A Conceptual Introduction to Hetero-Functional Graph Theory for Systems-of-Systems

A defining feature of 21st century engineering challenges is their inherent complexity, demanding the convergence of knowledge across diverse disciplines. Establishing consistent methodological foundations for engineering systems remains a challenge—one that both systems engineering and network science have sought to address. Model-based systems engineering (MBSE) has recently emerged as a practical, interdisciplinary approach for developing complex systems from concept through implementation. In contrast, network science focuses on the quantitative analysis of networks present within engineering systems. This article introduces hetero-functional graph theory (HFGT) as a conceptual bridge between these two fields, serving as an entry point for both communities. For systems engineers, HFGT preserves the heterogeneity of conceptual and ontological constructs in MBSE, including system form, function, and concept. For network scientists, it provides multiple graph-based data structures enabling matrix-based quantitative analysis. The modeling process begins with ontological foundations, where an engineering system is defined as an abstraction and represented with a model. Model fidelity is assessed using four linguistic properties: soundness, completeness, lucidity, and laconicity. A meta-architecture is introduced to manage the convergence challenges between domain-specific reference architectures and case-specific instantiations. Unlike other meta-architectures, HFGT is rooted in linguistic structures, modeling resources as subjects, system processes as predicates, and operands—such as matter, energy, organisms, information, and money—as objects. These elements are integrated within a system meta-architecture expressed in the Systems Modeling Language (SysML). The article concludes by offering guidance for further reading. Significance and Practitioner Points: This article introduces hetero-functional graph theory (HFGT) as a methodological bridge between the graphical modeling of Model-Based Systems Engineering (MBSE) and mathematical models founded in network science, dynamic systems, and operations research. For researchers, HFGT provides a rigorous methodological foundation that addresses the complex and heterogeneous interdependencies in systems of systems, overcomes the ontological limitations of multilayer networks, and reconciles the methods for structural analysis, dynamic simulation, and optimization. It addresses a critical gap in the systems science literature by enabling the mathematical synthesis of structural and functional viewpoints of complex engineering systems. For practitioners, HFGT offers a scalable and automatable methodology to translate complex architectural diagrams into computable mathematical models. This allows for proactive and self-consistent design and analysis of the structural and dynamic properties of a system-of-systems from the earliest phases of the system life cycle to its final implementation. Ultimately, the paper provides a conceptual introduction to HFGT, demonstrating a foundational language for engineering systems that ensures architectural descriptions are not just visual aids, but mathematically actionable blueprints.