Abstract

Critical infrastructure ensures the stability of society, economic growth, and national security. This paper presents a tensor-based model for assessing infrastructure resilience, considering technical, economic, environmental, social, and managerial aspects. The proposed model represents infrastructure subsystems - generation, transportation, and resource consumption-across different operational phases: pre-disaster, crisis, and recovery. Tensor analysis enables a comprehensive evaluation of interactions between system components, multiple threat impacts, and resilience criteria such as functionality, recovery time, threat resistance, adaptability, and economic efficiency. By defining threat vectors for various disruptions, including cyberattacks, natural disasters, and technological failures, the model identifies vulnerabilities and provides insights for strengthening infrastructure resilience. The findings support strategic management, risk mitigation, and policy development in national security and engineering planning.

Keywords

• resilience of critical infrastructures
• tensor analysis
• resilience criteria
• threat modeling
• risk management
• strategic planning
• multidimensional analysis
• system adaptability
• cyberattacks
• environmental sustainability
• 1

1. Introduction

Critical infrastructures, including energy, transportation, communications, water supply, and healthcare, are essential for national security and economic stability. However, these systems face increasing threats from natural disasters, cyberattacks, technological failures, and geopolitical conflicts. Their interconnected nature makes resilience a key priority.

Resilience defines a system’s ability to maintain functionality, resist external impacts, and recover after crises. It involves technical, organizational, social, and economic factors that ensure adaptability to dynamic threats. A structured approach to assessing resilience requires clear criteria, mathematical modeling, and analytical methods.

This paper explores tensor analysis as a tool for modeling resilient critical infrastructure. Tensor models enable the evaluation of complex interactions between system components, threat scenarios, and adaptive strategies. The proposed approach integrates technical, economic, and managerial aspects to enhance infrastructure security and operational stability.

2. Literature Review

The article [ 1 ] presents an innovative approach to assessing the resilience of critical infrastructure under conditions of multi-level threats, particularly for transportation facilities as key components of critical infrastructure. The authors have developed an adaptive methodology that integrates various risk parameters, providing a robust foundation for decision-making during crises. This approach serves as a basis for further research in infrastructure resilience and risk management.

The following study [ 2 ] focuses on a novel framework for evaluating the resilience of multicomponent critical infrastructure. Specifically, it demonstrates how modern approaches in engineering systems management can enhance resilience to complex threats. To achieve this, the authors pay significant attention to mathematical models for analyzing inter-system dependencies, which are critical for ensuring the uninterrupted functioning of infrastructure during emergencies.

In article [ 3 ], a quantitative method for evaluating the resilience of interdependent infrastructures is proposed. The mathematical model developed by the authors enables the assessment of functionality losses and recovery rates after emergency events. This approach is particularly relevant as it considers interconnections between infrastructure components, making it valuable for practical risk management solutions.

The analysis of urban critical infrastructure resilience, exemplified by Ahvaz, Iran, is presented in study [ 4 ]. The authors employ an indicative approach to assess vulnerabilities in urban networks such as water supply, electricity, and transportation. This method emphasizes the integration of environmental, social, and technical factors, thereby enhancing the overall resilience of urban systems to crises.

The work [ 5 ] proposes metrics and frameworks for analyzing the resilience of engineering and infrastructure systems. The research focuses on methods for evaluating systems' ability to withstand external impacts, quickly adapt, and restore functionality. This study provides a theoretical foundation for further works in resilience and risk management for infrastructure.

Research [ 6 ] emphasizes the impact of dynamic cost changes on assessing infrastructure resilience. It highlights the importance of incorporating time factors to improve the accuracy of resilience forecasting, developing methodologies for adaptive risk assessment and infrastructure management.

The development of metrics and methods for quantitative assessment of the resilience of power systems is presented in study [ 7 ]. The authors propose an integrative approach to analyzing operational and structural resilience. The suggested metrics allow the formulation of strategies for risk minimization and enhancing the reliability of energy supply systems.

Article [ 8 ] presents a comprehensive framework for evaluating the resilience of critical infrastructure components by incorporating technical, organizational, and social dimensions. This multidimensional approach aids in designing measures to enhance resilience across different sectors of critical infrastructure, offering a holistic view of how various factors interact.

Study [ 9 ] introduces a methodology for assessing the resilience of networked infrastructures, with a focus on the interdependencies among system elements and the effects of their potential failure. The findings provide a foundation for creating effective risk management strategies to maintain and strengthen critical infrastructure resilience.

In article [ 10 ], operational models for analyzing infrastructure resilience are examined. The study focuses on identifying vulnerabilities in networks and finding optimal solutions for ensuring continuous infrastructure operation during crises, making this work a significant step in developing resilience strategies.

Article [ 11 ] explores the resilience of power systems, considering approaches to assessing critical infrastructure resilience and criteria established by government policies. The work presents an interdisciplinary approach integrating technical and regulatory aspects to enhance the reliability of energy infrastructure.

Research [ 12 ] highlights the use of expert judgment to assess the resilience of critical infrastructures. The authors propose a model that incorporates subjective expert evaluations for quantitatively determining resilience levels. This study is useful for rapid risk analysis in situations with limited data.

In article [ 13 ], the concept of resilience curves for infrastructure is presented. The study identifies new performance metrics and data aggregation methods to evaluate the effectiveness of infrastructure systems during emergencies. The proposed approach allows for a detailed analysis of recovery dynamics and functionality losses.

Study [ 14 ] introduces a framework for evaluating the resilience of both infrastructural and economic systems. It offers an in-depth analysis of methods for modeling interdependencies among system components, facilitating a clearer understanding of their responses to different types of threats.

Work [ 15 ] presents a scenario-based methodology for assessing the resilience of critical infrastructures, with a particular focus on the seismic resilience of seaports. The authors develop a multi-level approach that integrates scenario modeling and impact assessment, enabling precise forecasting of potential risks.

Article [ 16 ] conducts a systematic review of quantitative resilience indicators for water infrastructure systems. The research focuses on developing metrics that quantify the ability of water systems to recover functionality after adverse impacts. This study serves as a foundation for strategic decision-making in water resource management.

Study [ 17 ] analyzes approaches to measuring the resilience of transportation infrastructure. The work focuses on developing indicators and methods for resilience assessment in the context of transport systems, although it provides limited information on the specifics of the infrastructure.

In article [ 18 ], the performance of green infrastructure is investigated through the lens of urban resilience. The authors propose an analytical methodology for assessing the impact of green infrastructure on the recoverability of urban systems, considering environmental and socioeconomic aspects.

Research [ 19 ] focuses on evaluating and enhancing organizational resilience in Slovakia's critical infrastructure. The work presents a multi-level approach to strengthening organizational capacity for adaptation and crisis response.

Article [ 20 ] examines principles and criteria for evaluating the resilience of energy systems in urban environments. This review study highlights key factors ensuring the reliability and continuity of energy supply under rapidly changing conditions.

In article [ 21 ], time-dependent resilience of urban infrastructural systems is assessed. The authors propose a methodology for resilience evaluation that accounts for dynamic changes in infrastructure functionality over time, enabling more precise planning for resilience improvements.

Study [ 22 ] offers a continuous and multidimensional assessment of resilience based on functional analysis for interconnected systems. This work emphasizes the importance of an integrated approach to infrastructure resilience assessment, where each element interacts with others, creating a complex network of interdependencies.

Article [ 23 ] explores metrics and methods for measuring resilience in transportation infrastructure. The work discusses the current state of development of criteria and methodologies for resilience assessment in transport systems, particularly in the context of climate change and extreme events.

Research [ 24 ] focuses on the assessment of infrastructure resilience, exploring different methods and approaches for measuring how infrastructure systems adapt to changing conditions.

Article [ 25 ] examines the integrity of infrastructure systems through a systemic perspective, highlighting the significance of integrating multiple components to ensure they can operate cohesively under unexpected circumstances.

In article [ 26 ], the concept of system resilience in the context of infrastructure is analyzed using Latvia as a case study. The research applies theoretical approaches to assessing infrastructure resilience in countries with transitional economies.

Study [ 27 ] proposes a unified approach to assessing the resilience and sustainability of urban infrastructure. This approach incorporates various parameters to evaluate infrastructure resilience under climate change and extreme events.

Article [ 28 ] proposes a qualitative methodology for evaluating the performance of IT infrastructure elements, considering technical characteristics and operating conditions. The author developed a model to assess the reliability and performance of IT system components, predict potential failures, and optimize operations, emphasizing the approach's versatility for various types of infrastructure.

Article [ 29 ] discusses a new approach to measuring the resilience of transportation infrastructure networks. The work focuses on developing methods for resilience assessment that consider various extreme event scenarios.

Research [ 30 ] addresses the assessment of resilience in interdependent infrastructure systems, emphasizing the modeling and analysis of joint recovery processes following damage.

Article [ 31 ] evaluates the resilience of interdependent infrastructures by examining different response strategies, with a focus on their capacity to recover during major disasters or crises.

3. Results

3.1. Key Criteria for Critical Infrastructure Resilience

Based on the analysis of the literature, the main criteria for the resilience of critical infrastructures have been identified (Table 1):          

Infrastructure Functionality – assessment of the ability of infrastructure to perform its core

functions during and after stress events (e.g., natural disasters or man-made catastrophes).

Recovery Capability – the ability of infrastructure to quickly recover after damage or functional disruptions. This criterion includes time, resources, and measures needed to return to normal conditions.

Resistance to External Threats – the ability of infrastructure to withstand extreme factors, such

as natural disasters, technological accidents, economic and social crises.

Flexibility and Adaptability – the capacity of infrastructure to adapt to new conditions and changes, such as climate change, technological advancements, or shifts in political and economic contexts.

Economic Cost Assessment – evaluation of recovery costs after a disaster, including direct

costs of restoration and indirect losses from service interruptions.

Integration with Other Systems – assessment of how infrastructure systems interact and depend on each other. This criterion is important for identifying how one failure may affect others.

Instant Analysis and Forecasting – utilization of data for real-time evaluation of the current state of infrastructure and prediction of potential issues.

Flexibility of Management Structures – the ability of management bodies and organizations responsible for infrastructure to quickly adapt to new conditions, organize effective responses, and coordinate recovery efforts.

Data Security and Protection – ensuring the security of data and information systems against

cyberattacks and other threats that may disrupt infrastructure operations.

Environmental Sustainability – assessment of the extent to which infrastructure complies with environmental standards and can adapt to changes in the surrounding environment. 112

Criterion Infrastructure

Functionality Recovery Capability

Resistance to External Threats Flexibility and Adaptability Economic Cost

Assessment Integration with

Other Systems Instant Analysis and

Forecasting Flexibility of Management

Structures Data Security and

Protection Environmental Sustainability

Criterion Description Sources (References)

Assessment of the ability of infrastructure [1], [6], [12], [17], to perform its core functions during and [19], [29] after stress events.

The ability of infrastructure to quickly [2], [9], [14], [17], recover after damage or functional [18], [29] disruptions, including time, resources, and measures needed for restoration.

The ability of infrastructure to withstand [3], [7], [15], [20], extreme factors, such as natural disasters, [24], [28] technological accidents, economic, and social crises.

The capacity of infrastructure to adapt to [4], [12], [14], [17],

new conditions and changes, such as [ 26 ], [ 30 ] climate change, technological advancements, or shifts in political and economic contexts.

Evaluation of recovery costs after a [8], [11], [18], [23], disaster, including direct restoration costs [28] and indirect losses from service interruptions.

Assessment of how infrastructure systems [5], [11], [15], [18], interact and depend on each other, [22], [29] identifying how one failure may impact others.

Utilization of data for real-time evaluation [6], [9], [19], [21], of the current state of infrastructure and [30] prediction of potential issues.

The ability of management bodies and [5], [14], [20], [29],

organizations responsible for [ 30 ] infrastructure to quickly adapt to new conditions, organize effective responses, and coordinate recovery.

Ensuring the security of data and [ 13 ], [ 15 ], [ 17 ], [ 28 ] information systems against cyberattacks and other threats that may disrupt infrastructure operations.

Assessment of the extent to which infrastructure complies with environmental standards and can adapt to changes in the surrounding environment. [ 4 ], [ 7 ], [ 10 ], [ 15 ], [ 27 ] utility systems), post-disaster).

   stages; statistical data.

Next, we will examine the tensor model of critical infrastructure resilience based on the identified resilience criteria.

3.2. Tensor Model of Resilient Critical Infrastructure

To construct a tensor model of resilient critical infrastructure based on the defined criteria, each criterion can be considered as a separate component interacting with others through specific parameters. A tensor model is a multidimensional mathematical object that allows for the description of interconnections between various characteristics of infrastructure and its resilience.

Let T represent the tensor describing the resilient critical infrastructure. Each element of the tensor corresponds to a specific resilience characteristic of the infrastructure, grouped according to its various parameters. The model can be represented as a third-order tensor: – index representing individual resilience criteria, – index representing subsystems or components of infrastructure (e.g., energy, transportation, – index representing the time aspect or stages of recovery (e.g., pre-disaster, during disaster,

Principles of the model operation are as follows:

the tensor , , defines all the interrelationships between resilience criteria, subsystems, and for each stage changes in infrastructure resilience can be assessed based on specific criteria, as well as interdependencies between subsystems; tensor parameters can be calculated based on expert assessments, mathematical models, or The proposed model can easily be expanded by introducing additional tensors. For example, we can introduce an additional threat tensor , which demonstrates the impact of specific threats on the resilience criteria of the system.

Let there be threats affecting the resilience of the system. Then, for each resilience criterion, we have the following threat vector , which reflects the impact of each threat on criterion :

The result of the impact of threats on the resilience of subsystems can be represented by tensor , composed of the corresponding matrices , which contain the result of multiplying by the corresponding threat element . Thus, for each criterion i we will obtain a matrix of size   × , where each element of this matrix is calculated as: = ( …

).

, , = , , × , where: , , – element of the matrix for criterion , subsystem and stage for threat . , , – element of the matrix for criterion , subsystem and stage .

– element of the threat vector for criterion and threat .

The general appearance of the matrix for each threat is as follows:

Additional tensors can also be introduced into the model, which, for example, describe vulnerabilities and protection mechanisms within the system. Additionally, further interaction and mutual influence rules between tensors can be defined.

3.3. Scenarios for model research

The model, built on multidimensional tensors for assessing system resilience under the influence of threats, allows for a series of experimental studies to analyze systems in different scenarios: 1. Analysis of the impact of different threats on system resilience   

Objective – to investigate how different types of threats affect different resilience criteria.

Experiment:   change the values of tensor to simulate varying threat intensities; analyze the resulting tensor to identify resilience criteria most affected by the threats.

Result – identification of the most vulnerable criteria or subsystems.

2. Evaluation of the effectiveness of protection measures

Objective – to verify how specific protective measures reduce the impact of threats.

Experiment:  add a correction tensor to representing the influence of protective measures;  recalculate and compare it to the baseline value.

Result – assessment of the effectiveness of specific measures. 3. Analysis of disaster scenarios           



Objective – to investigate how changes in one or more elements of affect the resulting .

Experiment:  gradually change the threat values (e.g., increase or decrease the impact of a specific threat on a specific criterion); analyze the dynamic changes in .

Objective – to model various disaster scenarios and assess system resilience at each stage.

Experiment:  for each stage (pre-disaster, during disaster, post-disaster), modify the corresponding values of tensor and analyze the changes in tensor ;  specifically, study how the system recovers after a disaster.

Result – identification of critical stages requiring the most attention.

4. Comparison of systems with different resilience characteristics

Objective – to assess which system has higher resilience under the same threat conditions.

Experiment:  use different initial values of tensor (e.g., for different organizations, regions, or system types);  analyze the resulting tensors and identify the systems with the best performance.

Result – ranking of systems based on resilience level. 5. Determination of system sensitivity to changes in threat intensity             

Result – identification of critical threats with the most significant impact on the system.

6. Determination of interrelationships between resilience criteria

Objective – to find out how the impact of one threat on a specific criterion can change other criteria.

Experiment:  analyze the matrices in tensor corresponding to different resilience criteria;  build correlations between the results.

Result – identification of interdependencies between criteria.

7. System parameter optimization 9. Model validation on real data

Objective – to assess how the system responds to prolonged periods of threat impact.

Experiment:  use a variable tensor to model long-term or periodic threats;  analyze changes in over time.

Result – forecasting the long-term resilience of the system.

Objective – to compare the model's results with real-world data.

Experiment:  use empirical data to build and .

 compare the resulting with actual performance indicators of systems.

Result – validation of the model and its potential use for real systems.

Objective – to determine optimal parameter values to reduce the impact of threats.

Experiment:  use optimization algorithms to find values of that maximize resilience with fixed .

Result – recommendations for improving the system.

8. Modeling long-term consequences

3.4. Example of using the model for the energy sector

To apply the tensor model of critical infrastructure resilience to the energy sector, it is necessary to define how each of the resilience criteria affects energy systems and determine the values for each criterion and subsystem (e.g., energy networks, generation, and electricity transmission).

We describe the resilience criteria of the model as follows:  infrastructure functionality – the ability of electrical networks and stations to perform their functions after natural disasters or man-made accidents.  recovery capability – the time required to restore power supply after an accident or disaster.  resilience to external threats – the ability of energy systems to withstand natural disasters (floods, snowstorms) or technological accidents.  flexibility and adaptability – the ability to adapt to changes in electricity demand or new technologies, such as the integration of renewable energy sources.  economic cost assessment – the cost of restoring energy infrastructure after a disaster. systems. to energy crises. integration with other systems

– the impact of energy disruptions on other critical infrastructures, such as transportation or utilities. real-time analysis and forecasting – the use of data to assess the current state of energy flexibility of management structures – the ability of management bodies to respond quickly data security and protection – protection of energy systems from cyberattacks. environmental resilience

– adaptation of energy systems to environmental requirements, particularly reducing CO2 emissions.

For each criterion (10 criteria), we have a 3x3 matrix, where each matrix represents a subsystem at different stages. Therefore, the overall form of the tensor will be as follows (1):

Let us assume there are three main subsystems of energy infrastructure: : energy generation; : electricity transmission; : electricity consumption (distribution and consumption).

We will evaluate the resilience of the energy system at three stages: : before the disaster (normal state); : during the disaster (damage); : after the disaster (recovery).

where each is a matrix that describes the corresponding criterion.

In particular, for each (2): = , ,

We have a three-dimensional structure where each element is a matrix of size 3  × 3 and represents a subsystem at different stages for each resilience criterion, the first index represents the resilience criterion (for = 1,2, … ,10), the second index (1 – generation, 2 – transportation, 3 – consumption), the third index (1 – before the disaster, 2 – during the disaster, 3 – after the disaster).

Let there be 4 threats that affect the system's resilience. Then, for each resilience criterion, we have the following threat vector , which reflects the impact of each threat on criterion (3): = ( … ). (3)

The result of the impact of threats on subsystem resilience can be represented by a tensor , consisting of the corresponding matrices , which contain the result of multiplying by the corresponding threat element . Thus, for each criterion , a matrix of size 3  × 3 will be obtained, where each element of this matrix is computed as (4):

, , = , , × , (4)

, , is the element of the matrix for the -th criterion, the -th subsystem, and the -th stage for infrastructure functionality ( ): recovery capability ( ): resilience to external threats ( ): for : for : for : = [0.9,0.8,0.7,0.6], = [0.85,0.75,0.65,0.55], = [0.8,0.7,0.6,0.5].

, , is the element of the matrix for the -th criterion, the -th subsystem, and the -th stage.

is the element of the threat vector for the -th criterion and the -th threat.

The general form of the matrix for each threat is as follows (5): allows assessing the resilience of the energy system to disasters by analyzing the impact of threats on key resilience criteria. Let us have the following matrices for three criteria:          

For each resilience criterion we calculate the matrices for all threats using formula (4).

Below are a few examples:

threat 1, criterion 1 ( Four main threats for the three criteria ( The results presented indicate the following: resilience reduction during a disaster ( ):  

0.7 ⋅ 0.85 = 0.6 ⋅ 0.85 0.4 ⋅ 0.85 0.6 ⋅ 0.85 0.55 ⋅ 0.85 0.3 ⋅ 0.85 0.5 ⋅ 0.85 0.5 ⋅ 0.85 = 0.2 ⋅ 0.85 for criterion , the most vulnerable stages are in the transportation subsystem ( ) and consumption subsystem ( ). for criterion , the impact of the disaster reduces the ability for quick recovery, particularly in the generation subsystem.

4. Discussions

The results displayed on the heatmaps (Figures 1-9) provide crucial information about changes in systems and threat levels at different stages of a disaster. Each heatmap shows the values of the tensor SS for a specific stage of the disaster (before, during, after) for each criterion, such as "Infrastructure Functionality," "Restoration Capability," and "Resilience to Threats." The color scale, such as "seismic," allows for visualization of contrasts between high and low values. Warm colors like red and orange may indicate high values or critical threat levels, while cool colors like blue and purple represent low values or the absence of threats.

The threat levels can have varying impacts on different subsystems, which is reflected in the heatmap. Generally, higher threat levels lead to significant changes in the values of tensor SS, indicating vulnerabilities in infrastructure or restoration capability. Different stages of the disaster also have varying impacts on these indicators. In the pre-disaster stage, values may be relatively stable, but weak points may already be identified that require attention. During the disaster stage, tensor values can change dramatically, indicating a high level of threats and potential degradation of system efficiency. After the disaster, the recovery or stabilization process may be visible, although tensor values may remain high due to the long-term effects of the disaster (Figures 3, 6, 9).

The heatmap also shows how each subsystem, such as generation, transportation, and consumption, responds to different threat levels at various stages. For example, the "Transportation" subsystem may be more vulnerable during the disaster, while "Generation" might show more stable results before the disaster. It is important to compare the heatmaps for different disaster stages to understand how system behavior changes over time. This helps identify trends, such as an increase in system vulnerability at certain times, depending on the type of disaster and threat level.

Overall, the heatmaps help visualize how various threat levels and disaster stages impact the functionality of systems and subsystems, as well as indicate areas where improvements are necessary to enhance resilience and infrastructure recovery.

5. Conclusions

This paper examines the conceptual and practical aspects of ensuring critical infrastructure resilience in an era of growing complexity and system interdependence. The proposed resilient critical infrastructure model, based on tensor analysis, captures the multidimensional nature of infrastructure, threat impacts, and subsystem dynamics across different operational stages.

The model formalizes key resilience criteria, including functionality, recovery speed, adaptability, economic efficiency, data security, and environmental sustainability. Tensor-based assessments enable precise identification of vulnerabilities and the evaluation of threat impacts, particularly in energy infrastructure.

The findings highlight the model’s potential for risk monitoring, forecasting, and strategy development to enhance infrastructure resilience. The approach is valuable for national security agencies, engineers, and researchers in risk management. Future work will expand the model to address intersectoral dependencies and assess the role of digitalization, AI, and blockchain in strengthening adaptive capabilities.

Declaration on Generative AI

During the preparation of this work, the authors used X-GPT-4 in order to: Grammar and spelling check. After using these tools/services, the authors reviewed and edited the content as needed and take full responsibility for the publication’s content.

References

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