Mathematical Flow Graph

Encode coupled linear equations as a weighted directed graph whose declared readback and path, loop, determinant, or elimination rules preserve and solve the represented system.

Version

v3 · 2026-09-06 · History

Domain-specific #

2235

Origin domain

systems mathematics

Subdomain

signal flow analysis

Aliases

Flow graph (mathematics)

Core Idea

A mathematical flow graph converts a coupled linear equation system into a weighted directed graph and then uses the graph as an exact algebraic working surface. Its nodes stand for variables, signals, or equation positions; directed branches encode coefficients, gains, transmittances, or transfer functions; and a declared construction convention says how to read the graph back into equations. Paths, loops, graph transformations, determinant expansions, or vertex elimination can then recover the same solutions as the original algebra.^[1][2]

For the Mason signal-flow family, a useful normal form is

\[ x = Tx + Bu, \]

where \(x\) is the vector of dependent node variables, \(u\) contains declared sources, and an edge \(x_j\to x_i\) has transmittance \(t_{ij}\). The node readback is

\[ x_i=\sum_j t_{ij}x_j+\sum_\ell b_{i\ell}u_\ell. \]

Thus parallel contributions add, gains along a directed path multiply, and cycles expose algebraic feedback. When \(I-T\) is nonsingular,

\[ x=(I-T)^{-1}Bu. \]

Mason’s path rule expresses a selected source-to-response gain as a sum over forward paths corrected by touching and non-touching loops. If \(P_k\) is a forward-path gain, \(\Delta\) is the alternating loop-gain determinant, and \(\Delta_k\) omits loops touching path \(k\), then

\[ \frac{y}{u}=\frac{\sum_k P_k\Delta_k}{\Delta}. \]

The formula is not a visual heuristic: its signs, loop-touching tests, and nonzero denominator are algebraic obligations.^[1][3]

The broader node also includes the closely related Coates flow graph family. Coates associates an oriented weighted graph with a matrix equation and derives topological expressions for determinants and solutions; vertex deletion corresponds to algebraic elimination.^[2][4] Coates and Mason graphs share equation–graph faithfulness and a graphical solution purpose, but their local construction conventions are not identical. “Mathematical flow graph” is therefore an umbrella with declared variants, not a reason to force every member into the Mason incoming-sum syntax.^[5]

Linear differential or difference equations enter only after their dynamic relations have been rendered algebraic—for example, by Laplace or z transforms. A transfer-function branch such as \(1/(s+1)\) carries a transformed linear relation. Zero initial conditions must be stipulated, or initial-condition terms must appear as additional sources. The graph does not silently make a nonlinear, time-varying, or incomplete system linear.

The word flow is historical and algebraic. Nothing need be conserved or transported. What moves along a branch is a weighted contribution to another variable, not a commodity constrained by edge capacity. The abstraction is successful when graph readback reproduces the equations and every allowed graph operation preserves the solution set or the selected transfer relation.

Structural Signature

Sig role-phrases:

• the equation system — a declared set of compatible linear algebraic relations, possibly obtained from a linear dynamic model in a transform domain
• the variable or signal nodes — graph vertices whose values have an explicit algebraic meaning
• the source or reference nodes — independent inputs or constant terms separated from dependent variables when the convention requires them
• the directed branches — oriented incidences stating which variable contributes to which equation or connection
• the branch or vertex weights — coefficients, gains, transmittances, or transfer functions preserving the magnitude and sign of each relation
• the construction convention — the Mason, Coates, or other declared rule that maps equations to graph roles
• the readback rule — the inverse interpretation that recovers equations or matrix entries from the graph
• the path-and-loop structure — directed routes and cycles that expose compound influence and algebraic feedback
• the graphical solution rule — Mason gain, Coates determinant expansion, vertex elimination, or another justified equation-preserving calculus
• the faithfulness boundary — checks for coefficient preservation, transform assumptions, consistency, and a nonzero determinant when a unique solution or transfer is claimed

The locked invariant is not one picture. Equivalent equation rearrangements may yield different graphs, and legal graph transformations may change the number of nodes or branches. What must remain fixed is the declared mapping: the transformed graph reads back to an equivalent linear system and preserves the requested solution or transfer.

In the Mason normal form, branch direction is indexed by contribution: \(x_j\to x_i\) carries the coefficient multiplying \(x_j\) in the equation for \(x_i\). This direction need not equal physical causation. In a Coates graph, the connection-matrix convention must be stated separately rather than inferred from a Mason drawing.

What It Is Not

• Not a generic directed graph. Direction and weight alone do not supply variable semantics, equation readback, or a solution rule.
• Not a commodity-flow network. There is no constitutive capacity, conservation, source/sink shipment, cost, or optimization objective.
• Not a control-flow graph. Compiler arcs indicate possible program execution transfers between basic blocks; they do not encode coefficients in simultaneous equations.^[6]
• Not a data-flow graph by default. A dependency diagram becomes a mathematical flow graph only when nodes and gains satisfy a declared linear equation convention and support its solution calculus.
• Not a block diagram. Blocks usually denote operators or subsystems and wires denote signals; a Mason graph instead puts signals at nodes and gains on branches. Conversion is possible, but identity is not.
• Not a matrix. A matrix and a flow graph may encode the same system, yet they expose different operations and roles.
• Not a graph of a function. A plotted set of ordered pairs has neither directed branches nor equation-network semantics.
• Not a dynamical flow. A continuous-time flow or group action describes evolution, whereas this node describes an equation representation.
• Not arbitrary nonlinear causal modeling. The standard graphical rules depend on linearity and superposition.
• Not proof that a physical system is stable. A loop is algebraic feedback; stability requires separate pole, eigenvalue, or domain-specific analysis.
• Not one standardized notation. Mason and Coates terminology overlaps historically, but the construction family must still be declared.

Scope of Application

Simultaneous linear equations. A coefficient matrix can be recast as a weighted equation graph. Sparse structure becomes visible as missing branches, and elimination or determinant expressions can be organized topologically. The graph is particularly useful when connectivity is more revealing than a dense array of mostly zero coefficients.^[2][4]

Linear control systems. Block-diagram relations can be translated into signal nodes and gain branches. Forward paths expose transmission from reference to output; loops expose feedback; Mason’s formula derives a closed-loop transfer function without repeatedly reducing blocks. Standard controls texts teach signal-flow graphs and Mason’s gain formula as system modeling tools.^[7]

Linear circuits and network equations. Currents, voltages, and transformed network variables can serve as nodes, while impedances, admittances, or transfer relations supply weights. Historically, both Mason’s feedback work and Coates’s circuit-theory treatment grew from this setting.^[1][2]

Signal processing and communication systems. Linear filters, cascades, feedback recursions, and multistage transmittances can be represented by gain branches when the node equations are explicit. The graph makes alternative paths, repeated substructures, and loop interactions inspectable.

Transformed differential and difference equations. Laplace- or z-domain relations can use rational functions as branch gains. The scope remains the transformed linear system and carries its convergence, initial-condition, and well-posedness assumptions.

Symbolic and structural analysis. Parameterized gains can be left symbolic. One can then see which paths and loops contribute to a transfer, which coefficient deletion breaks a route, or why a determinant factor appears. This is explanatory structure, not a promise that path enumeration will always be computationally cheap.

The scope stops when no faithful linear readback exists. A nonlinear model can sometimes be linearized around an operating point, but the resulting graph represents that local linearization—not the original nonlinear system over all states.

Clarity

The graph answers three questions that a bare arrow diagram leaves open:

1. What does a node mean? A named variable, signal, source, or declared matrix position.
2. What does an arrow mean? A weighted contribution under a stated construction convention.
3. What inference does the drawing license? Equation readback and a specified transformation or solution rule.

These questions make “flow” precise. An edge labeled \(-2\) from \(x_2\) to \(x_1\) in a Mason graph means “the equation for \(x_1\) contains the contribution \(-2x_2\).” It does not mean two negative units physically travel between locations. The same test distinguishes the node from Network Flow Models, where edge variables are quantities routed subject to conservation and capacity.^[8]

The construction convention also clarifies direction. Solving \(x_1+2x_2=5\) for \(x_1\) creates an incoming branch \(x_2\to x_1\) with gain \(-2\). Solving it for \(x_2\) would produce a different orientation and weights. Neither direction alone establishes physical causality; both are algebraic representations whose faithfulness can be checked by readback.

Manages Complexity

A system of many coupled equations distributes dependence across numerous coefficients. The flow graph compresses this into local node relations and a global topology. Zero coefficients disappear as absent branches, cascaded influences become paths, and circular dependence becomes loops. The analyst can isolate which routes affect one output without inspecting every matrix entry uniformly.

The representation also localizes model changes. Changing one coefficient changes one branch weight; adding an input adds a source branch; eliminating a variable becomes a graph operation whose effect on neighboring relations can be traced. For sparse or modular systems, this can reduce bookkeeping and make structural errors visible.

That benefit has limits. Mason’s formula may require enumeration of many forward paths and combinations of non-touching loops, which can grow combinatorially. A graph with dozens of dense feedback connections may be harder to solve visually than its matrix. The abstraction manages complexity by exposing structure and choosing a relevant subproblem, not by abolishing the algebraic cost.

Abstract Reasoning

Use this recognition and inference sequence:

1. Declare the linear equations and the variables or transformed signals.
2. Choose and name the graph convention; do not mix Mason and Coates local rules silently.
3. Create nodes and weighted directed branches under that convention.
4. Read the graph back into equations and compare every sign, coefficient, source term, and transform-domain assumption.
5. Select a justified graph calculus—path/loop gain, determinant expansion, or equation-preserving elimination.
6. Check existence and uniqueness conditions before claiming a solution.
7. Translate the result back to the original variable or transfer quantity.

The structure then supports predictions. Removing every directed path from a declared input to an output eliminates that input’s contribution to the output under the represented linear model. Multiplying one branch gain scales every path product containing that branch, but not paths that bypass it. Adding a feedback branch creates new loops and can change the denominator of a source-to-output transfer. If an alleged graph transformation changes the readback equations, it is invalid regardless of visual simplicity.

The graph also supports counterfactual reasoning: set one gain to zero and identify which paths and loops vanish; substitute a symbolic parameter and track where it enters the determinant; compare two equation rearrangements by verifying that both yield the same transfer. None of these inferences requires treating the arrows as literal physical channels.

Knowledge Transfer

Transfer is strong within the bounded family of linear systems. The same roles—variable node, weighted contribution, path product, loop correction, and faithful readback—move literally among circuit equations, control models, filter structures, and simultaneous linear algebra. A practitioner trained to check missing input branches and sign errors in a control graph can use the same checks in a circuit graph because the algebraic roles remain identical.

Transfer outside that family is only structural residue. Representation contributes the target–medium–mapping–faithfulness discipline. Network contributes nodes, directed edges, and topology. Linear Combination contributes weighted addition. Those primes can be recognized in a compiler CFG or a logistics network, but the Mason/Coates equation conventions and solution rules do not transfer with them. Importing “loop gain” into a program CFG or “flow conservation” into a signal-flow graph is a category error, not a creative analogy.

Examples

Canonical — Converting and solving three simultaneous equations

Consider

\[ \begin{bmatrix} 1&2&0\\ 0&1&1\\ 5&-1&-1 \end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \begin{bmatrix}5\\5\\0\end{bmatrix}. \]

For a Mason-family construction, solve each row for its chosen dependent variable and introduce a unit source \(q=1\):

\[ x_1=5q-2x_2,\qquad x_2=5q-x_3,\qquad x_3=5x_1-x_2. \]

The graph has variable nodes \(x_1,x_2,x_3\) and source node \(q\). Its branches are \(q\to x_1\) with gain \(5\), \(x_2\to x_1\) with gain \(-2\), \(q\to x_2\) with gain \(5\), \(x_3\to x_2\) with gain \(-1\), \(x_1\to x_3\) with gain \(5\), and \(x_2\to x_3\) with gain \(-1\).

The readback immediately verifies every equation. Substituting the third node relation into the second gives

\[ x_2=5-(5x_1-x_2), \]

so \(x_1=1\). The first relation then gives \(x_2=2\), and the third gives \(x_3=3\). Direct matrix substitution checks \((1+4,\;2+3,\;5-2-3)=(5,5,0)\).

Mapped back: the equation system is the displayed matrix relation; \(x_1,x_2,x_3\) are variable nodes; \(q\) is the declared unit source; the six branches carry exactly the rearranged coefficients; the Mason incoming-sum convention is the construction and readback rule; the cycle among dependent variables exposes coupling; substitution is the equation-preserving graphical elimination rule; and the direct matrix check verifies faithfulness and the unique solution \((1,2,3)\).

Applied / In Practice — Closed-loop controller and plant

A controls engineer models a negative-feedback loop with reference \(r\), error \(e\), controller output \(v\), and plant output \(y\):

\[ e=r-Hy,\qquad v=Ce,\qquad y=Pv. \]

The signal-flow graph uses branches \(r\to e\) with gain \(1\), \(y\to e\) with gain \(-H\), \(e\to v\) with gain \(C\), and \(v\to y\) with gain \(P\). There is one forward path \(r\to e\to v\to y\) with gain \(CP\), and one loop \(e\to v\to y\to e\) with gain \(-CPH\). Mason’s determinant is therefore

\[ \Delta=1-(-CPH)=1+CPH, \]

and the closed-loop transfer is

\[ \frac{Y}{R}=\frac{CP}{1+CPH}. \]

For a proportional controller \(C(s)=4\), first-order plant \(P(s)=1/(s+1)\), and unity feedback \(H(s)=1\),

\[ \frac{Y(s)}{R(s)}=\frac{4}{s+5}. \]

The result has DC gain \(4/5\) and one pole at \(-5\). Those control interpretations follow only after the graph has delivered the correct transfer; the graph loop alone did not prove stability.^[7]

Mapped back: \(r\) is the independent source; \(e,v,y\) are signal nodes; \(1,-H,C,P\) are transfer-function branch gains; the four equations are recovered by Mason incoming sums; the reference-to-output route is the forward path; the negative-feedback route is the loop; Mason’s path/loop rule is the solution calculus; the nonzero \(1+CPH\) is the well-posedness boundary; and substitution of \(C,P,H\) translates the symbolic graph result into the engineer’s concrete closed-loop model.

Structural Tensions

Topology versus algebraic labels. The picture makes connectivity salient, but signs and gains—not adjacency alone—determine the equations. Two graphs with identical topology can have opposite feedback or different solutions. Diagnostic: read every node equation back from the labeled graph and compare it coefficient by coefficient with the source system.

Graphical insight versus combinatorial expansion. Paths and loops expose structure in sparse systems, yet Mason enumeration can explode in dense, highly cyclic graphs. Diagnostic: estimate forward-path and non-touching- loop counts before choosing the graphical formula; switch to sparse matrix or symbolic elimination when enumeration obscures rather than compresses.

Equivalent equations versus canonical direction. Rearranging which variable is isolated can reverse branches and change gains while preserving the solution set. A direction that looks causal may be only an algebraic choice. Diagnostic: distinguish “contributes to this chosen node equation” from “physically causes,” and validate equivalence by substitution or matrix comparison.

Feedback visibility versus well-posedness and stability. A loop exposes circular dependence, but it neither guarantees a unique algebraic solution nor diagnoses dynamical stability by itself. Diagnostic: check the relevant determinant for nonzero value, then perform the separate pole or eigenvalue analysis required by the application.

Transform convenience versus omitted conditions. Laplace and z transforms turn dynamics into branch gains, but unrepresented initial conditions or invalid transform assumptions change the problem. Diagnostic: record the transform convention, region/regularity assumptions as needed, and either state zero initial conditions or add their source terms.

Umbrella unity versus variant drift. Mason and Coates graphs share an equation-graph solution identity, while differing in node/branch conventions and formulas. Diagnostic: name the construction family and test claims against that family before generalizing them to the umbrella.

Reduction convenience versus semantic preservation. Combining branches or eliminating a node can simplify a graph, but an invalid local rewrite may change an external transfer or the solution set. Diagnostic: prove the rewrite rule under the declared convention and verify boundary-node behavior before and after reduction.

Autonomy versus reduction to parent primes. Representation, Network, and Linear Combination explain the portable skeleton, tempting the curator to treat this node as their routine conjunction. That reduction loses variable- node and gain-branch roles, equation readback, Mason/Coates variant boundaries, and graphical solution obligations. Diagnostic: ask whether the parent primes alone tell an analyst how to construct, validate, and solve the graph; if they do not, retain the domain node while routing only the portable residue to its parents.

Structural–Framed Character

Mathematical Flow Graph is mixed-structural with a strongly structural center.

1. Vocabulary travel. Node, branch, path, loop, and gain travel across circuits, control, signal processing, and linear algebra, but retain a specialist equation-system sense. Outside those domains, “flow graph” is too ambiguous for unqualified transfer.
2. Evaluative loading. The abstraction is nearly value-neutral. A loop, negative gain, or dense graph is not inherently good or bad; correctness is determined by equation faithfulness and the stated analytic goal.
3. Institutional dependence. Mason and Coates historically established named conventions, but no institution creates the underlying algebraic facts. The determinant and solution consequences follow once a convention and equations are fixed.
4. Human-practice dependence. People choose variables, rearrangements, and graph layout, yet the resulting readback, singularity, and transfer formula do not depend on social acceptance. Practice shapes the representation, not the truth of its algebra.
5. Import-versus-recognize behavior. Within linear-systems work the node is recognized literally. In logistics or compiler analysis, only the parent structures are recognizable; importing branch-gain algebra without an equation mapping would fabricate a false instance.

Its character: a formal representational calculus whose graph structure is objective once constructed, whose notation is historically family-bound, and whose full mechanism travels reliably inside linear systems but not as a domain-neutral prime.

Structural Core vs. Domain Accent

Skeletal structural core. Representation supplies a target equation system, graph medium, mapping, readback, and faithfulness test. Network supplies nodes, directed edges, paths, and loops. Linear Combination supplies weighted sums and superposition. These three primes explain why the graph can compress coupled relations and why branch contributions can be combined.

Irreducible domain accent. The node fixes variables or signals as graph roles; coefficients, transmittances, or transfer functions as gains; a Mason, Coates, or comparably explicit construction convention; equation-preserving graph transformations; source-to-response solution formulas; determinant and uniqueness cautions; and transform-domain boundaries. Those commitments are not supplied by a generic representation, network, or weighted sum.

Why it is not a prime. Its complete vocabulary does not recur literally across materially different domains. A compiler CFG, social network, and shipping network preserve directed connectivity but not node equations, branch transmittances, or Mason/Coates solution rules. Generalizing until those terms disappear leaves the already-owned parents.

Why it is not a mere composite. The parents do not jointly entail which graph objects correspond to equations, how a graph is validated, when branch gains may be transfer functions, how Mason and Coates variants differ, or which transformations preserve a selected solution. The domain node adds a stable role package and licenses repeatable graph-to-solution inference.

Instantiates / Related Primes

• Representation — instantiates by strict subsumption. Every mathematical flow graph faithfully maps an equation target into a graph medium with a declared readback. Representations need not be weighted equation graphs.
• Network — instantiates by strict subsumption. Every member has directed nodes and weighted edges whose topology matters. Networks need not carry variable equations or a graphical solution rule.
• Linear Combination — presupposes by strict composition. Mason node equations add weighted predecessor variables, while Coates starts from linear matrix relations. Linear combinations can exist without graphs.
• Problem Representation — related, direct edge declined. Some uses represent a control or circuit problem, but the live prime’s full state/operator/cost/goal package is not required by every equation graph.
• Network Flow Models — collision neighbor, not parent. Both use weighted digraphs; only network-flow models constitutively route conserved or optimized quantities under capacity and balance constraints.
• Feedback — related, direct edge declined. Cyclic signal-flow graphs expose feedback, but acyclic equation graphs remain valid members.
• Matrix — equivalent neighboring representation, not parent. Coefficient matrices and mathematical flow graphs can encode the same system without either being taxonomically every instance of the other.

Relationships to Other Abstractions

Current abstraction Mathematical Flow Graph Domain-specific

Parents (3) — more general patterns this builds on

• Mathematical Flow Graph is a kind of Network Prime

  Network — instantiates by strict subsumption. Every member has directed nodes and weighted edges whose topology matters.

• Mathematical Flow Graph is a kind of Representation Prime

  Representation — instantiates by strict subsumption. Every mathematical flow graph faithfully maps an equation target into a graph medium with a declared readback.

• Mathematical Flow Graph presupposes Linear Combination Prime

  Linear Combination — presupposes by strict composition. Mason node equations add weighted predecessor variables, while Coates starts from linear matrix relations.

Hierarchy paths (3) — routes to 3 parentless roots

• Mathematical Flow Graph → Network → Reservoir-Flux Network → Conservation Laws → Invariance

Show alternative paths (2)

Neighborhood in Abstraction Space

Mathematical Flow Graph sits in a sparse region of the domain-specific corpus (82^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Formal Languages, Types & Programs (41 abstractions)

Nearest neighbors

• Specification language — 0.82
• Unity-of-Command Breakdown — 0.81
• Cross-reference Relation — 0.81
• Enumeration Algorithm — 0.81
• Social system — 0.81

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With

• Signal-flow graph as an unqualified exact synonym. In much engineering usage this denotes specifically the Mason family rather than the umbrella. Tell: Mason nodes are signals/variables and each dependent node sums incoming branch contributions.
• Coates flow graph. It is a sibling construction family within the umbrella, with a connection-matrix and topological determinant convention. Tell: the source explicitly invokes Coates’s matrix-associated graph or vertex-deletion calculus rather than Mason node equations.
• Control-flow graph. It represents possible execution transfers among program basic blocks or instructions.^[6] Tell: following an edge predicts the next executable block, not a weighted term in a linear equation.
• Data-flow graph. It represents production, use, or dependency of values and operations. Tell: unless every edge has a declared linear gain and the graph reads back into solvable equations, it is not this node.
• Flow network / Network Flow Models. It routes a quantity through capacity-, conservation-, cost-, or objective-bearing arcs.^[8] Tell: ask whether the edge label transforms a variable contribution or bounds/prices an amount being shipped.
• Generic directed or weighted graph. It supplies only a graph carrier. Tell: deleting the variable/equation readback leaves the object fully defined, so mathematical-flow-graph identity was never present.
• Block diagram. Blocks are operators or subsystems and wires are signals. Tell: if the gain is inside a box and the signal travels on a line, the representation is a block diagram even if it can be converted to a Mason graph.
• Graph of a function. It is a point set such as \(\{(x,f(x))\}\). Tell: there are no directed gain branches, node equations, paths, or loops.
• Dynamical flow. It maps a state through time or a parameterized action. Tell: the defining operation is temporal composition, not equation-to- graph construction.
• Causal graph. Its arrows assert causal structure under a causal-model semantics. Tell: intervention or conditional-independence claims are licensed; algebraic branch direction alone licenses neither.
• Matrix. It arranges coefficients by rows and columns and is solved by matrix operations. Tell: graph topology is absent even when the same linear system is encoded.
• Nonlinear computational graph. It may compose arbitrary nonlinear operations and automatic-differentiation rules. Tell: nodes are operations or tensors and the chain rule, not linear path/loop gain, is the governing calculus.

References

[1] Mason, S. J. (1953). “Feedback Theory—Some Properties of Signal Flow Graphs.” Proceedings of the IRE, 41(9), 1144–1156. https://doi.org/10.1109/JRPROC.1953.274449. Primary presentation of system equations as directed signal-flow graphs, graph transformations, paths and loops, and the gain formula. registry ↩a ↩b ↩c

[2] Coates, C. L. (1959). “Flow-Graph Solutions of Linear Algebraic Equations.” IRE Transactions on Circuit Theory, CT-6(2), 170–187. https://doi.org/10.1109/TCT.1959.1086537. Primary source for matrix-associated Coates flow graphs, topological expressions, determinants, and vertex elimination. registry ↩a ↩b ↩c ↩d

[3] Mason, S. J. (1956). “Feedback Theory—Further Properties of Signal Flow Graphs.” Proceedings of the IRE, 44(7), 920–926. https://doi.org/10.1109/JRPROC.1956.275147. Primary extension of the Mason graph calculus. registry ↩

[4] Chen, Wai-Kai. (1967). “On Flow Graph Solutions of Linear Algebraic Equations.” SIAM Journal on Applied Mathematics, 15(1), 136–142. https://doi.org/10.1137/0115010. Specialist formal treatment of flow-graph solutions and their relation to linear algebraic equations. registry ↩a ↩b

[5] Thulasiraman, Krishnaiyan. (2021). “Signal Flow Graphs.” In Mathematics for Circuits and Filters. CRC Press. https://www.taylorfrancis.com/chapters/edit/10.1201/9781315214023-8/signal-flow-graphs-krishnaiyan-thulasiraman. Current specialist synthesis treating Mason and Coates as closely related graph-theoretic methods for linear equations. registry ↩

[6] LLVM Flang Project. “Control Flow Graph.” https://flang.llvm.org/docs/ControlFlowGraph.html. Authoritative compiler definition of basic-block nodes and execution-transfer arcs, used to fix the control-flow-graph boundary. registry ↩a ↩b

[7] Phillips, Charles L., and John M. Parr. Feedback Control Systems, 5^th ed. Pearson. Official catalog and contents: https://www.pearson.com/en-us/subject-catalog/p/Phillips-Feedback-Control-Systems-5th-Edition/P200000003157?view=educator. Authoritative controls treatment placing signal-flow graphs and Mason’s gain formula in physical-system modeling. registry ↩a ↩b

[8] Bertsekas, Dimitri P. Linear Network Optimization: Algorithms and Codes. MIT Press, 1991; paperback 2003. https://mitpress.mit.edu/9780262514439/linear-network-optimization/. Authoritative treatment of shortest-path, max-flow, assignment, transportation, and minimum-cost flow models, used to fix the optimization-network boundary. registry ↩a ↩b