL	Economic	WM	Environment	Inter-domain	0.998558
DL	Economic	WR	Critical Infrastructure	Inter-domain	0.912930
ER	Economic	ER	Economic	Intra-domain	1.000000
ER	Economic	CR	Critical Infrastructure	Inter-domain	0.997688

Found 106 unique relationships where Stakeholder = 0 and Data-Driven ≥ 0.75. Intra-domain: 41; Inter-domain: 65.

). Of these, 41 were intra-domain cases, meaning missing dependencies within the same domain (e.g., Economic → Economic), while 65 were inter-domain discrepancies, spanning domains such as Health, Environment, Infrastructure, Economics, Governance, and Leadership. These hidden dependencies represent structurally significant couplings that stakeholders did not recognize or affirm.

Examples include QH → AQ (Health and Well-being influencing Environment), BS → CR (Economic factors influencing Critical Infrastructure), and BI → GG (Infrastructure informing Leadership and Strategy). Many of these pairs aligned with high-betweenness indicators identified in the network analysis, particularly Rule of Law, Air Quality, and Human Capital, suggesting stakeholder perceptions under-recognize pathways essential for systemic stability.

These overlooked dependencies highlight potential early-stage vulnerabilities in the urban resilience system. Their identification supports hybrid model validation: strengthening stakeholder calibration, improving cross-domain awareness, and guiding targeted interventions.

In addition to the bottleneck indicators identified earlier, our masking evaluation revealed latent dependencies not captured by stakeholder inputs. Specifically, the 106 high-weight empirical relationships (≥0.75) that stakeholders assigned a weight of zero demonstrate a measurable misalignment between expert perception and data-driven system behavior. These mismatches are not random; rather, they cluster around domains previously identified as high-betweenness and structurally critical, including Rule of Law, Air Quality, Human Capital, Public Health, and Transportation.

By integrating the masking analysis with betweenness and edge-weight diagnostics, we show that several unacknowledged dependencies occur along key cross-domain bridges. For example, links such as QH → AQ (Healthcare to Air Quality), BS → CR (Budget/Subsidy to Critical Services), and BI → GG (Built Infrastructure to Good Governance) reveal multidimensional ties between health, infrastructure, socio-economics, and governance systems. These represent precisely the type of latent cross-domain couplings that the 75% deviation framework aims to uncover. The existence of these strong yet unrecognized relationships suggests that stakeholder mental models under-represent systemic interdependencies central to shock transmission and resilience performance.

Recognizing these hidden relationships strengthens the hybrid validation workflow proposed for future phases of the Mosaic III project. Stakeholders can use this information to recalibrate local judgments, refine influence matrices, and prioritize interventions in areas that both the data-driven model and network topology indicate are structurally sensitive. This supports more accurate scenario modeling, particularly under cascading failure conditions or chronic stress pathways.

3.5. Network-Based Resilience Metrics

3.5.1. System-Level Dependency Structure (Weighted Dependency Density-WDD)

The value of WDD for the City of Vancouver is calculated as 0.5673. This value indicates that more than half of the network’s maximum possible weighted connectivity is active. The system therefore exhibits a relatively high degree of structural coupling, with dependencies broadly distributed across domains rather than concentrated in isolated clusters. The network is neither sparse nor loosely organized; instead, a substantial portion of potential inter-indicator relationships carries meaningful influence.

Such density implies that disturbances are likely to propagate beyond their immediate point of origin through multiple alternative pathways. System risk is therefore embedded not only in a few dominant corridors of influence but within a wider architecture of interdependence. While high connectivity may support coordination under stable conditions, it also increases exposure to cascading effects during periods of stress.

3.5.2. Indicator Embeddedness (Cumulative Edge Criticality Score—CECS)

The Cumulative Edge Criticality Score (CECS) as presented in

Table 6. Top and Bottom 5 Indicators by CECS (node embeddedness).

Index	Indicator	CECS
0	IE	27.66356910
1	ED	26.75927235
2	HV	25.94399954
3	EN	25.38543553
4	PF	25.22175078
32	PH	14.14703247
33	FL	13.85444293
34	RL	12.73613903
35	QH	8.16475894
36	AQ	4.83022090

, reveals a clear hierarchy of indicator embeddedness within the resilience network.

Indicators such as IE, ED, HV, EN, and PF exhibit the highest embeddedness, reflecting extensive weighted connectivity across multiple domains. High CECS values indicate participation in numerous dependency relationships and suggest that these indicators are structurally integrated within the broader resilience system.

In contrast, PH, FL, RL, QH, and AQ display comparatively lower CECS values, indicating more limited participation in dominant dependency structures. These indicators occupy relatively peripheral structural positions and are less influential in cross-domain cascading dynamics.

Figure 2 illustrates the structural distribution of the metric across the network, while Figure 3 presents its statistical distribution in the form of a histogram, providing complementary perspectives on the same underlying data.

3.5.3. Edge-Level Leverage Points (Edge Criticality Score—ECS)

The Edge Criticality Score (ECS) identifies relationships that disproportionately contribute to the strongest dependency paths in the network.

Table 7. Top Edges by ECS (policy leverage points on strongest dependency paths).

Index	Edge	ECS
0	FS–HA	278.34514029
1	FS–IN	277.77045339
2	DL–IN	258.16394197
3	ED–IE	256.91323014
4	DL–ED	251.93158240
5	HA–PN	248.88479636
6	BS–PN	243.73880490
7	GI–IE	227.24389897
8	BS–II	216.20522631
9	HC–II	205.73043893

shows the top 10 edges ranked by ECS values.

Edges such as FS–HA, FS–IN, DL–IN, and ED–IE emerge as dominant structural connectors. These relationships contribute significantly to maximum-strength dependency paths, indicating that resilience dynamics are sensitive to the integrity of specific inter-indicator linkages rather than solely to individual node performance.

The ECS distribution (Figure 4 and Figure 5) demonstrates that a relatively small subset of edges accounts for a disproportionate share of path influence.

3.5.4. Dominant Dependency Chains (Maximum Dependency Path Strength—MDPS)

The Maximum Dependency Path Strength (MDPS) analysis captures the strongest transitive relationships between indicator pairs (

Table 8. Top 10 Strongest Dependency Paths by MDPS.

Index	Source	Target	MDPS
0	FL	PH	1.00000000
1	FL	IC	1.00000000
2	IC	PH	1.00000000
3	PF	TN	0.99999922
4	IN	WM	0.99985092
661	AQ	RL	0.37857051
662	QH	AQ	0.30520043
663	AQ	FL	0.29716362
664	AQ	IC	0.29716362
665	AQ	PH	0.29716362

).

Several indicator pairs exhibit near-unity path strength, indicating that resilience in certain areas is highly conditional on upstream performance elsewhere, even when direct edges are absent.

The MDPS distribution (Figure 6) and heatmap (Figure 7) reveal strong transitive-dependency clusters spanning governance-, institutional-, and infrastructure-related indicators. These findings suggest that coordination requirements extend beyond direct pairwise interactions and are embedded within multi-step dependency chains.

3.5.5. Cross-Domain Coupling (Inter-Domain Dependency Ratio—IDDR)

The Inter-Domain Dependency Ratio (IDDR) quantifies the balance between cross-domain and intra-domain connectivity (

Table 9. Domains by IDDR (cross-domain reliance vs. within-domain cohesion).

	Domain	Intra-Strength	Inter-Strength	IDDR
0	Leadership and Strategy	2.687056	65.590817	24.409915
1	Health and Well-Being	2.024156	44.207479	21.839954
2	Institutional	4.403911	69.590057	15.801875
3	Social and Demographic	5.661216	74.497573	13.159288
4	Critical Infrastructure	12.727689	102.308054	8.038227
5	Environment	13.863616	92.945503	6.704276
6	Economic	39.127003	145.565228	3.720327

).

Leadership and Strategy, Health and Well-Being, and Institutional domains exhibit the highest IDDR values, indicating that their structural influence is derived primarily from cross-domain linkages rather than internal cohesion. In contrast, the Economic domain demonstrates comparatively stronger intra-domain connectivity and lower reliance on cross-domain relationships.

Figure 8 and Figure 9 visualize inter-domain ratios and the relative magnitude of intra- versus inter-domain strength.

3.6. Sensitivity Analysis of Indicator Weighting

A sensitivity analysis was conducted to evaluate the robustness of the indicator value formulation presented in Equation (4), where indicator scores are calculated as a weighted combination of average normalized sub-indicator values and their average betweenness centrality within each indicator’s sub-network. The baseline configuration assigns 30% weight to data-driven performance (Avg-SubValue) and 70% weight to structural importance (Avg-Betweenness). To test the stability of the framework, an alternative scenario increasing the structural emphasis (0.1 × Avg-SubValue + 0.9 × Avg-Betweenness) was evaluated.

The comparison as depicted in

Table 10. (a) Largest raw increases in indicator weight. (b) Largest decreases in normalized standing.

(a)
Indicator	Weight (0.3/0.7)	Weight (0.1/0.9)	Change
Good Governance	4.5680	5.6802	+1.1122
Digital Infrastructure	1.9218	2.2703	+0.3484
Disaster Management	1.3508	1.5792	+0.2284
Budget and Subsidy	0.7754	0.8848	+0.1095
Access to Quality Healthcare	0.6369	0.6634	+0.0265
(b)
Indicator	Norm (0.3/0.7)	Norm (0.1/0.9)	Change
Public Health	0.0657	0.0176	−0.0481
Flooding	0.0657	0.0176	−0.0481
Institutional Collaboration	0.0657	0.0176	−0.0481
Access to Quality Healthcare	0.2300	0.1896	−0.0404
Community Engagement & Preparedness	0.1364	0.1089	−0.0276

a,b, shows that increasing the betweenness weight shifts indicator importance toward nodes with higher structural centrality in the network. Indicators that play bridging or intermediary roles gain relative weight, while indicators primarily supported by strong sub-indicator scores lose relative influence. This behavior confirms the intended interpretation of the formulation: the model increasingly prioritizes structural dependencies and potential system bottlenecks as the betweenness contribution grows.

Despite these shifts in indicator-level weights, the system-level network metrics remained stable across both scenarios (

Table 11. Change in network resilience measures derived by indicator weight coefficient.

Final Metric	Scenario 1 (0.3/0.7)	Scenario 2 (0.1/0.9)	Change
WDD	0.5673	0.5673	0.0000
MDPS pairs	666	666	0
MDPS = 1 pairs	3	3	0
MDPS = 1 (%)	0.45%	0.45%	0.00%
Top CECS indicator	IE (27.6636)	IE (27.6636)	No change
Lowest CECS indicator	AQ (4.8302)	AQ (4.8302)	No change
Top ECS edge	FS–HA (278.3451)	FS–HA (278.3451)	No change
Highest IDDR domain	Leadership and Strategy (24.4099)	Leadership and Strategy (24.4099)	No change

). Key resilience measures, including Cumulative Edge Criticality Score (CECS), Edge Criticality Score (ECS), Maximum Dependency Path Strength (MDPS), Inter-Domain Dependency Ratio (IDDR), and Weighted Domain Density (WDD), remained unchanged between the two configurations. The purpose of the weighting sensitivity analysis is to evaluate the topological stability of the network rather than to determine an optimal weighting configuration. This stability occurs because the network topology and influence matrices that generate these metrics remain unchanged in the sensitivity test; only the indicator weighting layer is affected. This supports the use of the 30/70 configuration as a stable and balanced representation of both empirical indicator performance and network structure, although it is not claimed to be optimal. Future work will explore data-driven or optimization-based approaches for calibrating these weights.

4. Discussion

This study advances urban resilience modeling by extending the Urban Systems Abstraction Hierarchy (USAH) into a fully operational, graph-based analytical framework supported by weighted network diagnostics. Unlike traditional resilience frameworks that treat indicators as parallel or sectoral components, the proposed model operationalizes interdependencies explicitly, enabling structural analysis of embeddedness, leverage points, transitive dependencies, and cross-domain coupling. The Vancouver case study demonstrates that resilience is not merely a function of indicator performance but of how indicators are structurally positioned within a densely interconnected socio-technical system.

At the system level, the Weighted Dependency Density (WDD) value of 0.5673 indicates a moderately high degree of structural coupling. More than half of all potential weighted interactions in the network are active, suggesting that dependencies are broadly distributed rather than concentrated in isolated clusters. This density implies that systemic risk is embedded in the overall architecture of interdependence rather than in a few isolated pathways. While such coupling can enhance coordination under stable conditions, it also increases the risk of cascading propagation when disturbances occur. Resilience, therefore, depends not only on strengthening individual indicators but also on managing structural connectivity itself.

The CECS results reveal a hierarchical distribution of indicator embeddedness. Indicators such as IE, ED, HV, EN, and PF function as system stabilizers due to their extensive weighted connectivity across domains. Their structural embeddedness suggests that disruptions affecting these nodes would reverberate across multiple sectors. Conversely, lower-ranked indicators, including PH, FL, RL, QH, and AQ, occupy more peripheral structural positions, indicating comparatively weaker participation in dominant dependency structures. This hierarchy reinforces the importance of distinguishing between substantively important indicators and structurally embedded indicators; the latter exert disproportionate systemic influence due to their connectivity patterns.

Edge-level analysis through ECS further refines this understanding by identifying relationships that disproportionately support dominant dependency pathways. Edges such as FS–HA, FS–IN, DL–IN, and ED–IE contribute most significantly to maximum-strength paths, indicating that resilience dynamics are often conditioned by specific inter-indicator interfaces rather than by individual node centrality alone. These findings shift attention from isolated indicators toward relational leverage points within the system.

The MDPS results extend this insight by revealing near-unity transitive dependencies among several indicator pairs. Strong dependency chains demonstrate that certain outcomes are structurally conditioned on upstream performance even when direct links appear weak. This underscores the presence of latent coordination requirements embedded in multi-step pathways, reinforcing the need to analyze resilience beyond pairwise interactions. The MDPS formulation, based on multiplicative path strength, inherently emphasizes shorter and stronger dependency chains, as values decay rapidly along longer paths. While this highlights dominant transitive relationships, it may underrepresent extended dependency structures. To mitigate this limitation, multiple complementary metrics (CECS, ECS, IDDR, and WDD) are used to capture different aspects of network structure. However, potential correlations among these metrics were not formally evaluated in this study. Future work will incorporate statistical analyses (e.g., correlation and multicollinearity assessments) to better understand metric independence and ensure a robust representation of structural properties, particularly in dense networks.

Even though the existing sensitivity analysis demonstrates the network's topological robustness to varying degrees of weighting, the impact of altering the configuration of structural metrics in response to changes in indicator representation (e.g., adding or removing certain indicators) has not been quantitatively assessed. As such, this represents a major area for continued work, evaluating how well the framework can be generalized across other combinations of indicators and urban contexts.

Cross-domain analysis using IDDR reveals pronounced asymmetries in structural coupling across domains. Leadership and Strategy, Health and Well-Being, and Institutional domains exhibit exceptionally high cross-domain reliance relative to intra-domain cohesion. These domains derive their structural influence primarily through external linkages, functioning as integrative layers within the system. In contrast, the Economic domain demonstrates comparatively stronger internal cohesion and lower relative dependence on cross-domain interactions. This pattern confirms that governance- and society-oriented domains operate less as insulated clusters and more as systemic connectors.

Importantly, the updated bottleneck analysis identifies Good Governance, Disaster Management, and Digital Infrastructure as the dominant structural mediators within the network. Good Governance emerges as the primary integrative backbone, mediating interactions across institutional, environmental, infrastructure, and socio-economic dimensions. Disaster Management serves as a critical conditioning node for adaptation to multi-hazard and chronic stressors, reflecting the system’s reliance on preparedness and coordinated response mechanisms. Digital Infrastructure’s prominence underscores the centrality of information flows, communication networks, and technological systems in sustaining cross-domain integration. Together, these bottlenecks reflect a structural shift toward governance and coordination-centric resilience rather than purely sectoral vulnerabilities.

The integration of CECS, ECS, MDPS, IDDR, and WDD metrics thus provides a multi-scalar diagnostic lens. Rather than relying solely on node betweenness or isolated indicator rankings, the model captures embeddedness (CECS), relational leverage (ECS), transitive conditioning (MDPS), domain coupling (IDDR), and system density (WDD). This layered structural perspective reveals resilience as an emergent property of network architecture.

Despite these advances, several limitations remain. Data heterogeneity, indicator normalization, and weighting assumptions influence structural outputs. Socio-economic and governance indicators often rely on composite or perception-based measures that may vary across jurisdictions. Additionally, high-density networks risk over-amplifying weak relationships unless thresholding and sensitivity testing are rigorously applied. Future work should incorporate robustness testing under alternative weighting schemes and longitudinal recalibration to assess the temporal stability of structural bottlenecks.

Additionally, edge weights in this study are derived from similarity-based measures using normalized differences between indicators. This approach reflects statistical alignment rather than causal dependency. In the absence of longitudinal data required for causality inference (e.g., Granger causality or structural equation modeling), similarity-based coupling provides a pragmatic alternative consistent with similarity-based network construction approaches [16,17]. The interpretation of strong coupling, therefore, indicates behavioral similarity rather than direct causal influence. Future work will incorporate causality-based methods as more comprehensive temporal datasets become available.

Overall, the extended USAH framework, combined with graph-based structural diagnostics, moves resilience assessment beyond static indicator inventories toward an operational systems model. By quantifying embeddedness, leverage points, transitive dependencies, and cross-domain coupling, the approach provides a more precise understanding of systemic fragility and integrative capacity. The Vancouver case study demonstrates that resilience emerges not only from infrastructure robustness but from governance coherence, coordinated disaster response, and digital connectivity, collectively shaping the structural backbone of the urban system.

From a policy perspective, the proposed framework enables a shift from sector-specific interventions toward system-level prioritization. High-centrality and bottleneck indicators (e.g., governance, disaster management, digital infrastructure) can be targeted as leverage points for maximizing system-wide resilience gains. Similarly, high ECS edges highlight critical interdependencies that require coordinated policy action across sectors, while IDDR results identify domains with strong cross-domain reliance that may benefit from integrated governance approaches. These outputs can support prioritization of investments, inter-agency coordination, and risk-informed planning. Detailed implementation pathways remain an area for future applied research.

5. Conclusions

This study sets out to address a key limitation in existing urban resilience frameworks: the lack of integrated, system-level analysis of socio-economic, infrastructural, and governance interdependencies. By extending the Urban System Abstraction Hierarchy (USAH) into a graph-based modeling environment, this research demonstrates how urban resilience can be operationalized as a networked, multi-domain system rather than a collection of isolated indicators.

Vancouver’s analysis shows that network structure, rather than performance within a specific sector, determines the structural basis of resilience. Although it has a moderate to high Weighted Dependency Density indicating that there is an extensive distribution of the risks from the various interdependencies throughout the system and not located primarily on one or two paths; more importantly three “bottleneck” nodes (Good Governance, Disaster Management, Digital Infrastructure) have been identified as having a greater influence over the overall resilience of the system due to their role as coordination points among institutional, economic, and environmental dimensions. Additionally, 106 high-weight dependencies were identified, which were not known to stakeholders and indicate that stakeholders’ mental models of urban risk systematically underestimate cross-domain coupling and therefore can also impact what interventions will be prioritized regarding increasing resilience.

Beyond methodological contributions, this study provides practical implications for urban planners and policymakers. The graph-based framework offers a decision-support tool that can identify hidden dependencies, diagnose systemic vulnerabilities, and guide targeted interventions. For Vancouver specifically, this means that investments in governance coherence, digital connectivity, and disaster preparedness yield disproportionate system-wide returns, not because these sectors are individually important, but because they structurally mediate resilience across all other domains. By shifting focus from siloed planning toward system-wide coordination, the approach supports more effective and adaptive resilience strategies in complex urban environments.

Future research should focus on extending this framework across multiple cities to enable comparative analysis. In parallel, incorporating longitudinal data will allow future work to explore causal dependency modeling through methods such as Granger causality or structural equation modeling [18], moving beyond the current similarity-based edge weight construction toward more theoretically grounded inference. Additional directions include incorporating temporal dynamics to assess how resilience evolves over time, and refining data inputs through improved stakeholder calibration and real-time data integration. These directions will further enhance the robustness and applicability of graph-based resilience modeling.

In conclusion, this research demonstrates that urban resilience is not only a function of what systems exist, but how they are connected. Understanding and managing these connections is essential for building cities that can adapt, withstand, and evolve in the face of increasing uncertainty.