Hetero-Functional Graphs

• Hetero-Functional Graphs are structured representations that combine resources, processes, and operands to capture engineering heterogeneity and functional dependencies.
• They enable precise modeling of complex systems, optimizing flows, state estimation, and resilience across diverse infrastructure and multi-domain applications.
• Multiple formulations exist, including tri-partite, process-centric, and capability-centric graphs, each tailored to different engineering analysis and decision support tasks.

Hetero-Functional Graph (HFG) denotes the graph representations that emerge from Hetero-Functional Graph Theory (HFGT), a modeling framework that combines network science with model-based systems engineering in order to represent system form, system function, and the allocation of function onto form. In this literature, HFGs are used to preserve heterogeneity that conventional topology-only graphs suppress: resources, processes, operands, buffers, capabilities, and functional dependencies can all appear explicitly, depending on the formulation. As a result, HFGs have been used for interdependent utility infrastructures, electric grids, multi-modal energy systems, watersheds, life-cycle analysis, project scheduling, economic input-output models, and megaproject management (Farid et al., 29 May 2025, Farid et al., 2021, Ghorbanichemazkati et al., 2024).

1. Ontological basis and scope of the term

HFGT is rooted in the linguistic structure “subject–predicate–object,” recast in engineering terms as resources, processes, and operands. Resources are the subjects or assets that participate in system behavior; processes are the verbs or activities; operands are the objects acted upon. A capability is the allocation of a process to a resource, often expressed as the sentence “Resource $r_v$ does process $p_w$.” This ontology is central in the HFGT tutorial literature and in later applications that treat HFGs as the graph-theoretic realization of such subject–verb–object structures (Farid et al., 29 May 2025, Hosseini et al., 29 May 2025).

The term “Hetero-Functional Graph” is not restricted to a single graph tuple across the literature. In one line of work, the central artifact is a typed, tri-partite graph built from resources, processes, and operands. In another, the HFG is the process-level dependency graph produced by HFGT, with processes as nodes and directed functional dependencies as edges. In another, the graph is capability-centric, so that nodes are process-resource pairings and edges are feasible logical sequences of capabilities. A more application-specific formulation appears in disaster-repair decision support, where the HFG is a typed directed graph over functionality instances with “Requires” edges among generation, transport, storage, and consumption functions (Farid et al., 29 May 2025, Munikoti et al., 2020, Thompson et al., 2020, Okeukwu-Ogbonnaya et al., 17 Jul 2025).

Formulation in the literature	Nodes	Edge or incidence semantics
Tri-partite HFG	Resources, processes, operands	Participation and action incidences
Process-centric HFG	Processes	Directed functional dependencies
Capability-centric HFG	Capabilities	Feasible logical sequences
Functionality HFG	Functionality instances	Typed “Requires” relations

This plurality is important because many misunderstandings arise from treating HFG as merely a synonym for a heterogeneous graph. In the cited literature, the defining feature is not heterogeneity alone, but heterogeneity organized around engineering semantics: what can act, what action is performed, what is acted upon, and what sequences or dependencies are feasible. The same papers also trace HFGT’s origins to mass-customized production systems and present it as suitable for convergent systems-of-systems and megaprojects because physical and informatic elements coexist and interact over long planning and operational horizons (Hosseini et al., 29 May 2025, Farid et al., 29 May 2025).

2. Formal structures and graph constructions

The most general conceptual formulation uses incidence relations among subjects, predicates, and objects. Let $S$ be the set of resources, $P$ the set of processes, and $O$ the set of operands. The tutorial formulation defines a subject–predicate incidence matrix $M_{SP}$, a predicate–object incidence matrix $M_{PO}$, and a derived subject–object association $A_{SO} = M_{SP} M_{PO}$. It also defines a typed supra-adjacency

$$\mathcal{A} = \begin{bmatrix} 0 & M_{SP} & A_{SO} \ M_{SP}^{\top} & 0 & M_{PO} \ A_{SO}^{\top} & M_{PO}^{\top} & 0 \end{bmatrix},$$

which preserves type distinctions while permitting network-science analysis (Farid et al., 29 May 2025).

A second formal family is tensor-based and Petri-net-based. Here the central objects are the positive and negative third-order hetero-functional incidence tensors

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

where $p_w$0 is the set of operands, $p_w$1 the set of buffers, and $p_w$2 the set of capabilities. $p_w$3 when capability $p_w$4 pulls operand $p_w$5 from buffer $p_w$6, and $p_w$7 when it injects operand $p_w$8 into buffer $p_w$9. After matricization, the hetero-functional incidence matrix is $S$0, and the associated engineering system net is $S$1 (Farid et al., 2021, Naderi et al., 27 May 2025).

From these incidence structures, several adjacency constructions are defined. In the multi-modal energy literature, aggregating over operands yields a formal graph adjacency $S$2, while capability-level matricization yields a hetero-functional adjacency $S$3. The first recovers buffer-to-buffer physical connectivity; the second preserves logical sequencing among capabilities (Thompson et al., 2022). In the tensor formulation, the hetero-functional adjacency matrix is also constructed from a sequence knowledge base and a system sequence constraints matrix, so that feasible capability strings are retained while infeasible ones are eliminated (Farid et al., 2021).

A third family is process-centric and weighted. The weighted HFGT robustness formulation defines a directed, weighted process graph $S$4, where nodes are processes, edges are functional dependencies, and $S$5 is the degree of dependency of target process $S$6 on source process $S$7. Weighted in-degree and out-degree,

$S$8

are then used as service and vulnerability descriptors (Munikoti et al., 2020).

Finally, the disaster-repair literature uses a deliberately simplified functionality graph. There the HFG is $S$9, with typed nodes for roles such as Generate, Transport, Store, and Consume, and edges $P$0 meaning that $P$1 requires $P$2 to be operational. This formulation is acyclic along functionality flow and induces a partial order on repair actions by reachability (Okeukwu-Ogbonnaya et al., 17 Jul 2025).

3. Dynamics, optimization, and estimation

HFGs are not only descriptive. A central development is the Hetero-functional Network Minimum Cost Flow optimization, which turns the engineering system net into a discrete-time optimization model. In the hydrogen–natural gas study, the decision variables include place markings, transition markings, and firing vectors over a time horizon, and the optimization is written in quadratic-program form with state-update, synchronization, boundary, duration, and capacity constraints. The same paper shows that HFGT can optimize flows across a multi-operand network, transform operands in the network, store operands over time, and analyze behavior for a quadratic cost function (Schoonenberg et al., 2021).

This optimization formalism is repeatedly specialized. One paper proves that hetero-functional graphs are a formal generalization of both linear graphs and bond graphs. Under a specialization with real-valued variables, null objective, $P$3, instantaneous capabilities, no operand nets for power operands, and no accumulation of power operands in buffers, continuity laws and constitutive laws of linear-graph and bond-graph models are recovered as HFG constraints (Ghorbanichemazkati et al., 2024). Another paper proves that process-based life-cycle analysis is a formal special case of the engineering system net state transition function under a one-to-one process-resource allocation, simulation horizon $P$4, $P$5, and instantaneous capabilities (Gohil et al., 30 May 2025). A later paper embeds economic input-output and Rectangular Choice of Technology models into the MBSE–HFGT workflow and states that the dynamics captured by basic EIO models and RCOT are exactly reproducible within that framework (Naderi et al., 16 Feb 2026). An RCPSP paper similarly proves that “The RCPSP specialization of the HFNMCF problem in Eqs. 16a–16f is a generalization of the RCPSP in Eqs. 1a–1e” (Hosseini et al., 21 Oct 2025).

The same structural apparatus also supports state estimation. For the American Multi-Modal Energy System, the Weighted Least Squares Error Hetero-functional Graph State Estimation formulation uses a linear measurement equation $P$6, a weighted quadratic objective, and conservation constraints $P$7. In that application, the estimator is a sparse convex quadratic program with monthly time steps and asset-level flows across electricity, natural gas, oil, and coal (Thompson et al., 23 Sep 2025). A watershed adaptation defines a specialized annual steady-state estimator with decision variables $P$8, mass-balance constraints, and a diagonal weighting matrix

$P$9

so that nutrient applications, routing, and end-of-tide loads can be estimated within an explicit hetero-functional routing structure (Harris et al., 2 Mar 2026).

A common theme across these formulations is that HFG-based dynamics are state-based and constraint-based rather than purely topological. This distinguishes them from graph models that terminate at adjacency analysis. It also clarifies a recurring limitation: several HFG papers explicitly state that functional dependency and feasibility can be modeled without solving full physical flow equations, whereas higher-fidelity dynamic or steady-state device models must be added through constitutive constraints or application-specific optimization layers (Munikoti et al., 2020, Naderi et al., 27 May 2025).

4. Dependency, partial orders, and operational decision support

One of the clearest operational uses of HFGs appears in infrastructure restoration. In a toy interdependent power-and-water system spanning three communities, the HFG encodes generation, transport, storage, and consumption functions and the “Requires” dependencies between them. Reachability induces a partial order: $O$0 and valid repair sequences are linear extensions of that poset. Any topological sort yields a technically valid sequence, but ambiguity remains among mutually unrelated bottom-layer consumption nodes (Okeukwu-Ogbonnaya et al., 17 Jul 2025).

The same study uses this ambiguity as the point where community preferences enter. Pairwise comparisons over damaged functionalities are collected from 200 synthetic personas generated with GPT-3.5-turbo, aggregated into soft labels $O$1, and learned by a neural comparator with hybrid cross-entropy and KL-divergence loss. A preference graph $O$2 is then built over repair options, with confidence-weighted edges, and PageRank is applied to obtain a global total order. Because candidate repair options are defined so that HFG prerequisites are already satisfied or included, the resulting order is consistent with HFG feasibility (Okeukwu-Ogbonnaya et al., 17 Jul 2025).

The reported results illustrate how an HFG can delimit the technically admissible space while a secondary ranking method resolves socially meaningful ambiguity. The learned global prioritization is identical for community-aware and community-unaware personas; around 50% of pairwise data yields Kendall’s $O$3 with the full-data ranking; and top-ranked items are comparatively stable across prompt variants, while lower-ranked items are more sensitive. The same paper also states that the HFG and experimental data/code for the toy network are available in the public CommRank-HFG repository (Okeukwu-Ogbonnaya et al., 17 Jul 2025).

This application is narrower than the broader HFGT literature, but it is technically significant because it shows that HFGs can be used as feasibility-constrained decision backbones rather than only as descriptive models. The graph imposes strict precedence where it exists and leaves underdetermined regions explicit, making it possible to layer learning, voting, or optimization methods on top without violating system constraints.

5. Structural robustness and resilience analysis

HFGs have been used extensively for robustness and resilience studies because they can represent functional dependencies that topology-only graphs miss. In the weighted HFGT robustness framework for interdependent urban utility networks, the graph includes electricity, water, district heating, natural gas, road transportation, and repair facilities, with weights on process dependencies to model partial degradation and graded service levels. Robustness is evaluated under complete/partial and random/targeted attacks using Largest Connected Component, Number of Connected Components, Flow Robustness, and Service Robustness. The reported synthetic case shows that targeted attacks disconnect the graph much faster—10–12 stages versus about 40 stages under random attacks—and that partial targeted attacks can be more damaging than complete random attacks (Munikoti et al., 2020).

A related resilience study of the future American electric grid uses HFGs to evaluate meshed distribution lines, distributed generation, and energy storage. There, HFGs reproduce conventional graph-theoretic vulnerability patterns when used like formal graphs, but also detect functionality added on existing assets. The paper reports that meshed distribution increases average degree centrality in both formal and hetero-functional analyses, whereas distributed generation and storage change the HFG but not the formal graph adjacency. Using Latent Engineering Resilience, it concludes that meshing, DG, and storage all enhance structural resilience and that “there is no structural trade-off between grid sustainability and resilience” (Thompson et al., 2020).

The resilience program is further generalized in a convergent systems-of-systems study that defines Actual Engineering Resilience, Latent Engineering Resilience, Dynamic Actual Engineering Resilience, and Dynamic Latent Engineering Resilience. That paper uses capabilities-as-nodes, third-order incidence tensors, operand nets, and path enumeration to measure both “survival” and “recovery.” In its hypothetical energy-water nexus case, normal operation yields 1840 delivery paths for the full system-of-systems service; disruption of the Wastewater Treatment Plant or Industrial Facility reduces both AER and LER to zero for the full service; and the paper reports Dynamic Resilience of 0.8706 for the scenario set (Farid, 2024).

These studies also clarify an interpretive point. In HFG-based resilience analysis, centrality is not always a reliable proxy for functional criticality. The convergent SoS paper explicitly states that eigenvalue centrality did not correlate with number of paths lost under disruption, because the decisive variable was each capability’s role in operand delivery paths rather than its graph-theoretic prominence alone (Farid, 2024). This contrasts with many homogeneous-network analyses and is one reason HFG-based resilience metrics were introduced.

6. Domains, tooling, and recurrent limitations

HFGs have been instantiated in a wide range of domains. In megaproject engineering management, HFGT is presented as a quantitative, cyber-physical meta-architecture that unifies physical and informatic elements and models stakeholders as digital resources with decision-making, measurement, and digital transport/transform processes (Hosseini et al., 29 May 2025). In Anthropocene systems-of-systems, MBSE and HFGT are used to simulate the Mono Lake system and are reported to produce trajectories nearly identical to System Dynamics, with nRMSE of approximately 0.15% for Mono Lake and approximately 0.000% for the Mono Aquifer (Naderi et al., 27 May 2025). In multi-energy modeling, HFGs have been built for New York, California, Texas, and the contiguous United States, with capability counts ranging from 43,766 in NY to 3,130,235 in the USA-wide model (Thompson et al., 2022). In watershed modeling, HFGT has been used to reconstruct opaque nutrient routing into explicit land-segment, outlet, and estuary capabilities, preserving near-perfect end-of-tide closure with relative error below 0.01% for both nitrogen and phosphorus in the 2024 progress scenario (Harris et al., 2 Mar 2026).

Implementation support has also been formalized. The HFGT Toolbox, written in MATLAB and tested with v9.6 (R2019a), constructs the HFGT model suite from a SysML-like XML input, including system concept matrices, hetero-functional incidence tensors, hetero-functional adjacency, controller matrices, service feasibility matrices, and projected system adjacency. The paper states that the toolbox has been fully validated against several peer-review HFGT publications and is openly available on GitHub together with a sample input file (Thompson et al., 2020).

Several limitations recur across the literature. First, multiple HFG formulations coexist, and the same term may refer to a tri-partite graph, a process graph, a capability graph, or an application-specific dependency graph; this improves flexibility but complicates direct comparison across papers (Farid et al., 29 May 2025, Munikoti et al., 2020, Thompson et al., 2020, Okeukwu-Ogbonnaya et al., 17 Jul 2025). Second, weights, thresholds, and dependency allocations are often heuristic or exogenous, especially in robustness studies, and several papers explicitly call for more data-driven estimation and empirical validation (Munikoti et al., 2020). Third, physical flow models are not always embedded: some HFG applications capture functional dependencies, shared-path constraints, or routing structure without solving power flow, hydraulic balance, or other domain equations unless additional device models are attached (Munikoti et al., 2020, Naderi et al., 27 May 2025). Fourth, behavioral and social layers remain difficult. The community-survey application states that it does not compare against real surveys and that synthetic preferences may diverge from real community behavior under stress (Okeukwu-Ogbonnaya et al., 17 Jul 2025).

Taken together, these limitations do not diminish the core contribution of HFGs. Rather, they delimit the present state of the field. HFGs provide a family of typed graph and net representations in which engineering heterogeneity is explicit, functional dependencies are analyzable, and system structure can be coupled to optimization, resilience analysis, simulation, state estimation, and decision support. The literature’s central claim is therefore not that HFG is a single immutable graph object, but that HFGT supplies a coherent formal vocabulary for representing and computing over heterogeneous engineering systems and systems-of-systems across domains (Farid et al., 2021, Farid et al., 29 May 2025, Ghorbanichemazkati et al., 2024).