Multidimensional Architecture Design for Cross-Domain Operational Systems: A Framework for Robustness and Adaptability Optimization

Multidimensional Architecture Design for Cross-Domain Operational Systems: A Framework for Robustness and Adaptability Optimization

1.    Introduction 
Cross-domain operational systems represent a new generation of complex systems in which heterogeneous resources, functional capabilities, information infrastructures, and organizational entities operate across multiple interconnected domains. Unlike conventional systems that are generally designed around a relatively stable operational environment and a clearly defined organizational structure, cross-domain systems must operate under conditions characterized by uncertainty, dynamic interactions, heterogeneous resources, incomplete information, and rapidly changing mission requirements.
From a systems-engineering perspective, the fundamental characteristic of a cross-domain operational system is not simply the coexistence of multiple domains, but rather the continuous interaction and dynamic coupling among heterogeneous system elements. Consequently, system effectiveness increasingly depends on the quality of interconnections, information exchange, organizational coordination, resource allocation, and structural adaptability.
The architecture of a complex operational system represents its highest-level structural description. It determines, to a considerable extent, the composition of system elements, their relationships, deployment configurations, organizational structures, information flows, and functional processes. An inappropriate architecture may result in excessive structural complexity, insufficient redundancy, weak information connectivity, poor resource utilization, and limited adaptability. Therefore, architecture design should be regarded as a fundamental stage preceding detailed component design and system implementation.
Several architecture description frameworks have been developed internationally, including the Department of Defense Architecture Framework (DoDAF), NATO Architecture Framework (NAF), MODAF, UPDM, and TOGAF. These frameworks provide structured approaches for representing complex systems from different viewpoints. Model-based systems engineering (MBSE), agent-based modeling, and simulation-based systems engineering have subsequently extended these approaches by connecting architecture descriptions with executable models and simulation environments.
However, conventional architecture design methods still face several limitations when applied to highly dynamic cross-domain systems. First, architecture descriptions frequently emphasize structural relationships but provide insufficient quantitative representation of robustness. Second, adaptability is often treated as a qualitative property rather than as an explicit design variable. Third, conceptual, physical, and logical architectures are sometimes developed as relatively independent views, which may result in inconsistencies between capability requirements, physical resources, organizational structures, and information processes.
To address these limitations, this study proposes an enhanced multidimensional architecture design framework in which robustness and adaptability are explicitly incorporated into architecture design and evaluation.
The main contributions of this study are as follows:
    A unified multidimensional architecture framework integrating conceptual, physical, and logical architectures is established.
    Architectural robustness is introduced as an explicit variable for measuring the ability of an architecture to maintain essential functions under component degradation, communication disruption, and environmental uncertainty.
    Architectural adaptability is introduced as an explicit variable for measuring the ability of an architecture to modify its structure, resources, and information relationships in response to changing requirements.
    A directed complex-network model is established to represent heterogeneous relationships among system elements.
    A multi-objective architecture optimization model is proposed to jointly consider effectiveness, robustness, adaptability, complexity, and resource consumption.
    A closed-loop architecture design process is established, connecting requirements analysis, architecture construction, evaluation, simulation, optimization, and iterative refinement.
Accordingly, the proposed framework extends conventional multidimensional architecture design from a primarily descriptive methodology toward a quantitative, evaluative, and optimization-oriented architecture design approach.
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2. Multidimensional Architecture Design Framework
2.1 Characteristics of Cross-Domain Operational Systems
Cross-domain operational systems possess several characteristics that distinguish them from conventional systems.
First, system elements are highly heterogeneous. Different elements may possess different functions, performance characteristics, communication protocols, organizational affiliations, and information-processing capabilities.
Second, the operational environment is dynamic. System requirements may change because of environmental disturbances, mission transitions, resource availability, or changes in system states.
Third, system relationships are strongly coupled. The failure or degradation of one component may influence multiple other components through information, functional, organizational, or resource dependencies.
Fourth, system-level effectiveness cannot always be obtained through the simple aggregation of individual component performance. Emergent behavior may arise from interactions among components.
Therefore, architecture design should simultaneously consider:
    functional capability;
    structural connectivity;
    organizational relationships;
    information exchange;
    resource allocation;
    robustness;
    adaptability;
    complexity;
    reconfiguration cost.
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2.2 Overall Architecture Design Process
The proposed architecture design process consists of six major stages:
Requirement analysis → Conceptual architecture → Physical architecture → Logical architecture → Robustness/adaptability evaluation → Multi-objective optimization
Unlike a purely sequential design process, the proposed framework adopts an iterative architecture-development mechanism.
Let the architecture state at iteration (k) be represented as
 
where:
    (AC) represents the conceptual architecture;
    (AP) represents the physical architecture;
    (AL) represents the logical architecture.
The architecture is evaluated according to
 where:
    (E) = system effectiveness;
    (R) = robustness;
    (D) = adaptability;
    (C) = architecture complexity;
    (Q) = resource or reconfiguration cost.
The design process seeks an architecture that satisfies required capability constraints while achieving an appropriate balance among these objectives.
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3. Conceptual Architecture Design
3.1 Definition
The conceptual architecture represents the highest-level description of the system. It does not focus on detailed component implementation but establishes the relationship between environmental conditions, mission requirements, capability requirements, and operational concepts.
The conceptual architecture therefore provides the basis for subsequent physical and logical architecture development.
The conceptual architecture includes four principal activities:
    environmental analysis;
    mission and stakeholder analysis;
    capability requirement analysis;
    operational concept generation.
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3.2 Environmental and Context Analysis
The first stage is the identification of the operational environment.
Instead of representing the environment as a static condition, the proposed framework describes it using a state vector:
 
where  represents an environmental variable relevant to system operation.
The environment can therefore be represented as
 
rather than as a constant state.
The architecture should remain functional under a range of plausible environmental states:
 
where  denotes the expected operational environment space.
This formulation provides a basis for evaluating architecture robustness under environmental uncertainty.
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4. Capability Requirement Analysis
Capability requirements are derived from mission objectives, environmental conditions, functional requirements, performance requirements, and system constraints.
Let the capability vector be
 
where each  represents a required system capability.
A requirement matrix can subsequently be defined as
 
where
 
This matrix establishes traceability between requirements and capabilities.
An important advantage of this approach is that subsequent physical and logical architecture elements can be traced back to their originating requirements.
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5. Operational Concept Generation
Based on the capability requirements, an operational concept is generated to describe how the system is expected to achieve its required functions.
The operational concept establishes:
    major functional activities;
    information dependencies;
    temporal relationships;
    organizational coordination;
    resource interactions;
    transition conditions between different system states.
The operational concept should not be treated as a fixed sequence. Instead, multiple feasible operational paths should be considered.
Let
 
represent the set of feasible functional pathways.
The architecture should maintain sufficient alternatives to prevent excessive dependence on a single pathway.
This characteristic directly contributes to architectural robustness.
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6. Physical Architecture Design
6.1 Definition
The physical architecture describes the actual system elements, their functional characteristics, interconnections, quantities, and deployment configurations.
The physical architecture can be represented as
 
where:
    (V) represents physical elements;
    (E) represents relationships among elements;
    (L) represents deployment locations or configuration states.
The elements may include sensing, processing, communication, command, support, and other functional entities, depending on the specific application.
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7. Physical Element Selection
Physical elements should be selected based on capability requirements rather than simply according to existing equipment inventories.
Let
 
 represent candidate physical elements capable of contributing to capability (c_i).
The selection problem can therefore be expressed as:
 
subject to:
 
where:
     is the importance weight of capability  
     indicates whether element (j) is selected;
     represents its resource requirement;
     represents the available resource budget.
This converts physical architecture construction into a constrained architecture-selection problem.
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8. Complex-Network Representation
Because a cross-domain architecture consists of numerous heterogeneous elements and relationships, graph theory provides an appropriate mathematical representation.
The architecture can be represented as a directed graph:
 
where
 
represents system elements and
 
represents directed relationships.
The adjacency matrix is defined as
 
where
 
For weighted relationships,
 
where  may represent communication reliability, latency, bandwidth, dependency strength, or another architecture-specific parameter.
This weighted representation is particularly useful for robustness and adaptability analysis.
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9. Architecture Connectivity and Complexity
Several network metrics can be used to evaluate the structural characteristics of an architecture.
9.1 Connectivity
The normalized connectivity index can be expressed as
 
for a directed architecture.
Higher connectivity generally improves information accessibility, but excessive connectivity may increase structural complexity and resource consumption.
9.2 Average Path Length
 
where  represents the shortest path between nodes  and  .
9.3 Network Density
 
which provides a measure of the relative density of system relationships.
9.4 Centralization
Architecture centralization can be evaluated using node centrality measures. Excessive dependence on a small number of highly central nodes may increase vulnerability to component degradation.
Therefore, high network connectivity should not automatically be interpreted as optimal architecture quality.
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10. Architectural Robustness
10.1 Definition
The first new variable introduced by this study is architectural robustness.
Robustness refers to the ability of an architecture to maintain essential functions when subjected to disturbances without requiring fundamental structural redesign.
Disturbances may include:
    component degradation;
    communication interruption;
    resource reduction;
    environmental uncertainty;
    temporary information loss;
    partial functional failure.
Let the nominal system effectiveness be
 
and the effectiveness after disturbance (s) be
 
A basic robustness index can be defined as
 
For multiple disturbance scenarios:
 
where  represents the probability or importance weight of disturbance scenario (s).
A higher (R) indicates that the architecture is less sensitive to disturbances.
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11. Architectural Redundancy
Robustness is strongly associated with functional redundancy.
For a function  , let  denote the number of independent elements capable of supporting the function.
A simple redundancy index can be defined as
 
 
The overall redundancy can then be expressed as
However, excessive redundancy increases cost and complexity. Therefore, redundancy should be optimized rather than maximized.
This establishes an important design principle:
The objective is not maximum redundancy, but sufficient redundancy under acceptable resource and complexity constraints.
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12. Architectural Adaptability
12.1 Definition
The second major variable introduced in the proposed framework is architectural adaptability.
Adaptability describes the ability of a system architecture to modify its organizational structure, information relationships, resource allocation, or functional configuration in response to changes in mission requirements and environmental conditions.
Let
 
represent the initial architecture and
 
represent the reconfigured architecture.
The architectural distance between the two configurations can be defined as
 
while the corresponding reconfiguration cost is
 
The adaptability index can therefore be expressed conceptually as
 
where  represents the improvement or preservation of system effectiveness following reconfiguration.
A high value indicates that the architecture can achieve significant functional adaptation with relatively low reconfiguration cost.
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13. Dynamic Reconfiguration Model
The architecture can be considered as a dynamic state:
 
When environmental or mission conditions change from
 
to
 
the architecture may need to transition from
 
to
 
The reconfiguration process can be represented as
 
where R is the architecture-reconfiguration operator.
This formulation makes adaptability an explicit architectural property rather than an informal qualitative description.
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14. Logical Architecture Design
Logical architecture describes the functional and organizational relationships that cannot be adequately represented by physical architecture alone.
It includes:
    organizational hierarchy;
    functional processes;
    information interaction;
    temporal relationships;
    information-quality propagation.
The logical architecture can be represented as
 
where:
    (F) = functional relationships;
    (I) = information relationships;
    (T) = temporal relationships;
    (O) = organizational relationships.
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15. Organizational Structure
Traditional centralized architectures generally adopt hierarchical structures:
 
Such structures provide clear authority and relatively simple coordination mechanisms, but may become less flexible under rapidly changing conditions.
A distributed architecture instead permits multiple nodes to interact through lateral relationships.
The proposed framework therefore considers organizational structure as a variable rather than a fixed design assumption.
The degree of organizational decentralization can be represented as
 
The optimal value of (OD) depends on mission requirements, information latency, resource constraints, and adaptability requirements.
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16. Information Interaction Architecture
Information interaction is represented by an information-flow graph:
 
Each edge represents an information exchange between two functional elements.
For each information link  a vector can be assigned:
 
where:
    qacc = information accuracy;
    qtim = timeliness;
    qrel = reliability.
An overall information quality function may then be expressed as
 
with
 
This provides a quantitative mechanism for evaluating the information dimension of the architecture.
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17. Information Propagation and Quality Degradation
Information quality may decrease as information passes through multiple processing and transmission stages.
For an information chain consisting of (n) stages, the resulting quality can be modeled as
 
where:
     = initial information quality;
     = degradation coefficient associated with stage (i).
This model makes it possible to identify architectures that contain excessive information-processing or transmission dependencies.
Reducing unnecessary information-processing stages can improve both efficiency and robustness.
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18. Multi-Objective Architecture Optimization
The central extension of the proposed research is to transform architecture design into a multi-objective optimization problem.
The architecture quality vector is defined as
 
The optimization problem is:
 
subject to:
 
A normalized weighted objective function can be expressed as
 where:
    (E) = effectiveness;
    (R) = robustness;
    (D) = adaptability;
    (Q) = information quality;
    (C) = architecture complexity;
    (K) = reconfiguration cost.
The weights satisfy
 
This formulation allows the architecture designer to adjust priorities according to mission requirements.
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19. Robustness–Adaptability Trade-off
An important finding of the proposed framework is that robustness and adaptability should not be treated as completely independent variables.
A highly redundant architecture may exhibit strong robustness but suffer from increased complexity.
Conversely, a highly flexible architecture may exhibit strong adaptability but incur greater reconfiguration costs.
Therefore, the architecture design problem can be regarded as a trade-off:
 
subject to:
 
and
 
The optimal architecture is therefore not necessarily the architecture with the maximum individual robustness or adaptability. Rather, it is an architecture located on an appropriate Pareto-optimal frontier.
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20. Architecture Evaluation Framework
The proposed architecture can be evaluated using five dimensions:
Dimension    Representative indicators
Effectiveness    Capability fulfillment, functional coverage
Robustness    Failure tolerance, redundancy, performance degradation
Adaptability    Reconfiguration cost, response time, structural flexibility
Information quality    Accuracy, timeliness, reliability
Complexity    Number of nodes, edges, dependencies, coordination burden
A comprehensive architecture score can be calculated as
 
This provides a standardized mechanism for comparing alternative architectures.
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21. Architecture Iteration and Optimization
The proposed framework adopts an iterative process:
Requirements → Conceptual Architecture → Physical Architecture → Logical Architecture → Simulation → Evaluation → Optimization → Architecture Update
If the evaluated architecture does not satisfy the required performance:
 
the architecture returns to an earlier design stage.
This feedback mechanism is important because architecture quality cannot always be determined during the initial design phase.
The iterative process can be expressed as
 
where  represents the architecture optimization operator.
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22. Discussion
The proposed multidimensional architecture framework extends conventional architecture design in three important ways.
First, it establishes explicit relationships among conceptual, physical, and logical architectures. The conceptual architecture determines the capability requirements, the physical architecture maps capabilities to system resources, and the logical architecture defines how these resources interact and perform functions.
Second, the proposed approach transforms robustness and adaptability from descriptive concepts into measurable architecture variables.
Third, the framework supports iterative optimization rather than one-time architecture construction.
From a systems-engineering perspective, this is particularly important because architecture quality emerges from interactions among multiple dimensions rather than from any single structural characteristic.
A system may contain highly capable individual components but still exhibit poor system-level performance if its information connectivity is weak. Similarly, a highly connected architecture may exhibit poor robustness if it depends excessively on a limited number of central nodes.
Consequently, architecture evaluation should focus on the relationship structure among components, rather than simply on individual component performance.
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23. Proposed Architecture Optimization Algorithm
The overall optimization procedure can be summarized as follows.
Step 1: Requirement initialization
Define mission requirements and system constraints.
Step 2: Conceptual architecture generation
Construct alternative conceptual architectures according to environmental and capability requirements.
Step 3: Physical architecture generation
Select candidate components and construct alternative physical configurations.
Step 4: Logical architecture generation
Construct organizational, functional, information, and temporal relationships.
Step 5: Network modeling
Convert the architecture into a directed weighted graph.
Step 6: Robustness evaluation
Apply representative disturbance scenarios and calculate
 
Step 7: Adaptability evaluation
Apply representative environmental or mission changes and calculate
 
Step 8: Multi-objective optimization
Calculate
 
and identify Pareto-optimal architectures.
Step 9: Simulation and validation
Evaluate candidate architectures under multiple representative scenarios.
Step 10: Iterative refinement
Return to the architecture generation stage if the required performance has not been achieved.
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24. Conclusions
This study proposed a multidimensional architecture design framework for cross-domain operational systems and extended the conventional conceptual–physical–logical architecture methodology by incorporating robustness and adaptability as explicit design variables.
The principal conclusions are as follows.
(1) A unified multidimensional architecture framework was established. The framework integrates conceptual architecture, physical architecture, and logical architecture into a continuous architecture-development process, thereby improving traceability between requirements, system elements, functions, and information relationships.
(2) Architectural robustness was introduced as a quantitative variable. By considering component degradation, communication disruption, environmental uncertainty, and resource reduction, the proposed robustness index can characterize the ability of an architecture to maintain essential functions under disturbances.
(3) Architectural adaptability was introduced as a second major variable. Adaptability measures the ability of an architecture to modify its configuration, organizational relationships, functional pathways, and information interactions when environmental or mission conditions change.
(4) A directed complex-network representation was developed to describe relationships among heterogeneous architecture elements. Network connectivity, centrality, density, path length, and redundancy can therefore be incorporated into architecture evaluation.
(5) A multi-objective optimization model was proposed to jointly optimize system effectiveness, robustness, adaptability, information quality, structural complexity, and reconfiguration cost. This approach avoids optimizing a single architecture characteristic at the expense of overall system performance.
(6) Robustness and adaptability should be considered jointly rather than independently. Excessive redundancy may improve robustness while increasing complexity, whereas excessive structural flexibility may improve adaptability while increasing reconfiguration costs. Consequently, the desired architecture should be selected from a Pareto-optimal solution set according to specific requirements and constraints.
(7) The proposed framework establishes a foundation for subsequent MBSE-based architecture modeling, simulation, quantitative evaluation, and optimization. Future research can further introduce dynamic network models, uncertainty quantification, machine-learning-assisted architecture optimization, and digital-twin-based validation to improve the accuracy and efficiency of architecture evaluation.
Overall, the proposed methodology transforms multidimensional architecture design from a primarily descriptive activity into a quantitative, iterative, and optimization-oriented systems-engineering process. By explicitly integrating robustness and adaptability into conceptual, physical, and logical architecture design, the framework provides a general methodological basis for designing complex cross-domain systems capable of maintaining essential functions under uncertainty while dynamically adapting to changing operational requirements.


Ethical Considerations
Not applicable. This study did not require ethical approval because it does not include human or animal subjects and does not involve any personal or sensitive data.



Acknowledgment: 
The author would like to express their sincere gratitude to The International Journal of Applied Sciences - Noor Al-Ilm for Publishing and Distribution for their generous support in waiving all publication fees and facilitating the publication of this manuscript free of charge. Their commitment to promoting scientific research and supporting researchers is highly appreciated.








Author Contribution: 
All authors contributed equally to the main contributor to this paper. All authors read and approved the final paper.




Declaration of generative AI and AI-assisted technologies in the writing process
The authors hereby declare that no generative artificial intelligence or AI-assisted technologies were used at any stage during the preparation of this manuscript, including language editing, proofreading, or content development. The authors take full responsibility for the originality and integrity of the work presented in this publication.









Funding: 
This research received no external financial funding. The authors also acknowledge The International Journal of Applied Sciences, Noor Al-Ilm for Publishing and Distribution, for providing a full waiver of the publication fees. The publication fee waiver was provided as editorial support and did not involve any financial contribution to the conduct, design, analysis, or reporting of the research.

Conflicts of Interest: 
“The authors declare no conflict of interest.”