Hetero-functional Graph Theory

• Hetero-functional Graph Theory is a formalism that distinctly defines resources, processes, operands, buffers, and capabilities for modeling heterogeneous systems.
• It integrates mathematical structures like matrices, tensors, and Petri nets to enable dynamic simulation, optimization, and state estimation across various domains.
• HFGT supports resilience, robustness, and multi-domain analysis in energy, infrastructures, and environment while addressing computational scaling challenges.

Hetero-functional Graph Theory (HFGT) is a mathematical and model-based systems engineering formalism for representing heterogeneous systems in terms of resources, processes, operands, buffers, and capabilities, rather than reducing them to homogeneous nodes and undifferentiated edges. Across the literature, it is presented as a bridge between MBSE and network science, preserving system form, system function, and system concept while enabling matrix-, tensor-, Petri-net-, optimization-, and resilience-based analysis of large-scale systems-of-systems (Farid et al., 2021, Farid et al., 29 May 2025). Its applications span electric power, interdependent urban utilities, multi-modal energy systems, hydrogen-natural gas infrastructure, hydrological and watershed systems, life cycle analysis, economic input-output models, and megaproject engineering management (Thompson et al., 2020, Munikoti et al., 2020, Thompson et al., 2022, Schoonenberg et al., 2021, Harris et al., 31 May 2025, Harris et al., 2 Mar 2026, Naderi et al., 16 Feb 2026, Hosseini et al., 29 May 2025).

1. Ontological basis and conceptual scope

HFGT is grounded in a subject–predicate–object meta-architecture. In that mapping, system resources act as subjects, system processes as predicates, and operands as objects. This linguistic structure is used to preserve ontological clarity when translating SysML models into mathematical objects suitable for analysis (Farid et al., 29 May 2025, Hosseini et al., 29 May 2025). The literature repeatedly emphasizes that conventional graphs conflate “what runs” with “what is connected,” whereas HFGT distinguishes physical form from functional behavior and from the allocation of behavior to physical or digital assets (Munikoti et al., 2020, Farid et al., 2021).

Several papers define the same core ontology with slightly different emphases. Resources $R$ are the assets or agents that execute processes; processes $P$ are transformation, transportation, holding, decision, measurement, or other activities; operands $L$ are the entities consumed, produced, stored, or transported; buffers $B_s \subseteq R$ are those resources that can store operands at a location; and capabilities are the bindings of resources to processes, typically written as tuples such as $(r,p)$ or, in some formulations, as triples involving the operand (Schoonenberg et al., 2021, Gohil et al., 30 May 2025, Naderi et al., 16 Feb 2026, Naderi et al., 27 May 2025). This means that HFGT does not privilege a single commodity or a single network layer; it is explicitly designed for multi-domain systems in which matter, energy, information, money, organisms, or sectoral products may coexist in the same analytical structure (Ghorbanichemazkati et al., 2024, Farid et al., 29 May 2025).

The conceptual literature also frames HFGT as a response to convergence problems in systems-of-systems. It is presented as a means of reconciling domain-specific reference architectures and case-specific instantiations, and as a way to support soundness, completeness, lucidity, and laconicity in system modeling (Farid et al., 29 May 2025). A recurring implication is that HFGT is not only a graph-theoretic device but also an ontological discipline for keeping form, function, and behavior aligned in a single representation.

2. Mathematical structures and graph representations

At the structural level, HFGT begins from a resource–process allocation. One common notation defines a binary system concept matrix

$$A_s \in \{0,1\}^{|P|\times|R|}, \quad A_s(w,v)=1 \iff \text{process } p_w \text{ can execute on resource } r_v,$$

while another uses a knowledge base matrix $J_s$ or $H_{0F}$ to encode the same allocation relation in different orientations (Schoonenberg et al., 2021, Munikoti et al., 2020, Thompson et al., 2020). These nonzero allocations define the available capabilities.

The most characteristic data structures in HFGT are the positive and negative hetero-functional incidence tensors. In a common formulation,

$$M^- \in \{0,1\}^{|L|\times|B_s|\times|E|}, \qquad M^+ \in \{0,1\}^{|L|\times|B_s|\times|E|},$$

where $M^-_{i,s,k}=1$ if capability $P$0 pulls operand $P$1 from buffer $P$2, and $P$3 if capability $P$4 injects operand $P$5 into buffer $P$6 (Schoonenberg et al., 2021, Gohil et al., 30 May 2025, Thompson et al., 2022). After matricization, these become second-order incidence matrices over buffer–operand pairs and capabilities, and their difference

$P$7

serves as a net incidence matrix (Gohil et al., 30 May 2025, Naderi et al., 16 Feb 2026).

Adjacency is then obtained by contracting the positive and negative incidence structures after matricization. Different expositions adopt different orientation conventions, including

$P$8

and

$P$9

but in both cases the goal is to determine when one capability can feed another through a compatible operand–buffer relation (Thompson et al., 2022, Harris et al., 31 May 2025). Tensor-based formulations extend this idea further by defining fourth-order adjacency tensors over resource–process–resource–process indices, then flattening them back into matrices when a standard graph representation is needed (Farid et al., 2021, Hosseini et al., 29 May 2025).

The weighted HFGT literature adds a dependency matrix

$L$0

with $L$1, $L$2 if $L$3, and $L$4 for each process with at least one incoming edge. This allows downstream processes to draw fractional inputs from multiple upstream processes and supports weighted robustness analysis in interdependent infrastructures (Munikoti et al., 2020).

A notable point of variation is representational emphasis. Some papers describe HFGT nodes as processes, others as capabilities, and others as the place–transition structure of an elementary Petri net. This suggests that HFGT is best understood as a family of tightly related data structures rather than a single fixed graph object.

3. Dynamics, Petri nets, optimization, and estimation

A major development in HFGT is the embedding of the structural model in Petri-net dynamics. In the engineering system net, places represent buffer–operand pairs and transitions represent capabilities. The state of the system is given by marking vectors over places and, when durations are modeled, over transitions as well. A standard discrete-time update is

$L$5

with a companion transition-marking equation used to represent in-transit or in-process quantities (Schoonenberg et al., 2021). Closely related formulations appear in MBSE-based LCA, hydrological systems, and project-scheduling specializations (Gohil et al., 30 May 2025, Harris et al., 31 May 2025, Hosseini et al., 21 Oct 2025).

These Petri-net dynamics are the foundation of the Hetero-functional Network Minimum Cost Flow formulation. In the hydrogen–natural gas case, all state equations, synchronization rules, boundary conditions, initial/final conditions, and capacity bounds are assembled into a linearly constrained convex quadratic program of the form

$L$6

subject to equality and inequality constraints (Schoonenberg et al., 2021). The paper characterizes this as the first HFGT-based optimization program and demonstrates optimization of flows across a multi-operand network that can transform operands, store operands over time, and analyze a quadratic cost function (Schoonenberg et al., 2021).

Later work specializes this same architecture in several directions. Process-based life cycle analysis is recovered as a special case of MBSE + HFGT under the assumptions $L$7, $L$8, $L$9, and instantaneous capabilities $B_s \subseteq R$0, yielding

$B_s \subseteq R$1

(Gohil et al., 30 May 2025). The RCPSP literature similarly shows that the classical project-scheduling model is recoverable as a special case of HFNMCF after collapsing the engineering-system net and interpreting the remaining operand net in activity-on-node terms (Hosseini et al., 21 Oct 2025). In watershed modeling, HFGT is extended from optimization to estimation through a Weighted Least Squares Error Hetero-functional Graph State Estimator with closed-form solution

$B_s \subseteq R$2

together with regularization terms on $B_s \subseteq R$3 and $B_s \subseteq R$4 (Harris et al., 2 Mar 2026).

This progression indicates that HFGT is not limited to one computational regime. The same structural objects support simulation, optimization, recovery of legacy models as special cases, and state estimation from uncertain data.

4. Resilience, robustness, and structural metrics

Resilience analysis is one of the earliest and most developed application areas for HFGT. In the American electric grid study, the formal graph and hetero-functional graph representations of the U.S. grid both exhibited exponential tail cumulative degree distributions,

$B_s \subseteq R$5

with $B_s \subseteq R$6 and $B_s \subseteq R$7 (Thompson et al., 2020). The same work argues that hetero-functional graphs more precisely describe functional changes associated with distributed generation and energy storage, because these additions create new capabilities even when they do not produce a large topological change in a purely formal graph (Thompson et al., 2020).

The resilience literature distinguishes structural survivability from recovery. A key formulation defines engineering performance as

$B_s \subseteq R$8

and then defines Actual Engineering Resilience as

$B_s \subseteq R$9

while Latent Engineering Resilience compares path counts before and after disruption (Farid, 2024). The latter can be reported in logarithmic form because the number of paths often grows exponentially in length (Farid, 2024). In the earlier grid study, LER is also defined as the total count of viable end-to-end service paths,

$(r,p)$0

with empirical findings that adding meshed lines yields approximately exponential growth in LER, whereas adding distributed generation or storage yields linear gains (Thompson et al., 2020).

Weighted robustness analysis extends HFGT to interdependent utility networks. Here attacks may be random or targeted, complete or partial; child processes are also removed when they lose all inputs or when cumulative incoming weight falls below a critical $(r,p)$1 threshold; and robustness is evaluated using Largest Connected Component, Number of Connected Components, Flow Robustness, and Service Robustness (Munikoti et al., 2020). In the reported synthetic case with $(r,p)$2, partial-attack degradation $(r,p)$3, and $(r,p)$4, complete targeted attacks by weighted out-degree reduced FR by 50% after just 1 removal, whereas complete random attacks required approximately 10 node removals (Munikoti et al., 2020).

More broadly, the structural-analysis literature uses sparsity, in-degree, out-degree, process-classified degree distributions, capabilities per square mile, and capabilities per capita to compare states and whole-country energy systems, concluding that geography and sustainable energy policies are reflected in the structure of multi-energy infrastructures (Thompson et al., 2022).

5. Relationship to adjacent modeling traditions

A recurring theme in the literature is that HFGT formally generalizes existing modeling paradigms while retaining their analytical content. One strand proves that hetero-functional graphs are a formal generalization of linear graphs and bond graphs. The relevant papers map linear-graph nodes to HFG buffers, linear-graph arcs to HFG capabilities, and show that the continuity and constitutive relations of the original models become null-objective HFNMCF instances under an absolute reference (Ghorbanichemazkati et al., 2024). The same paper states that any bond-graph causal DAE yields exactly the same state-space ODE when recast as the corresponding linear-graph instance of HFNMCF (Ghorbanichemazkati et al., 2024).

A second strand establishes that MBSE + HFGT is a formal generalization of process-based LCA (Gohil et al., 30 May 2025). A third embeds economic input-output analysis in the MBSE-HFGT workflow, recasting the Leontief equilibrium

$(r,p)$5

as the steady-state solution of

$(r,p)$6

with an objective that matches the RCOT cost-minimization linear program. In the reported RCOT example, the HFGT formulation recovers

$(r,p)$7

matching the RCOT solution (Naderi et al., 16 Feb 2026).

A fourth strand compares HFGT with System Dynamics. In the Mono Lake case, both SD and HFGT reproduce identical trajectories for lake and aquifer volumes, with normalized RMSE values of 0.15% for lake volume and 0.000% for aquifer volume (Naderi et al., 27 May 2025). The same study argues that MBSE and HFGT provide a broader set of systems thinking abstractions, explicit system boundaries, and native support for hybrid, discrete-event, stochastic, and optimization constraints (Naderi et al., 27 May 2025). This comparison is not presented as a rejection of SD; rather, it positions HFGT as a more general formalism that can reproduce SD behavior in that example while retaining additional structural detail.

The tensor-based literature also contrasts HFGT with multi-layer networks. It states that the usual multi-layer adjacency tensor is not lucid or complete with respect to HFGT’s notions of processes and transportation resources, whereas HFGT’s incidence and adjacency constructions are explicitly designed to preserve those distinctions (Farid et al., 2021).

6. Applications, implementation infrastructure, and recognized limitations

HFGT has been instantiated in a wide range of systems. Electric-power studies examine decarbonization measures such as meshed distribution lines, distributed generation, and storage in the American grid (Thompson et al., 2020). Synthetic interdependent urban-network studies combine electrical power, water, district heating, natural gas, and road transportation with repair services (Munikoti et al., 2020). Multi-modal energy-system studies cover coal, oil, natural gas, electricity, and biomass across New York, California, Texas, and the USA as a whole (Thompson et al., 2022). Optimization studies analyze hydrogen–natural gas infrastructure with ten nodes and a 20-day horizon (Schoonenberg et al., 2021). Environmental applications include hydrological examples involving lake and land segments, the Mono Lake system, and a Chesapeake Bay watershed instantiation (Harris et al., 31 May 2025, Naderi et al., 27 May 2025, Harris et al., 2 Mar 2026). Socioeconomic applications include process-based LCA, EIO/RCOT models, and megaproject engineering management (Gohil et al., 30 May 2025, Naderi et al., 16 Feb 2026, Hosseini et al., 29 May 2025).

Implementation work has accompanied these applications. The MATLAB-based HFGT Toolbox provides a two-stage workflow, XML2LFES followed by raw2FullLFES, and stores the complete model in the myLFES data structure with knowledge bases, constraints, tensors, controller models, service nets, and assembled adjacency matrices (Thompson et al., 2020). The paper states that the toolbox has been fully validated against several peer-review HFGT publications and reproduces exact sparse-matrix and tensor results across published test cases (Thompson et al., 2020).

The literature also records limitations. In the LCA extension, the size of $(r,p)$8 grows as $(r,p)$9, so sparse-tensor techniques or hierarchical decompositions are needed for very large systems (Gohil et al., 30 May 2025). In the Mono Lake study, manual incidence-matrix computation is described as feasible only for small systems (Naderi et al., 27 May 2025). Uncertainty in time-varying process weights is identified as a reason to pursue stochastic or scenario-based Petri-net extensions (Gohil et al., 30 May 2025). These limitations are methodological rather than conceptual: the papers generally treat them as computational and data-engineering challenges that arise when HFGT is scaled to increasingly heterogeneous systems-of-systems.

Taken together, this body of work presents HFGT as a domain-agnostic, MBSE-grounded, graph-theoretic formalism whose distinctive feature is the elevation of function allocation and operand-specific flow structure to first-class analytical status. Its enduring contribution is the claim—supported across resilience, optimization, estimation, environmental, and economic studies—that heterogeneous systems cannot be adequately characterized by topological form alone when the central scientific questions concern what resources can do, on which operands, in which sequences, under which constraints.