Convergent Anthropocene Systems-of-Systems: Overcoming the Limitations of System Dynamics with Hetero-functional Graph Theory

Amro M. Farid; Ehsanoddin Ghorbanichemazkati; John C. Little; Megan Harris; Mohammad Mahdi Naderi

REVIEW 4 major objections 5 minor 132 references

Convergent Anthropocene Systems-of-Systems: Overcoming the Limitations of System Dynamics with Hetero-functional Graph Theory

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read SysML plus hetero-functional graph theory reproduces System Dynamics results for Mono Lake while adding structural expressiveness.

desk verdict A tidy translation of a two-stock SD model into HFGT, with an overreaching title: the equivalence is by construction, not a validation of general superiority. read the letter →

arxiv 2505.21793 v1 pith:PUDMPYHO submitted 2025-05-27 eess.SY cs.SY

Mohammad Mahdi Naderi , Megan Harris , Ehsanoddin Ghorbanichemazkati , John C. Little , Amro M. Farid This is my paper

classification eess.SYcs.SY

keywords systemdynamicshetero-functionalgraphtheorymodel-basedsystemsengineeringSysMLAnthropocenesystems-of-systemsMonoLakeHFNMCFthinkingabstractions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

System Dynamics is the standard way to simulate coupled human-natural systems, but it builds every model from stocks, flows, and parameters, draws no explicit system boundary, and only evolves continuous real-valued states. This paper argues that Model-Based Systems Engineering stated in SysML, quantified with Hetero-functional Graph Theory (HFGT), can serve as a general alternative that keeps those limitations from binding. The evidence is a side-by-side simulation of Mono Lake, California: the same hydrological model is expressed as an SD model and as a SysML block-and-activity model translated into an HFGT minimum-cost-flow problem, and the two simulations produce nearly identical trajectories. The intended consequence is that an analyst can adopt MBSE-HFGT without losing the conclusions an SD model would give, while gaining an explicit boundary, separation of form and function, and support for discrete-event and heterogeneous systems-of-systems.

What carries the argument

The carrying object is the hetero-functional graph, a Petri-net-like representation whose places are operand–buffer pairs and whose transitions are capabilities, where a capability is the sentence 'resource $r_v$ does process $p_w$.' The dynamics are governed by the engineering system net state transition function $Q[k+1] = \Phi(Q[k], U^-[k], U^+[k])$, with buffer markings updated as $Q_B[k+1] = Q_B[k] + M^+ U^+[k]\Delta T - M^- U^-[k]\Delta T$; the third-order hetero-functional incidence tensors $f^+$ and $f^-$ record which capabilities inject or pull each operand into each buffer and matricize into $M = M^+ - M^-$. The HFNMCF problem minimizes a convex time-separable cost subject to these state transitions together with synchronization, boundary, initial/final, capacity, and device-model constraints. For Mono Lake many constraints collapse as trivial because there is one operand with no state change and instantaneous transitions, leaving continuity equations plus the constitutive laws for evaporation, percolation, and surface area as device-model constraints.

What would settle it

Run the translation recipe on the food-energy-water Qazvin system shown in the paper as an SD example, which has multiple operands, nonlinear feedback, and sector couplings, and check whether the HFGT simulation reproduces the SD trajectories; if it diverges or cannot represent the feedback structure, the claimed equivalence and generality fail.

Watch

Extended reading notes

Core claim

The paper's central claim is the three-part statement that MBSE and HFGT '1) exhibit a broader set of systems thinking abstractions, 2) reproduce the analytical conclusions produced with an SD approach, and 3) ultimately overcome many of the methodological limitations' of SD for Anthropocene systems-of-systems. On the Mono Lake test case, the paper translates the two-stock SD model (lake volume and aquifer volume, with precipitation, runoff, evaporation, percolation, groundwater discharge, and withdrawal) into an HFGT model: a SysML Block Definition Diagram and Activity Diagram instantiate the HFGT meta-architecture, and the SD equations become the constraints of the HFNMCF problem with a zero objective. Simulated over 1990–2090 with identical exogenous inputs, the HFGT trajectories match the SD trajectories to within a normalized RMSE of about 0.15% for the lake and 0.000% for the aquifer. The paper therefore concludes that no analytical insight is lost by switching to MBSE-HFGT, and that the framework gives greater insight for the same example through its explicit system boundary, encapsulation of attributes in blocks, classification and instantiation relationships, and allocation of functions to resources.

Load-bearing premise

The load-bearing premise is that the two-stock, single-operand Mono Lake model is representative enough of Anthropocene systems-of-systems that a successful reproduction there demonstrates HFGT's general superiority over SD.

Editorial extensions

If this is right

MBSE-HFGT preserves SD's analytical conclusions: on Mono Lake the simulated lake and aquifer volumes match the SD results with normalized RMSE of about 0.15% and 0.000%.
Modelers gain an explicit system boundary with labeled input and output parameter nodes, so open-system assumptions, exogenous inputs, and conservation checks become visible by inspection rather than implicit.
Separating system form (BDD) from system function (ACT), and using decomposition, classification, instantiation, and allocation, gives a way to structure large heterogeneous models that flat stock-flow diagrams do not provide.
Because the HFGT formulation is built on Petri nets and an optimization problem, it can in principle represent discrete-time, discrete-event, and hybrid dynamics, and can carry policy objectives and constraints into the simulation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A decisive test beyond this paper would apply the same translation to a system whose SD model has strong nonlinear feedback and multiple interacting stocks, rather than the two-stock linearized Mono Lake case; the claimed advantages would be most convincing where SD is known to struggle.
The equivalence result suggests a practical migration path: existing SD models can be converted to SysML and HFGT without re-deriving their behavior, which would let a research community keep its accumulated stock-flow knowledge while moving to a more expressive modeling language.
The paper's own argument implies that the bottleneck for Anthropocene system-of-systems modeling then shifts from the mathematics to the SysML modeling effort and to the availability of automated HFGT tooling; whether that tooling scales to millions of places and transitions is a testable engineering question rather than a proven fact.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper before citing it as evidence that HFGT can replace System Dynamics. The narrow result — that a two-stock SD model of Mono Lake can be re-expressed as an HFNMCF optimization and produce nearly identical trajectories — is true, but it is true by construction. The authors carry every SD equation into the HFGT constraint set, set the objective to Z=0, and disable most of HFGT's machinery: synchronization constraints, operand nets, final conditions, and capacity bounds are all declared trivial or relaxed. The ~0.15% nRMSE then only checks that the transcription and solver are arithmetically consistent. It is not an independent test.

What is genuinely useful is the exposition. The systems-thinking comparison in Sections II-III is a clear, organized way to talk about what SD can and cannot represent, and the equation-by-equation mapping in Section V-E is a concrete recipe anyone wanting to port an SD model into HFGT could follow. That contributes some tutorial value to the subfield.

The soft spots are real. The headline claim — that MBSE-HFGT "overcomes" SD limitations for Anthropocene systems-of-systems — is not demonstrated by a case in which most HFGT-specific features are switched off. The generality rests on a qualitative abstraction table and the authors' prior HFGT results, not on this paper. There is also a genuine sign inconsistency: Eq. 28 writes percolation as a loss from the aquifer while Eq. 48 treats it as an inflow, so even the "direct translation" is not literally exact. Parameters, initial conditions, and the Markov-chain driver details are not fully reported, no code or data is provided, and the paper leans heavily on self-citations for its framework. Those are fixable in revision.

Who is this for? Someone already sympathetic to MBSE who wants a worked SD-to-HFGT translation template, or a reviewer wanting to see how the authors plan to unify these literatures. It is not a general comparative study of modeling paradigms. I would not cite it as evidence of HFGT's advantages, but I would send it to a competent referee rather than desk-reject: the explicit mapping is a checkable artifact, and a referee can push the authors to either slim the claims or add a case that exercises discrete events, multiple operands, and a real objective.

Recommendation: engage, with a request for major revision and a demand to rescope the claims to what the Mono Lake example can support.

Referee Report

4 major / 5 minor

Summary. The paper argues that Model-Based Systems Engineering (MBSE) expressed in SysML, together with Hetero-functional Graph Theory (HFGT), can serve as an alternative to System Dynamics (SD) for modeling Anthropocene systems-of-systems. It compares the two approaches in terms of systems thinking abstractions, translates a published SD model of Mono Lake, California, into an HFGT-based Hetero-functional Network Minimum Cost Flow (HFNMCF) formulation, and reports that the two simulations produce nearly identical results. The authors then claim that MBSE-HFGT offers a broader set of abstractions and overcomes methodological limitations of SD. The paper's central evidence is the Mono Lake equivalence, supported by conceptual arguments about system boundary, form, and function.

Significance. The paper addresses a timely and important problem: how to model interdependent human-natural systems with an approach that is expressive enough to capture heterogeneity, structure, and function. The conceptual comparison in Sections II and III is well organized and gives a clear, readable account of why SD's flat stock-and-flow primitives may be limiting. The worked translation of the Mono Lake SD model into HFNMCF constraints (Section V) is a useful pedagogical artifact: it shows step by step how an SD equation set maps onto HFGT incidence matrices and device-model constraints. However, the paper's load-bearing demonstration is constructed as a transcription rather than an independent test. The HFNMCF model is obtained by copying the SD equations as equality constraints, setting the objective to Z=0, and disabling or relaxing every HFGT-specific constraint. Consequently, the numerical agreement between the two models is a consistency check on the transcription, not evidence that HFGT can reproduce SD conclusions in general or that it can handle greater complexity.

major comments (4)

[Sec. V-E, Eqs. 47-64] The HFGT model is constructed by rewriting the SD equations (27-46) as HFNMCF equality constraints (48-64), setting the objective to Z=0 (Eq. 47), and then declaring the HFGT-specific constraints trivial or eliminating them: Eqs. 9, 11-13 are collapsed, Eqs. 14-15 are called trivial, Eq. 19 is dropped, and Eq. 20 is relaxed to unbounded. The near-identical trajectories in Figs. 9-10 therefore only confirm that the transcription is internally consistent; they cannot establish the Sec. I-A claim that MBSE-HFGT 'reproduce[s] the analytical conclusions produced with an SD approach' in any non-tautological sense, nor the Sec. VI claim that HFGT 'offers a valuable tool for modeling systems with considerably greater complexity.' A supporting example that actually activates discrete-event, multiple-operand, or optimization features would be needed to support these claims.
[Sec. V-C, Eq. 28] The printed aquifer balance equation is inconsistent. Eq. 28 reads VAqui[k+1] - VAqui[k] - (VPerc + VDis + VGWWith)ΔT = 0, which implies that groundwater discharge and withdrawal increase the aquifer volume. The surrounding text (and the physical description) treat percolation as inflow and discharge/withdrawal as outflows, and the corresponding HFGT row in Eq. 48 (second row: [0 0 0 -1 1 -1]) is physically correct. This is not a cosmetic typo: the paper explicitly claims a direct translation from SD equations to HFNMCF constraints (Sec. V-E), yet Eq. 48 is not the literal translation of Eq. 28. The authors should correct Eq. 28 or explain the intended sign convention.
[Sec. VI and Fig. 8] The central quantitative result is not reproducible as reported. The paper states that both models were calibrated and simulated over 1990-2090, but it does not provide numerical values for the percolation coefficient lambda_Perc, the freshwater evaporation rate lambda_FW, the elevation-volume and surface-area-volume linearization coefficients (Eqs. 53-54), the temperature and specific-gravity regression coefficients (Eqs. 62-63), the export withdrawal V_LA, the initial lake and aquifer volumes, the total dissolved solids mass, or the Markov-chain driver parameters for precipitation, temperature, and runoff. Without these values, a reader cannot reproduce Figs. 9-10 or verify the reported nRMSE values.
[Sec. VI] The claimed advantages of MBSE-HFGT over SD are not demonstrated by the Mono Lake example. The system has a single operand, no state-changing transformations, no discrete events, no active objective function, and unbounded capacity constraints. All of the HFGT-specific machinery that would differentiate it from SD is disabled in the translation. The qualitative arguments in Sec. VI recapitulate the abstraction comparison from Sections II-III without providing any comparative experiment or case study. Thus the paper's broader claim that MBSE-HFGT 'overcome[s] many of the methodological limitations' of SD is unsupported by the evidence presented.

minor comments (5)

[Sec. V-C, Eqs. 31-32] The initial conditions are written with a universal quantifier 'for all k in {0,...,K}', which is not meaningful for an initial condition; these equations should be stated for the single time index at which the volume is prescribed.
[Sec. IV-B, Eqs. 25-26] The definitions of Q^-_SL and U^+_SL contain confusing notation and likely typos (e.g., 'U^-_{Sl|L|}' in Eq. 25). Please rewrite these concatenation definitions with consistent subscripts.
[Sec. VI, first paragraph] The word 'Anthropoecene' is a typo for 'Anthropocene'.
[Abstract and Sec. V-A] Mono Lake is described as being in Northern California; the lake is in eastern California (Mono County). Please correct the geographic description.
[Global] The paper uses 'exogeneous' in several places (e.g., Fig. 8 caption, Sec. VI); the standard spelling is 'exogenous'.

Circularity Check

1 steps flagged · score 6.0 of 10

Mono Lake equivalence is constructed by transcription: the HFGT model is the SD equations re-expressed with Z=0 and HFGT-specific constraints disabled, so the match is forced by construction rather than independent confirmation.

self definitional [Sec. V-E (Eqs. 47-48; bullets on Eqs. 9, 11-15, 19, 20)]
"Eq. 7 simplifies to Eq. 47 because there are no defined objectives ... Consequently, a dummy objective function (Z = 0) is defined. ... Eq. 8 simplifies to Eq. 48, which is a translation of Eq. 27 and 28 from SD model. ... Eqs. 9, 11, 12, and 13 collapse to triviality ... Eqs. 14 and 15 also collapse to triviality ... Eq. 19 is a final condition constraint and is eliminated ... Eq. 20 are eliminated by relaxation."

The HFGT simulation is built by copying every SD equation into HFNMCF equality constraints (48-64 correspond to 27-46), setting the objective to a dummy Z=0, and disabling the constraints that distinguish HFGT from SD (operand-net and synchronization constraints, final condition, capacity bounds). Consequently, any feasible trajectory of the HFGT model is, by construction, a trajectory of the SD model; the observed nRMSE of ~0.15% only verifies arithmetic transcription and solver numerics. The paper's headline claim that MBSE-HFGT 'reproduce[s] the analytical conclusions produced with an SD approach' (Sec. I-A) is therefore not an independent test of HFGT but a restatement of how the HFGT model was defined.

full rationale

The core quantitative exhibit is a translation, not an independent prediction. Because the HFGT model is assembled from the SD equations (with Z=0 and HFGT-specific constraints relaxed), the equality of trajectories is predetermined; this is the one place where the paper's central claim reduces by construction to its own input. The broader conceptual comparison of systems-thinking abstractions is not circular: it is an argument, grounded in the cited SysML/HFGT literature, that does not depend on the Mono Lake simulation. Nor is the heavy use of the authors' prior HFGT publications circular in itself, since HFGT is a published framework and no uniqueness theorem is being invoked to forbid alternatives. The sign inconsistency between Eq. 28 and Eq. 48 (percolation/withdrawal terms) is an internal-consistency typo rather than a circular step. Overall, partial circularity: the 'reproduces SD' claim is forced by construction, while the 'broader abstractions' claim retains independent content, so score 6 rather than 8-10.

Assumptions & free parameters 9 free parameters · 3 assumptions · 0 invented entities

The central demonstration inherits all of its numerical content from Ford's SD model [124] and prior HFGT publications by the same group; no new parameters are estimated and the equivalence is built by transcription. The only axioms beyond standard math are domain assumptions about the completeness of the abstraction taxonomy and the representativeness of the case study.

free parameters (9)

Percolation coefficient lambda_Perc = 0.01 (per Eq. 58)
Relates lake volume to percolation outflow; taken from prior SD model [124], not estimated in this paper.
Freshwater evaporation rate lambda_FW = 3.75 ft/yr (Eq. 55)
Base evaporation rate used in Eq. 37/55; calibrated in prior work.
Lake elevation-volume linearization coefficients = slope 0.0265, intercept 6288.5 (Eq. 53)
Piecewise-linear relation H = f_H(V) from prior SD model; no data or source values provided.
Surface area-volume linearization coefficients = slope 0.008, intercept 15.44 (Eq. 54)
AS = f_AS(V) relation from prior SD model.
Temperature effect regression coefficients = 0.06 and -0.08 (Eq. 62)
Linear regression of evaporation temperature effect; fitted values inherited.
Specific gravity effect regression coefficients = -0.9 and 1.9 (Eq. 63)
Linear regression of evaporation salinity effect; inherited.
Export withdrawal to Los Angeles V_LA = 16 KAF/yr (Eq. 64)
Fixed exogenous withdrawal, assumed constant.
Initial lake and aquifer volumes = not stated in text
Initial conditions in Eq. 50 are required for simulation but their values are not reported; they come from the prior SD model.
Markov-chain driver parameters for precipitation, temperature, runoff = not specified
Future trajectories in Fig. 8 are generated from a Markov Chain fitted to 1960-1990 data; transition probabilities and states not given.

assumptions (3)

domain assumption The systems-thinking abstractions from [89], [90] (boundary, form, function and their sub-types) provide a complete and valid basis for comparing modeling methods.
The entire comparison in Secs. II-III and Table I rests on this taxonomy being the right yardstick; it is taken from prior systems engineering literature without independent justification.
domain assumption Mono Lake as modeled is representative of Anthropocene systems-of-systems complexities.
The paper generalizes from a two-stock hydrologic model to large multi-domain SoS in Sec. VI; no evidence that this case exercises the claimed advantages.
ad hoc to paper Setting the HFNMCF objective to Z=0 and relaxing all bounds preserves the SD simulation semantics.
Eq. 47 and the eliminations following it are introduced to make HFGT reproduce SD; if an actual objective or bounds were needed, the equivalence would be a different problem.

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Cite this review

Pith. "Pith review of Convergent Anthropocene Systems-of-Systems: Overcoming the Limitations of System Dynamics with Hetero-functional Graph Theory." pith.science (2026). https://pith.science/paper/PUDMPYHO

@misc{pith2026250521793,
  author       = {Pith},
  title        = {Pith review of: Convergent Anthropocene Systems-of-Systems: Overcoming the Limitations of System Dynamics with Hetero-functional Graph Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUDMPYHO}},
  note         = {Machine review of arXiv:2505.21793}
}

read the original abstract

Understanding the complexity and interdependence of systems in the Anthropocene is essential for making informed decisions about societal challenges spanning geophysical, biophysical, sociocultural, and sociotechnical domains. This paper explores the potential of Hetero-functional Graph Theory (HFGT) as a quantification tool for converting Model-based Systems Engineering (MBSE), stated in the Systems Modeling Language (SysML), into dynamic simulations-offering a comprehensive alternative to System Dynamics (SD) for representing interdependent systems of systems in the Anthropocene. The two approaches are compared in terms of systems thinking abstractions, methodological flexibility, and their ability to represent dynamic, multi-functional systems. Through a comparative study, the Mono Lake system is simulated in Northern California using both SD, and MBSE and HFGT, to highlight technical, conceptual and analytical differences. The simulations show equivalent results. However, MBSE and HFGT provide distinct advantages in capturing the nuances of the system through a broader set of systems thinking abstractions and in managing adaptive, multi-functional system interactions. These strengths position MBSE and HFGT as a powerful and flexible approach for representing, modeling, analyzing, and simulating heterogeneous and complex systems-of-systems in the Anthropocene.

Figures

Figures reproduced from arXiv: 2505.21793 by the authors.

**Figure 2.** A SysML BDD Diagram: the meta-architecture of the system form (adapted from [ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** A SysML ACT Diagram: the meta-architecture of the system function (adapted from [ [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Conceptual hydrological and water resources model of Mono Lake in California. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Integrated SD CLD and SFD of Mono Lake system. [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Block Definition Diagram (BDD) of Mono Lake. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: ACT diagram of Mono lake showing the structural movement of objects throughout the entire system. [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: Exogeneous parameters used in SD and HFGT simulations for 1990-2090: (a) groundwater withdrawal from [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: Simulated Mono Lake Volume by SD and HFGT. [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: Simulated Mono Aquifer Volume by SD and HFGT. [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.

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