submitted for publication:(doi) 1 a tensor-based
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A Tensor-Based Formulation ofHetero-functional Graph Theory
Amro M. Farid, Dakota Thompson, Prabhat Hegde, Wester Schoonenberg
Abstract
Recently, hetero-functional graph theory (HFGT) has developed as a means to mathematically modelthe structure of large flexible engineering systems. In that regard, it intellectually resembles a fusion ofnetwork science and model-based systems engineering. With respect to the former, it relies on multiplegraphs as data structures so as to support matrix-based quantitative analysis. In the meantime, HFGTexplicitly embodies the heterogeneity of conceptual and ontological constructs found in model-based sys-tems engineering including system form, system function, and system concept. At their foundation, thesedisparate conceptual constructs suggest multi-dimensional rather than two-dimensional relationships.This paper provides the first tensor-based treatment of some of the most important parts of hetero-functional graph theory. In particular, it addresses the “system concept”, the hetero-functional adjacencymatrix, and the hetero-functional incidence tensor. The tensor-based formulation described in this workmakes a stronger tie between HFGT and its ontological foundations in MBSE. Finally, the tensor-basedformulation facilitates an understanding of the relationships between HFGT and multi-layer networks.
I. Introduction
One defining characteristic of twenty-first century engineering challenges is the breadth of their scope.The National Academy of Engineering (NAE) has identified 14 “game-changing goals” [1].
1) Advance personalized learning2) Make solar energy economical3) Enhance virtual reality4) Reverse-engineer the brain5) Engineer better medicines6) Advance health informatics7) Restore and improve urban infrastructure8) Secure cyber-space9) Provide access to clean water
10) Provide energy from fusion11) Prevent nuclear terror12) Manage the nitrogen cycle13) Develop carbon sequestration methods14) Engineer the tools of scientific discoveryAt first glance, each of these aspirational engineering goals is so large and complex in its own right thatit might seem entirely intractable. However, and quite fortunately, the developing consensus across anumber of STEM (science, technology, engineering, and mathematics) fields is that each of these goalsis characterized by an “engineering system” that is analyzed and re-synthesized using a meta-problem-solving skill set [2].Definition 1: Engineering system [3]: A class of systems characterized by a high degree of technicalcomplexity, social intricacy, and elaborate processes, aimed at fulfilling important functions in society. �
The challenge of developing abstract and consistent methodological foundations for engineering sys-tems is formidable. Consider the engineering systems taxonomy presented in Table I [3]. It classifies
Amro M. Farid is an Associate Professor of Engineering with the Thayer School of Engineering at Dartmouth and an AdjunctAssociate Professor of Computer Science with the Department of Computer Science, Dartmouth College, Hanover, NH, [email protected]
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TABLE IA Classification of Engineering Systems by Function and Operand [3]
Function/Operand LivingOrganisms
Matter Energy Information Money
Transform Hospital Blast Furnace Engine, electricmotor
Analytic engine,calculator
Bureau of Print-ing & Engraving
Transport Car, Airplane,Train
Truck, train, car,airplane
Electricity grid Cables, radio,telephone, andinternet
Banking Fedwireand Swift transfersystems
Store Farm, ApartmentComplex
Warehouse Battery, flywheel,capacitor
Magnetic tape &disk, book
U.S. BuillonRepository (FortKnox)
Exchange Cattle auction, (il-legal) human traf-ficking
eBay trading sys-tem
Energy market World Wide Web,Wikipedia
London Stock Ex-change
Control U.S. Constitution& laws
NationalHighwayTraffic SafetyAdministration
NuclearRegulatoryCommission
Internet engineer-ing task force
United StatesFederal Reserve
engineering systems by five generic functions that fulfill human needs: 1.) transform 2.) transport 3.) store,4.) exchange, and 5.) control. On another axis, it classifies them by their operands: 1.) living organisms(including people), 2.) matter, 3.) energy, 4.) information, 5.) money. This classification presents a broadarray of application domains that must be consistently treated. Furthermore, these engineering systemsare at various stages of development and will continue to be so for decades, if not centuries. And sothe study of engineering systems must equally support design synthesis, analysis, and re-synthesis whilesupporting innovation; be it incremental or disruptive.
A. Background Literature
In that regard, two fields are of particular relevance: systems engineering and network science. Systemsengineering, and more recently model-based systems engineering (MBSE), emerged as a practical andinterdisciplinary engineering discipline that enables the successful realization of complex systems fromconcept, through design, to full implementation [4]. It is well-equipped to deal with systems of ever-greater complexity; be they for the greater interaction within these systems or because of the expandingheterogeneity they demonstrate in their structure and function. Despite its many achievements, model-based systems engineering, however, relies till today on graphical modeling languages that providelimited quantitative insight (on their own) [5], [6].
In contrast, network science has emerged as a scientific discipline for quantitatively analyzing networksthat appear in a wide variety of engineering systems. And yet, network science, despite its methodologicaldevelopments in multi-layer networks, has often been unable to address the explicit heterogeneity oftenencountered in engineering systems [7], [8]. In a recent comprehensive review Kivela et. al [8] write:
“The study of multi-layer networks . . . has become extremely popular. Most real and engineered systemsinclude multiple subsystems and layers of connectivity and developing a deep understanding of multi-layer systems necessitates generalizing ‘traditional’ graph theory. Ignoring such information can yieldmisleading results, so new tools need to be developed. One can have a lot of fun studying ‘biggerand better’ versions of the diagnostics, models and dynamical processes that we know and presumablylove – and it is very important to do so but the new ‘degrees of freedom’ in multi-layer systems alsoyield new phenomena that cannot occur in single-layer systems. Moreover, the increasing availabilityof empirical data for fundamentally multi-layer systems amidst the current data deluge also makes itpossible to develop and validate increasingly general frameworks for the study of networks.. . . Numerous similar ideas have been developed in parallel, and the literature on multi-layer
networks has rapidly become extremely messy. Despite a wealth of antecedent ideas in subjects likesociology and engineering, many aspects of the theory of multi-layer networks remain immature, and
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the rapid onslaught of papers on various types of multilayer networks necessitates an attempt to unifythe various disparate threads and to discern their similarities and differences in as precise a manneras possible.. . . [The multi-layer network community] has produced an equally immense explosion of disparate
terminology, and the lack of consensus (or even generally accepted) set of terminology and mathematicalframework for studying is extremely problematic.”
In many ways, the parallel developments of the model-based systems engineering and network sciencecommunities intellectually converge in hetero-functional graph theory (HFGT) [7]. With respect to the latter,it relies on multiple graphs as data structures so as to support matrix-based quantitative analysis. In themeantime, HFGT explicitly embodies the heterogeneity of conceptual and ontological constructs foundin model-based systems engineering including system form, system function, and system concept. Morespecifically, the explicit treatment of function and operand facilitates a structural understanding of thediversity of engineering systems found in Table I. Although not named as such originally, the first workson HFGT appeared as early as 2006-2008 [9]–[12]. Since then, HFGT has become multiply establishedand demonstrated cross-domain applicability [7], [13]; culminating in the recent consolidating text [7].
The primary benefit of HFGT, relative to multi-layer networks, is the broad extent of its ontologicalelements and associated mathematical models. In their recent review, Kivela et. al showed that all of thereviewed works have exhibited at least one of the following modeling constraints [8]:
1) Alignment of nodes between layers is required [8], [14]–[62]2) Disjointment between layers is required [8], [51], [56], [63]–[80]3) Equal number of nodes for all layers is required [8], [14]–[51], [59], [61], [65], [67], [71], [72], [81],
[82]4) Exclusively vertical coupling between all layers is required [8], [14]–[51], [59], [61], [66], [82]–[86]5) Equal couplings between all layers are required [8], [16]–[41], [45]–[51], [59], [61], [66], [82]–[86]6) Node counterparts are coupled between all layers [16]–[20], [24]–[41], [45]–[51], [59], [61], [66],
[82]–[86]7) Limited number of modelled layers [47]–[49], [51], [56], [59], [61], [63]–[93]8) Limited number of aspects in a layer [14]–[51], [56], [59], [61], [63]–[80], [82]
To demonstrate the consequences of these modeling limitations, the HFGT text [7] developed a verysmall, but highly heterogeneous, hypothetical test case system that exhibited all eight of the modelinglimitations identified by Kivela et. al. Consequently, none of the multi-layer network models identifiedby Kivela et. al. would be able to model such a hypothetical test case. In contrast, a complete HFGTanalysis of this hypothetical test case was demonstrated in the aforementioned text [7]. The same text
a) Topology of (Electrified) Water Distribution System b) Topology of Electric Power System c) Topology of Electrified Transportation System
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Fig. 1. A Topological Visualizaiton of the Trimetrica Smart City Infrastructure Test Case [7].
provides the even more complex hypothetical smart city infrastructure example shown in Fig. 1. It not
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only includes an electric power system, water distribution system, and electrified transportation systembut it also makes very fine distinctions in the functionality of its component elements.
Given the quickly developing “disparate terminology and the lack of consensus”, Kivela et. al.’s [8] statedgoal “to unify the various disparate threads and to discern their similarities and differences in as precise amanner as possible” appears imperative. While many may think that the development of mathematicalmodels is subjective, in reality, ontological science presents a robust methodological foundation. Asbriefly explained in Appendix A, and as detailed elsewhere [7], [94], [95], the process of developinga mathematical model of a given (engineering) system is never direct. Rather, a specific engineeringsystem (which is an instance of a class of systems) has abstract elements in the mind1 that constitute anabstraction A (which is an instance of a domain conceptualization C). C is mapped to a set of primitivemathematical elements called a language L, which is in turn instantiated to produce a mathematicalmodel M. The fidelity of the mathematical model with respect to an abstraction is determined by thefour complementary linguistic properties shown in Figure 2 [95]: soundness, completeness, lucidity, andlaconicity [96] (See Defns. 24, 25, 26, 27). When all four properties are met, the abstraction and themathematical model have an isomorphic (one-to-one) mapping and faithfully represent each other. Forexample, the network science and graph theory literature assumes an abstract conceptualization of nodesand edges prior to defining their 1-to-1 mathematical counterparts. Consequently, as hetero-functionalgraph and multi-layer network models of engineering systems are developed, there is a need to reconcileboth the abstraction and the mathematical model on the basis of the four criteria identified above (SeeSection A.).
(a) Soundness (b) Completeness
(c) Lucidity (d) Laconicity
Abstraction Model Abstraction Model
Abstraction Model Abstraction Model
Fig. 2. Graphical Representation of Four Ontological Properties As Mapping Between Abstraction and Model: a Soundness, bCompleteness, c Lucidity, and d Laconicity [95].
The ontological strength of hetero-functional graph theory comes from the “systems thinking” foun-
1It is likely that modeling abstract elements in the mind is unfamiliar to this journal’s readership. This is purely an issueof nomenclature. Most physicists and engineers would agree on the indispensable role that intuition – itself a mental model– has to the development of mathematical models of systems. For example, the shift from Newtonian mechanics to Einstein’srelativity constituted first an expansion in the abstract elements of the mental model and their relationships well before thatmental model could be translated into its associated mathematics. Similarly, the “disparate terminology and lack of consensus”identified by Kivela et. al [8] suggests that a reconciliation of this abstract mental model is required (See Section A).
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dations in the model-based systems engineering literature [97]. In effect, and very briefly, all systemshave a “subject + verb + operand” form where the system form is the subject, the system function is theverb + operand (i.e. predicate) and the system concept is the mapping of the two to each other. The keydistinguishing feature of HFGT (relative to multi-layer networks) is its introduction of system function.In that regard, it is more complete than multi-layer networks if system function is accepted as part ofan engineering system abstraction. Another key distinguishing feature of HFGT is the differentiationbetween elements related to transformation and transportation. In that regard, it takes great care to notoverload mathematical modeling elements and preserve lucidity.
B. Original Contribution
This paper provides a tensor-based formulation of several of the most important parts of hetero-functional graph theory. More specifically, it discusses the system concept, the hetero-functional adjacencymatrix, and the hetero-functional incidence tensor. Whereas the hetero-functional graph theory text [7] isa comprehensive discussion of the subject, the treatment is based entirely on two-dimensional matrices.The tensor-based formulation described in this work makes a stronger tie between HFGT and its onto-logical foundations in MBSE. Furthermore, the tensor-based treatment developed here reveals patterns ofunderlying structure in engineering systems that are less apparent in a matrix-based treatment. Finally,the tensor-based formulation facilitates an understanding of the relationships between HFGT and multi-layer networks (“despite its disparate terminology and lack of consensus”). In so doing, this tensor-basedtreatment is likely to advance Kivela et. al’s goal to discern the similarities and differences between thesemathematical models in as precise a manner as possible.
C. Paper Outline
The rest of the paper is organized as follows. Section II discusses the system concept as an alloca-tion of system function to system form. Section III discusses the hetero-functional adjacency matrixemphasizing the relationships between between system capabilities (i.e. structural degrees of freedom asdefined therein). Section IV, then, discusses the hetero-functional incidence tensor which describes therelationships between system capabilities, operands, and physical locations in space (i.e. system buffersas defined later). Section V goes on to discuss this tensor-based formulation from the perspective of layersand network descriptors. Section VI brings the work to a close and offers directions for future work. Giventhe multi-disciplinary nature of this work, several appendices are provided to support the work withbackground material. Appendix A provides the fundamental definitions of ontological science that wereused to motivate this work’s original contribution. Appendix B describes the notation conventions usedthroughout this work. The paper assumes that the reader is well grounded in graph theory and networkscience as it is found in any one of a number of excellent texts [98], [99]. The paper does not assumeprior exposure to hetero-functional graph theory. It’s most critical definitions are tersely introduced inthe body of the work upon first mention. More detailed classifications of these concepts are compiledin Appendix C for convenience. Given the theoretical treatment provided here, the interested reader isreferred to the hetero-functional graph theory text [7] for further explanation of these well-establishedconcepts and concrete examples. Furthermore, several recent works have made illustrative comparisonsbetween (formal) graphs and hetero-functional graphs [100], [101]. Finally, this work makes extensiveuse of set, Boolean, matrix, and tensor operations; all of which are defined unambiguously in AppendicesD, E, F, and G respectively.
II. The System Concept
At a high-level, the system concept AS describes the allocation of system function to system form as thecentral question of engineering design. This dichotomy of form and function is repeatedly emphasized inthe fields of engineering design and systems engineering [97], [102]–[104]. More specifically, the allocationof system processes to system resources is captured in the “design equation” [10], [94]:
P = JS �R (1)
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where R is set of system resources, P is the set of system processes, JS is the system knowledge base,and � is matrix Boolean multiplication (Defn. 47).
Definition 2 – System Resource: [4] An asset or object rv ∈ R that is utilized during the execution of aprocess. �
Definition 3 – System Process [4], [105]: An activity p ∈ P that transforms a predefined set of inputoperands into a predefined set of outputs. �
Definition 4 – System Operand: [4] An asset or object li ∈ L that is operated on or consumed duringthe execution of a process. �
Definition 5 – System Knowledge Base [9]–[13], [94]: A binary matrix JS of size σ (P ) × σ (R) whoseelement JS(w,v) ∈ {0,1} is equal to one when action ewv ∈ ES (in the SysML sense) exists as a systemprocess pw ∈ P being executed by a resource rv ∈ R. The σ () notation gives the size of a set. �
In other words, the system knowledge base forms a bipartite graph between the set of system processesand the set of system resources [13].
Hetero-functional graph theory further recognizes that there are inherent differences within the setof resources as well as within the set of processes. R = M ∪B∪H where M is the set of transformationresources (Defn. 28), B is the set of independent buffers (Defn. 29), and H is the set of transportationresources (Defn. 30). Furthermore, the set of buffers BS = M ∪ B (Defn. 31) is introduced for laterdiscussion. Similarly, P = Pµ∪ Pη where Pµ is the set of transformation processes (Defn. 32) and Pη is theset of refined transportation processes (Defn. 33). The latter, in turn, is determined from the Cartesianproduct (�) (Defn. 43) of the set of transportation processes Pη (Defn. 34) and the set of holding processesPγ (Defn. 35).
Pη = Pγ�Pη (2)
MagicDraw, 1-1 /Users/f002n19/Dropbox (LIINES)/1-MEPS/1-WesterSchoonenberg/IEM-WorkDocuments/04-Journals/D02-SpringerBrief/FiguresRaw/MagicDraw/Chapter5-Trimetrica.mdzip LFES-ResourceArchitecture Jun 3, 2018 4:51:16 PM
LFES-ResourceArchitecturepackage Model [ ]
+Transform Operand()operations
Transformation Resources M
operations+Transport Operand()+Hold Operand()
Resources R
Transportation Resources H
Buffers B_S
Independent Buffers B
Fig. 3. The Hetero-functional Graph Theory Meta-Architecture drawn using the Systems Markup Language (SysML). It consistsof three types of resources R =M ∪B∪H that are capable of two types of process Pη = Pγ�Pη [7].
This taxonomy of resources, processes, and their allocation is organized in the HFGT meta-architectureshown in Figure 3. The taxonomy of resources R and processes P originates from the field of productionsystems where transformation processes are viewed as “value-adding”, holding processes support thethe design of fixtures, and transportation processes are cost-minimized. Furthermore, their existenceis necessitated by their distinct roles in the structural relationships found in hetero-functional graphs.
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Consequently, subsets of the design equation 1 can be written to emphasize the relationships betweenthe constitutent classes of processes and resources [9]–[13].
Pµ = JM �M (3)
Pγ = Jγ �R (4)
Pη = JH �R (5)
Pη = JH �R (6)
where JM is the transformation knowledge base, Jγ is the holding knowledge base, JH is the transportationknowledge base, and JH is the refined transportation knowledge base [10], [13], [106]–[109].
The transportation knowledge base JH is best understood as a matricized 3rd-order tensor JH wherethe element JH (y1, y2,v) = 1 when the transportation process pu ∈ Pη defined by the origin bsy1 ∈ BS andthe destination bsy2 ∈ BS is executed by the resource rv ∈ R.
JH = FM (JH , [2,1], [3]) (7)
JH = F −1M (JH , [σ (BS ),σ (BS ),σ (R)], [2,1], [3]) (8)
where FM and F −1M are the matricization and tensorization functions (Defns. 55 and 56) respectively. Here,
FM() serves to vectorize the dimensions of the origin and destination buffers into the single dimensionof transportation processes.
The JH tensor reveals that the transportation knowledge base is closely tied to the classical under-standing of a graph ABS where point elements of form called nodes, herein taken to be the set of buffersBS , are connected by line elements of form called edges. Such a graph in hetero-functional graph theory(and model-based systems engineering) is called a formal graph [97] because all of its elements describethe system form and any statement of function is entirely implicit.
ABS (y1, y2) =σ (R)∨v
JH (y1, y2,v) =σ (R)∨v
JH (u,v) ∀y1, y2 ∈ {1, . . . ,σ (BS )},u = σ (BS )(y1 − 1) + y2,v ∈ {1, . . . ,σ (R)}
(9)
ABS = JH �3 1σ (R) = vec−1
(JH �1σ (R), [σ (BS ),σ (BS )]
)T(10)
AT VBS =(JH �3 1
σ (R))T V
= JH �1σ (R) (11)
where the∨
notation is the Boolean analogue of the∑
notation (Defn. 44), �n is the n-mode Booleanmatrix product (Defn. 61), vec−1() is inverse vectorization (Defn. 58) and ()V is shorthand for vectorization(Defn. 57). Furthermore, the notation 1n is used to indicate a ones-vector of length n. The transportationsystem knowledge base JH replaces the edges of the formal graph ABS with an explicit description offunction in the transportation processes Pη . The multi-column nature of the transportation knowledgebase JH contains more information than the formal graph ABS and allows potentially many resourcesto execute any given transportation process. Consequently, the OR operation across the rows of JH (orthe third dimension of JH ) is sufficient to reconstruct the formal graph ABS . In short, a single columntransportation knowledge base is mathematically equivalent to a vectorized formal graph ABS .
Similarly, the refined transportation knowledge base is best understood as a matricized 4th order tensorJH where the element JH (g,y1, y2,v) = 1 when the refined transportation process pϕ ∈ Pη defined by theholding process pγg ∈ Pγ , the origin bsy1 ∈ BS and the destination bsy2 ∈ BS is executed by the resourcerv ∈ R.
JH = FM (JH , [3,2,1], [4]) (12)
JH = F −1M
(JH , [σ (Pγ ),σ (BS ),σ (BS ),σ (R)], [3,2,1], [4]
)(13)
The JH tensor reveals that the refined transportation knowledge base is closely tied to the classicalunderstanding of a multi-commodity flow network ALBS [110]–[112]. Mathematically, it is a 3rd-ordertensor whose element ALBS (i,y1, y2) = 1 when operand li ∈ L is transported from buffer bsy1 to bsy2 . Again,
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the multi-commodity flow network ALBS is purely a description of system form and any statement offunction is entirely implicit. In the special case2 of a system where the set of operands L maps 1-to-1 theset of holding processes Pγ (i.e. i = g):
ALBS (i,y1, y2) =σ (R)∨v
JH (g,y1, y2,v) ∀g ∈ {1, . . . ,σ (Pγ )}, y1, y2 ∈ {1, . . . ,σ (BS )},v ∈ {1, . . . ,σ (R)} (14)
=σ (R)∨v
JH (ϕ,v) ∀ϕ = σ2(BS )(g − 1) + σ (BS )(y1 − 1) + y2,v ∈ {1, . . . ,σ (R)} (15)
ALBS = JH �4 1σ (R) = vec−1
(JH �1σ (R),
[σ (BS ),σ (BS ),σ (Pγ )
])T(16)
AT VLBS =(JH �4 1
σ (R))T V
= JH �1σ (R) (17)
The refined transportation system knowledge base JH replaces the operands and edges of the multi-commodity flow network AMP with an explicit description of function in the holding processes Pγ andtransportation processes Pη . The multi-column nature of the refined transportation knowledge base JHcontains more information than the multi-commodity flow network ALBS and allows potentially manyresources to execute any given refined transportation process. Consequently, the OR operation acrossthe rows of JH (or the fourth dimension of JH ) is sufficient to reconstruct the multi-commodity flownetwork ALBS . In short, a single column of the refined transportation knowledge base is mathematicallyequivalent to a vectorized multi-commodity flow network.
The transformation, holding, transportation and refined transportation knowledge bases (JM , Jγ , JH andJH ) readily serve to reconstruct the system knowledge base JS . First, the refined transportation knowledgebase is the Khatri-Rao product of the holding and transportation knowledge bases.
JH (σ (Pη)(g − 1) +u,v) = Jγ (g,v) · JH (u,v) ∀g ∈ {1, . . . ,σ (Pγ )},u = σ (BS )(y1 − 1) + y2,v ∈ {1, . . . ,σ (R)} (18)
JH = Jγ ~ JH (19)
=[Jγ ⊗1σ (Pη )
]·[1σ (Pγ ) ⊗ JH
](20)
where · is the Hadamard (or scalar) product (Defn. 50), ~ is the Khatri-Rao product (Defn. 53) and ⊗is the Kronecker product (Defn. 52). From this point, the system knowledge base JS is straightforwardlyreconstructed [9]–[13]:
JS =[JM | 0
JH
](21)
Hetero-functional graph theory also differentiates between the existence and the availability of physicalcapabilities in the system [10], [107]. While the former is described by the system knowledge base thelatter is captured by the system constraints matrix (which is assumed to evolve in time).
Definition 6 – System Constraints Matrix [9]–[13], [94]: A binary matrix KS of size σ (P )×σ (R) whoseelement KS(w,v) ∈ {0,1} is equal to one when a constraint eliminates event ewv from the event set. �
The system constraints matrix is constructed analogously to the system knowledge base [9]–[13].
KS =[KM | 0
KH
](22)
In this regard, the system constraints matrix has a similar meaning to graph percolation [113], [114] andtemporal networks [115].
Once the system knowledge base JS and the system constraints matrix KS have been constructed, thesystem concept AS follows straightforwardly.
2By Defn. 35, holding processes are distinguished by three criteria: 1.) different operands, 2.) how they hold those operands,and 3.) if they change the state of the operand. The special case mentioned above is restricted to only the first of these threeconditions.
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Definition 7 – System Concept [9]–[13], [94]: A binary matrix AS of size σ (P ) × σ (R) whose elementAS(w,v) ∈ {0,1} is equal to one when action ewv ∈ ES (in the SysML sense) is available as a system processpw ∈ P being executed by a resource rv ∈ R.
AS = JS KS = JS · KS (23)
where is Boolean subtraction (Defn. 48) and KS =NOT (KS ). �
Every filled element of the system concept indicates a system capability of the form: “Resource rv doesprocess pw”. The system constraints matrix limits the availability of capabilities in the system knowledgebase to create the system concept AS . The system capabilities are quantified by the structural degrees offreedom.Definition 8 – Structural Degrees of Freedom [9]–[13], [94]: The set of independent actions ES thatcompletely defines the available processes in a large flexible engineering system. Their number is givenby:
DOFS = σ (ES ) =σ (P )∑w
σ (R)∑v
[JS KS ] (w,v) (24)
=σ (P )∑w
σ (R)∑v
AS(w,v) (25)
= 〈JS , KS〉F (26)
�
As has been discussed extensively in prior publications, the term structural degrees of freedom is bestviewed as a generalization of kinematic degrees of freedom (or generalized coordinates) [9]–[13], [116].Note that the transformation degrees of freedom DOFM and the refined transportation degrees of freedomDOFH are calculated similarly [9]–[12]:
DOFM =σ (Pµ)∑j
σ (M)∑k
[JM KM ] (j,k) (27)
DOFH =σ (Pη )∑ϕ
σ (R)∑v
[JH KH ] (u,v) (28)
III. Hetero-functional Adjacency Matrix
Once the system’s physical capabilities (or structural degrees of freedom have been defined), the hetero-functional adjacency matrix Aρ is introduced to represent their pair-wise sequences. [13], [108], [117]–[119].Definition 9 – Hetero-functional Adjacency Matrix [13], [108], [117]–[119]: A square binary matrix Aρof size σ (R)σ (P )×σ (R)σ (P ) whose element Jρ(χ1,χ2) ∈ {0,1} is equal to one when string zχ1,χ2
= ew1v1ew2v2
∈Z is available and exists, where index χi ∈ [1, . . . ,σ (R)σ (P )]. �
In other words, the hetero-functional adjacency matrix corresponds to a hetero-functional graph G ={ES ,Z} with structural degrees of freedom (i.e. capabilities) ES as nodes and feasible sequences Z asedges.
Much like the system concept AS , the hetero-functional adjacency matrix Aρ arises from a Booleandifference [13], [108], [117]–[119].
Aρ = Jρ Kρ (29)
where Jρ is the system sequence knowledge base and Kρ is the system sequence constraints matrix.Definition 10 – System Sequence Knowledge Base [13], [108], [117]–[119]: A square binary matrix Jρ ofsize σ (R)σ (P )×σ (R)σ (P ) whose element Jρ(χ1,χ2) ∈ {0,1} is equal to one when string zχ1,χ2
= ew1v1ew2v2
∈ Zexists, where index χi ∈ [1, . . . ,σ (R)σ (P )]. �
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Definition 11 – System Sequence Constraints Matrix [13], [108], [117]–[119]: A square binary con-straints matrix Kρ of size σ (R)σ (P ) × σ (R)σ (P ) whose elements K(χ1,χ2) ∈ {0,1} are equal to one whenstring zχ1χ2
= ew1v1ew2v2
∈ Z is eliminated. �
The definitions of the system sequence knowledge base Jρ and the system sequence constraints matrixKρ feature a translation of indices from ew1v1
ew2v2to zχ1χ2
. This fact suggests that these matrices havetheir associated 4th order tensors Jρ, Kρ and Aρ.
Jρ = FM(Jρ, [1,2], [3,4]
)(30)
Kρ = FM(Kρ, [1,2], [3,4]
)(31)
Aρ = FM(Aρ, [1,2], [3,4]
)(32)
Each of these are discussed in turn. Jρ and its tensor-equivalent Jρ create all the potential sequencesof the capabilities in AS .
Jρ(w1,v1,w2,v2) = AS(w1,v1) ·AS(w2,v2) ∀w1,w2 ∈ {1 . . .σ (P )},v1,v2 ∈ {1 . . .σ (R)} (33)
Jρ(χ1,χ2) = AVS (χ1) ·AVS (χ2) ∀χ1,χ2 ∈ {1 . . .σ (R)σ (P )} (34)
Jρ = AVS AV TS (35)
Jρ =[JS · KS
]V [JS · KS
]V T(36)
Jρ = F −1M
(Jρ, [σ (P ),σ (R),σ (P ),σ (R)], [1,2], [3,4]
)(37)
Of these potential sequences of capabilities, the system sequence constraints matrix Kρ serves toeliminate the infeasible pairs. The feasibility arises from five types of constraints:
I: PµPµ. Two transformation processes that follow each other must occur at the same transformationresource. m1 =m2.
II: PµPη . A refined transportation process that follows a transformation process must have an originequivalent to the transformation resource at which the transformation process was executed. m1−1 =(u1 − 1)/σ (BS ) where / indicates integer division.
III: PηPµ. A refined transportation process that precedes a transformation process must have a destinationequivalent to the transformation resource at which the transformation process was executed. m2−1 =(u1 − 1)%σ (BS ) where % indicates the modulus.
IV: PηPη . A refined transportation process that follows another must have an origin equivalent to thedestination of the other. (u1 − 1)%σ (BS ) = (u2 − 1)/σ (BS )
V: P P . The type of operand of one process must be equivalent to the type of output of another process.In other words, the ordered pair of processes Pw1
Pw2is feasible if and only if AP (w1,w2) = 1 where
AP is the adjacency matrix that corresponds to a functional graph in which pairs of system processesare connected.
In previous hetero-functional graph theory works, the system sequence constraints matrix Kρ was calcu-lated straightforwardly using for FOR loops to loop over the indices χ1 and χ2 and checking the presenceof the five feasibility constraints identified above.
Here, an alternate approach based upon tensors is provided for insight into the underlying mathemat-ical structure. For convenience, Kρ =NOT (Kρ) captures the set of all feasibility conditions that pertain tovalid sequences of system capabilities. This set requires that any of the first four constraints above andthe last constraint be satisfied.
Kρ =(KρI ⊕ KρII ⊕ KρIII ⊕ KρIV
)· KρV (38)
where ⊕ is Boolean addition (Defn. 45) and KρI , KρII , KρIII , KρIV , KρV are the matrix implementationsof the five types of feasibility constraints identified above. Their calculation is most readily achievedthrough their associated 4th-order tensors.
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For the Type I constraint, KρI is constructed from a sum of 4th-order outer products (Defn. 54) ofelementary basis vectors.
KρI =σ (Pµ)∑w1=1
σ (M)∑v1=1
σ (Pµ)∑w2=1
v1∑v2=v1
eσ (P )w1 ◦ e
σ (R)v1 ◦ eσ (P )
w2 ◦ eσ (R)v2 (39)
where the eni notation places the value 1 on the ith element of a vector of length n. KρI is calculatedstraightforwardly by matricizing both sides and evaluating the sums.
KρI =σ (Pµ)∑w1=1
σ (M)∑v1=1
σ (Pµ)∑w2=1
v1∑v2=v1
(eσ (R)v1 ⊗ eσ (P )
w1
)⊗(eσ (R)v2 ⊗ eσ (P )
w2
)T(40)
KρI =σ (M)∑v1=1
(eσ (R)v1 ⊗
[1σ (Pµ)
0σ (Pη )
])(eσ (R)v1 ⊗
[1σ (Pµ)
0σ (Pη )
])T(41)
Similarly, for the Type II constraint:
KρII =σ (Pµ)∑w1=1
σ (M)∑v1=1
v1∑y1=v1
σ (R)∑v2=1
eσ (P )w1 ◦ e
σ (R)v1 ◦
0σ (Pµ)
Xσ (Pη )y1
◦ eσ (R)v2 (42)
Here, the Xσ (PH )y1 vector has a value of 1 wherever a refined transportation process pw2
originates at the
transformation resource mv1. Drawing on the discussion of the 3rd-order tensor JH in Section II, Xσ (PH )
y1 ,itself, is expressed as a vectorized sum of 3rd-order outer products.
Xσ (PH )y1 =
σ (Pγ )∑g=1
σ (BS )∑y2=1
(eσ (Pγ )g ◦ eσ (BS )
y1 ◦ eσ (BS )y2
)T V=
(1σ (Pγ ) ⊗ eσ (BS )
y1 ⊗1σ (BS ))
(43)
KρII is then calculated straightforwardly by matricizing both sides of Eq. 42 and evaluating the sums.
KρII =σ (Pµ)∑w1=1
σ (M)∑v1=1
v1∑y1=v1
σ (R)∑v2=1
(eσ (R)v1 ⊗ eσ (P )
w1
)⊗eσ (R)v2 ⊗
0σ (Pµ)
Xσ (PH )y1
T (44)
KρII =σ (M)∑v1=1
(eσ (R)v1 ⊗
[1σ (Pµ)
0Pη
])1σ (R) ⊗ 0σ (Pµ)
1σ (Pγ ) ⊗ eσ (BS )v1 ⊗1σ (BS )
T (45)
Similarly, for the Type III constraint:
KρIII =v2∑
y2=v2
σ (R)∑v1=1
σ (Pµ)∑w2=1
σ (M)∑v2=1
0σ (Pµ)
Xσ (Pη )y2
◦ eσ (R)v1 ◦ eσ (P )
w2 ◦ eσ (R)v2 (46)
Here, the Xσ (PH )y2 vector has a value of 1 wherever a refined transportation process pw1
terminates at the
transformation resource mv2. Xσ (PH )
y2 , itself, is expressed as a vectorized sum of 3rd-order outer products.
Xσ (PH )y2 =
σ (Pγ )∑g=1
σ (BS )∑y1=1
(eσ (Pγ )g ◦ eσ (BS )
y1 ◦ eσ (BS )y2
)T V=
(1σ (Pγ ) ⊗1σ (BS ) ⊗ eσ (BS )
y2
)(47)
KρIII is then calculated straightforwardly by matricizing both sides of Eq. 46 and evaluating the sums.
KρIII =v2∑
y2=v2
σ (R)∑v1=1
σ (Pµ)∑w2=1
σ (M)∑v2=1
eσ (R)v1 ⊗
0σ (Pµ)
Xσ (PH )y2
⊗ (eσ (R)v2 ⊗ eσ (P )
w2
)T(48)
KρIII =σ (M)∑v2=1
1σ (R) ⊗ 0σ (Pµ)
1σ (Pγ ) ⊗1σ (BS ) ⊗ eσ (BS )v2
(eσ (R)v2 ⊗
[1σ (Pµ)
0Pη
])T(49)
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Similarly, for the Type IV constraint:
KρIV =σ (BS )∑y2=1
σ (R)∑v1=1
y2∑y1=y2
σ (R)∑v2=1
0σ (Pµ)
Xσ (PH )y2
◦ eσ (R)v1 ◦
0σ (Pµ)
Xσ (PH )y1
◦ eσ (R)v2 (50)
KρIV is then calculated straightforwardly by matricizing both sides of Eq. 50 and evaluating the sums.
KρIV =σ (BS )∑y2=1
σ (R)∑v1=1
y2∑y1=y2
σ (R)∑v2=1
eσ (R)v1 ⊗
0σ (Pµ)
Xσ (PH )y2
⊗ eσ (R)v2 ⊗
0σ (Pµ)
Xσ (PH )y1
T (51)
KρIV =σ (BS )∑y2=1
1σ (R) ⊗ 0σ (Pµ)
1σ (Pγ ) ⊗1σ (BS ) ⊗ eσ (BS )y2
1σ (R) ⊗ 0σ (Pµ)
1σ (Pγ ) ⊗ eσ (BS )y2 ⊗1σ (BS )
T (52)
The Type V constraint must make use of the functional graph adjacency matrix AP . Consequently, thefourth-order tensor KρV is calculated first on a scalar basis using the Kronecker delta function δi (Defn.49) and then is matricized to KρV .
KρV (w1,v1,w2,v2) = δv1v1· δv2v2
·AP (w1,w2) ∀w1,w2 ∈ {1, . . . ,σ (P )}, v1,v2 ∈ {1, . . . ,σ (R)} (53)
KρV =σ (R)∑v1=1
σ (R)∑v2=1
(eσ (R)v1 ⊗ eσ (R)T
v2
)⊗AP (54)
KρV =(1σ (R) ⊗1σ (R)T
)⊗AP (55)
Once the system sequence knowledge base and constraints matrix have been calculated, the numberof sequence-dependent degrees of freedom follow straightforwardly.
Definition 12 – Sequence-Dependent Degrees of Freedom [13], [108], [117]–[119]: The set of inde-pendent pairs of actions zχ1χ2
= ew1v1ew2v2
∈ Z of length 2 that completely describe the system language.The number is given by:
DOFρ = σ (Z) =σ (R)σ (P )∑
χ1
σ (R)σ (P )∑χ2
[Jρ Kρ](χ1,χ2) (56)
=σ (R)σ (P )∑
χ1
σ (R)σ (P )∑χ2
[Aρ](χ1,χ2) (57)
�
For systems of substantial size, the size of the hetero-functional adjacency matrix may be challengingto process computationally. However, the matrix is generally very sparse. Therefore, projection operatorsare used to eliminate the sparsity by projecting the matrix onto a one’s vector [108], [119]. This isdemonstrated below for JVS and Aρ:
PSJVS = 1σ (ES ) (58)
PSAρPTS = Aρ (59)
where PS is a (non-unique) projection matrix for the vectorized system knowledge base and the hetero-functional adjacency matrix [108], [119]. Note that the number of sequence dependent degrees of freedomfor the projected hetero-functional adjacency matrix can be calculated as:
DOFρ = σ (Z) =σ (ES )∑ψ1
σ (ES )∑ψ2
[Aρ](ψ1,ψ2) (60)
where ψ ∈ [1, . . . ,σ (ES )].
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IV. Hetero-functional Incidence Tensor
To complement the concept of a hetero-functional adjacency matrix Aρ and its associated tensor Aρ,the hetero-functional incidence tensor Mρ describes the structural relationships between the physicalcapabilities (i.e. structural degrees of freedom) ES , the system operands L, and the system buffers BS .
Mρ = M+ρ −M−ρ (61)
Definition 13 – The Negative 3rd Order Hetero-functional Incidence Tensor M−ρ : The negative hetero-functional incidence tensor Mρ
− ∈ {0,1}σ (L)×σ (BS )×σ (ES ) is a third-order tensor whose element M−ρ(i,y,ψ) = 1when the system capability εψ ∈ ES pulls operand li ∈ L from buffer bsy ∈ BS . �
Definition 14 – The Positive 3rd Order Hetero-functional Incidence Tensor M+ρ : The positive hetero-
functional incidence tensor M+ρ ∈ {0,1}σ (L)×σ (BS )×σ (ES ) is a third-order tensor whose element M+
ρ(i,y,ψ) = 1when the system capability εψ ∈ ES injects operand li ∈ L into buffer bsy ∈ BS . �
The calculation of these two tensors depends on the definition of two more matrices.
Definition 15 – The Negative Process-Operand Incidence Matrix M−LP : A binary incidence matrixM−LP ∈ {0,1}
σ (L)×σ (P ) whose element M−LP (i,w) = 1 when the system process pw ∈ P pulls operand li ∈ L asan input. It is further decomposed into the negative transformation process-operand incidence matrixM−LPµ (Defn. 36) and the negative refined transformation process-operand incidence matrix M−LPη (Defn.37) which by definition is in turn calculated from the negative holding process-operand incidence matrixM−LPγ (Defn. 38).
M−LP =[M−LPµ M−LPη
]=
[M−LPµ M−LPγ ⊗1
σ (Pη )T]
(62)
�
Definition 16 – The Positive Process-Operand Incidence Matrix M+LP : A binary incidence matrix M+
LP ∈{0,1}σ (L)×σ (P ) whose element M+
LP (i,w) = 1 when the system process pw ∈ P injects operand li ∈ L as anoutput. It is further decomposed into the positive transformation process-operand incidence matrix M+
LPµ(Defn. 39) and the positive refined transformation process-operand incidence matrix M+
LPη(Defn. 40)
which, by definition, is, in turn, calculated from the positive holding process-operand incidence matrixM+LPγ
(Defn. 41)
M+LP =
[M+LPµ
M+LPη
]=
[M+LPµ
M+LPγ⊗1σ (Pη )T
](63)
�
With the definitions of these incidence matrices in place, the calculation of the negative and positivehetero-functional incidence tensors M−ρ and M+
ρ follows straightforwardly as a third-order outer product.For M−ρ :
M−ρ =σ (L)∑i=1
σ (BS )∑y1=1
eσ (L)i ◦ eσ (BS )
y1 ◦PS((X−iy1
)V )(64)
where
X−iy1=
M−TLPµeσ (L)i e
σ (M)Ty1 | 0
M−TLPγ eσ (L)i ⊗
(eσ (BS )y1 ⊗1σ (BS )
)⊗1σ (R)T
(65)
The X−iy1matrix is equivalent in size to the system concept AS . It has a value of one in all elements where
the associated process both withdraws input operand li and originates at the buffer bsy1 . Consequently,when X−iy1
is vectorized and then projected with PS , the result is a vector with a value of one only wherethe associated system capabilities meet these criteria.
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For M+ρ :
M+ρ =
σ (L)∑i=1
σ (BS )∑y2=1
eσ (L)i ◦ eσ (BS )
y2 ◦PS((X+iy2
)V )(66)
where
X+iy2
=
M+TLPµeσ (L)i e
σ (M)Ty2 | 0
M+TLPγeσ (L)i ⊗
(1σ (BS ) ⊗ eσ (BS )
y2
)⊗1σ (R)T
(67)
The X+iy2
matrix is equivalent in size to the system concept AS . It also has a value of one in all elementswhere the associated process both injects output operand li and terminates at the buffer bsy2 . Consequently,when X+
iy2is vectorized and then projected with PS , the result is a vector with a value of one only where
the associated system capabilities meet these criteria.It is important to note that the definitions of the 3rd order hetero-functional incidence tensors M−ρ ,
and M+ρ are provided in projected form as indicated by the presence of the projection operator PS in
Equations 64 and 66 respectively. It is often useful to use the un-projected form of these tensors.
M−ρ =σ (L)∑i=1
σ (BS )∑y1=1
eσ (L)i ◦ eσ (BS )
y1 ◦(X−iy1
)V(68)
M+ρ =
σ (L)∑i=1
σ (BS )∑y2=1
eσ (L)i ◦ eσ (BS )
y2 ◦(X+iy2
)V(69)
The third dimension of these unprojected 3rd order hetero-functional incidence tensors can then besplit into two dimensions to create 4th order hetero-functional incidence tensors.
M+P R = vec−1
(M+
ρ , [σ (P ),σ (R)],3)
(70)
M−P R = vec−1(M−ρ , [σ (P ),σ (R)],3
)(71)
These fourth order tensors describe the structural relationships between the system processes P , thephysical resources R that realize them, the system operands L that are consumed and injected in theprocess, and the system buffers BS from which these are operands are sent and the system buffers BS towhich these operands are received. They are used in the following section as part of the discussion onlayers.
MP R =M+P R −M
−P R (72)
Definition 17 – The Negative 4th Order Hetero-functional Incidence Tensor M−P R: The negative 4th
Order hetero-functional incidence tensor M−P R ∈ {0,1}σ (L)×σ (BS )×σ (P )×σ (R) has element M−P R(i,y,w,v) = 1
when the system process pw ∈ P realized by resource rv ∈ R pulls operand li ∈ L from buffer bsy ∈ BS . �
Definition 18 – The Positive 4th Order Hetero-functional Incidence Tensor M−P R: The positive 4th
Order hetero-functional incidence tensor M−P R ∈ {0,1}σ (L)×σ (BS )×σ (P )×σ (R) has element M−P R(i,y,w,v) = 1
when the system process pw ∈ P realized by resource rv ∈ R injects operand li ∈ L into buffer bsy ∈ BS . �
Returning back to the hetero-functional incidence tensor Mρ, it and and its positive and negativecomponents M+
ρ , M−ρ , can also be easily matricized.
Mρ = FM(Mρ, [1,2], [3]
)(73)
M−ρ = FM(M−ρ , [1,2], [3]
)(74)
M+ρ = FM
(M+
ρ , [1,2], [3])
(75)
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The resulting matrices have a size of σ (L)σ (BS ) × σ (ES ) which have a corresponding physical intuition.Each buffer bsy has σ (L) copies to reflect a place (i.e. bin) for each operand at that buffer. Each of theseplaces then forms a bipartite graph with the system’s physical capabilities. Consequently, and as expected,the hetero-functional adajacency matrix Aρ can be calculated as a matrix product of the positive andnegative hetero-functional incidence matrices M+
ρ and M+ρ .
Aρ =M+Tρ M−ρ (76)
Such a product systematically enforces all five of the feasibility constraints identified in Section III.
V. Discussion
Given the discussion on multi-layer networks in the introduction, it is worthwhile reconciling the gapin terminology between multi-layer networks and hetero-functional graph theory. First, the concept oflayers in hetero-functional graphs is discussed. Second, an ontological comparison of layers in hetero-functional graphs and multi-layer networks is provided. Third, a discussion of network descriptors inthe context of layers is provided. Given the “disparate terminology and the lack of consensus” in themulti-layer network literature, the discussion uses the multi-layer description provided De Dominico et.al [14].
A. Layers in Hetero-functional GraphsDefinition 19 – Layer: A layer Gλ = {ESλ,ZSλ} of a hetero-functional graph G = {ES ,ZS} is a subset ofa hetero-functional graph, Gλ ⊆ G, for which a predefined layer selection (or classification) criterionapplies. A set of layers in a hetero-functional graph adhere to a a classification scheme composed of anumber of selection criteria. �
Note that this definition of a layer is particularly flexible because it depends on the nature of theclassification scheme and its associated selection criteria. Nevertheless, and as discussed later, it isimportant to choose a classification scheme that leads to a set of mutually exclusive layers that arealso collectively exhaustive of the hetero-functional graph as a whole.
To select out specific subsets of capabilities (or structural degrees of freedom), HFGT has used theconcept of “selector matrices” of various types [7], [120]. Here a layer selector matrix is defined.Definition 20: Layer Selector Matrix: A binary matrix Λλ of size σ (P )×σ (R) whose element Λλ(w,v) = 1when the capability ewv ⊂ ESλ. �
From this definition, the calculation of a hetero-functional graph layer follows straightforwardly. First,a layer projection operator Pλ is calculated:
PλΛVλ = 1σ (ES ) (77)
Next, the negative and positive hetero-functional incidence tensors M−ρλ and M+ρλ for a given layer λ are
calculated straightforwardly.
M−ρλ =σ (L)∑i=1
σ (BS )∑y1=1
eσ (L)i ◦ eσ (BS )
y1 ◦Pλ((X−iy1
)V )= M−ρ �3Pλ (78)
M+ρλ =
σ (L)∑i=1
σ (BS )∑y2=1
eσ (L)i ◦ eσ (BS )
y2 ◦PS((X+iy2
)V )= M+
ρ �3Pλ (79)
From there, the positive and negative hetero-functional incidence tensors for a given layer can be matri-cized and the adjacency matrix of the associated layer Aρλ follows straightforwardly.
M+ρλ = FM(M+
ρλ, [1,2], [3]) (80)
M−ρλ = FM(M−ρλ, [1,2], [3]) (81)
Aρλ = M+Tρλ M
−ρλ (82)
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This approach of separating a hetero-functional graph into its constituent layers is quite generic becausethe layer selector matrix Λλ can admit a wide variety of classification schemes. Three classificationschemes are discussed here:
1) An Input Operand Set Layer2) An Output Operand Set Layer3) A Dynamic Device Model Layer
0
2
4
6
8
10
12
0 2 4 6 8 10 12
(Sequence-dependent) Degrees of Freedom of Trimetrica
Legend:
Degree of Freedom w/ operand “Electric Power at 132kV”
Sequence-dependent Degree of Freedom
0
2
4
6
8
10
120
2
4
6
8
10
120
2
4
6
8
10
120
2
4
6
8
10
12
Degree of Freedom w/ operand “Potable Water”
Degree of Freedom w/ operands “Potable Water and Electric Power at 132kV”
Degree of Freedom w/ operand “Electric Vehicle”
Degree of Freedom w/ operands “Electric Vehicle and Electric Power at 132kV”
Potable Water Topology
Electrified PotableWater Topology
Electrified PowerTopology
Charging TopologyTopology
Transportation Topology
Fig. 4. The Trimetric Smart City Infrastructure Test Case Visualized as Five Layers Defined by Input Operand Sets: ThePotable Water Topology, The Electrified Potable Water Topology, the Electric Power Topology, the Charging Topology, and theTransportation Topology
Definition 21: Input Operand Set Layer: A hetero-functional graph layer for which all of the node-capabilities have a common set of input operands Lλ ⊆ L. �
This definition of an Operand Set Layer was used in the HFGT text [7] to partition the Trimetrica testcase (first mentioned in Figure 1) into the multi-layer depiction in Figure 4. In this classification scheme,any system contains up to 2σ (L) possible layers. For completeness, an index λD ∈ {1, . . . ,2σ (L)} is used todenote a given layer. In reality, however, the vast majority of physical systems exhibit far fewer than2σ (L) layers. Consequently, it is often useful to simply assign an index λ to each layer and create a 1-1mapping function (i.e. lookup table) fλ back to the λD index.
fλ : λ→ λD (83)
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The utility of the λd index (stated as a base 10 number) becomes apparent when it is converted intoa binary (base 2) number λv ∈ {0,1}σ (L) which may be used equivalently as a binary vector of the samelength.
λv = bin(λD ) (84)
The resulting binary vector λv has the useful property that λv(li) = 1, iff operand li ∈ Lλ. Consequently,a given value of λv serves to select from L the operands that pertain to layer λ. The associated layerselector matrix follows straightforwardly:
Λλ(w,v) ={
1 if λv =M−LP (:,w) ∀rv ∈ R0 otherwise
(85)
It is also worth noting that the layer selector matrix Λ above is effectively a third order tensor whosevalue Λ(λ,w,v) = 1 when the capability ewv is part of layer λ.
One advantage of a classification scheme based on sets of input operands is that they lead to thegeneration of a mutually exclusive and collectively exhaustive set of layers. Because no process (andconsequently capability) has two sets of input operands, it can only exist in a single layer (mutualexclusivity). In the meantime, the presence of 2σ (L) assures that all capabilities fall into (exactly) onelayer (exhaustivity). It is worth noting that a classification scheme based on individual operands wouldnot yield these properties. For example, a water pump consumes electricity and water as input operands.Consequently, it would have a problematic existence in both the “water layer” as well as the “electricitylayer”. In contrast, a classification scheme based on operand sets creates an “electricity-water” layer.
Analogously to Defn. 21, an output-operand set layer can be defined and its associated layer selectormatrix calculated.
Definition 22: Output Operand Set Layer: A hetero-functional graph layer for which all of the node-capabilities have a common set of output operands Lλ ⊆ L. �
Λλ(w,v) ={
1 if λv =M+LP (:,w) ∀rv ∈ R
0 otherwise(86)
The third classification scheme is required when developing dynamic equations of motion from thestructural information of a hetero-functional graph [117]. Every process is said to have a “dynamic devicemodel” that is usually described as a set of differential, algebraic, or differential-algebraic equations[117]. The simplest of these are the constitutive laws of basic dynamic system elements (e.g. resistors,capacitors, and inductors). Some processes, although distinct, may have device models with the samefunctional form. For example, two resistors at different places in an electrical system have the sameconstitutive (Ohm’s) law, but have different transportation processes because their origin and destinationsare different. Consequently, layers that distinguish on the basis of dynamic device model (i.e. constitutivelaw) are necessary.
Definition 23: Dynamic Device Model Layer: A hetero-functional graph layer for which all of the node-capabilities have a dynamic device model with the same functional form. �
In such a case, the layer selector matrix Λλ straightforwardly maps capabilities to their layer and dynamicdevice model interchangeably. A sufficient number of layers need to be created to account for all of thedifferent types of dynamic device models in the system. This classification scheme may be viewed as ageneralization of the well-known literature on “linear-graphs” [121] and “bond graphs” [122].
B. Finding Commonality between Multilayer Networks and Hetero-functional Graphs
The above discussion of layers in a hetero-functional graph inspires a comparison with multi-layernetworks. The multi-layer adjacency tensor (AMLN ) defined by De Dominico et. al. [14] is chosen tofacilitate the discussion. This fourth order tensor has elements AMLN (α1,α2,β1,β2) where the indicesα1,α2 denote “vertices” and β1,β2 denote “layers”. De Dominico et. al write that this multilayer adjacencytensor is a [14]: “. . . very general object that can be used to represent a wealth of complicated relationships amongnodes.” The challenge in reconciling the multi-layer adjacency tensor AMLN and the hetero-functional
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adjacency tensor Aρ is an ontological one. Referring back to the ontological discussion in the introductionand more specifically Figure 2 reveals that the underlying abstract conceptual elements (in the mind) towhich these two mathematical models refer may not be the same.
Consider the following interpretation of AMLN (α1,α2,β1,β2) =ABS l1(y1, y2, i1, i2) where the multi-layernetwork’s vertices are equated to the buffers BS and the layers are equated to the operands L. Thisinterpretation would well describe the departure of an operand li1 from buffer bsy1
and arriving as li2 atbsy2
. The equivalence of vertices to buffers is effectively a consensus view in the literature. In contrast,the concept of a “layer” in a multi-layer network (as motivated in the introduction) remains relativelyunclear. The equivalence of layers to operands warrants further attention.
Theorem 1: The mathematical model ABS l1 is neither lucid nor complete with respect to the systemprocesses P (as an abstraction). �
Proof 1: By contradiction. Assume that ABS l1 is both lucid and complete network model with respect tosystem processes P . Consider an operand l1 that departs bs1, undergoes process p1, and arrives as l1 atbs2. Now consider the same operand l1 that departs bs1, undergoes process p2, and arrives as l1 at bs2.Both of these scenarios would be denoted by ABS l1(1,2,1,1) = 1. Consequently, this modeling element isoverloaded and as such violates the ontological property of lucidity. Furthermore, because ABS l1 makesno mention of the concept of system processes, then it violates the completeness property as well. �
The counter-example provided in the proof above is not simply a theoretical abstraction but ratherquite practical. For several decades, the field of mechanical engineering has used “linear graphs” [121] toderive the equations of motion of dynamic systems with multi-domain physics. Consider the RLC circuitshown in Figure 5 and its associated linear graph. As parallel elements, the inductor and capacitor bothtransfer electrical power (as an operand) between the same pair of nodes. However, the constitutive law(as a physical process) of a capacitor is distinct from that of the inductor. Consequently, the interpretationABS l1 of a multi-layer network is inadequate even for this very simple counter-example3.
Rs
L Cs(t) +
-
Rs
s(t) L C
Fig. 5. A Simple RLC Circuit shown as a circuit diagram on left and as a linear graph model on right. Each resistor, capacitorand inductor can be said to be part of its own layer by virtue of their distinct constitutive laws.
Another possible interpretation of a multi-layer network is AMLN (α1,α2,β1,β2) = ABS l2(y1, y2,w1,w2)where the multi-layer network’s vertices are equated to the buffers BS and the layers are equated to theprocesses P . This interpretation would well describe the execution of a process pw1
that is realized bybuffer bsy1
followed by a process pw2that is realized by buffer bsy2
. The equivalence of layers to processeswarrants further attention as well.
Theorem 2: The mathematical model ABS l2 is neither lucid nor complete with respect to the system’stransportation resources H (as an abstraction). �
3Although electric power systems and circuits have served as a rich application domain for graph theory and network science,these approaches usually parameterize the circuit components homogenously as a fixed-value impedance/admittance at constantfrequency. When the constant frequency assumption is relaxed, the diversity of constitutive laws for resistors, capacitors, andinductors must be explicitly considered.
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Proof 2: By contradiction. Assume that ABS l2 is both a lucid and complete network model with respect tosystem’s transportation resources H . Consider transportation process p between a buffer bs1 and a distinctbuffer bs2. If such a transportation process were realized by any buffer bs ∈ BS , then by definition it wouldno longer be a buffer but rather a transportation resource. Consequently,ABS l2 is not complete with respectthe system’s transportation resources H . Now consider a process p1 that is realized by buffer bs1 followedby a process p2 that is realized by a distinct buffer bs2. This is denoted by ABS l2(1,2,1,2) = 1. Given thedistinctness of bs1 and bs2, a transportation process must have happened in between p1 and p2 althoughit is not explicitly stated by the mathematical statement ABS l2(1,2,1,2) = 1. Such a transportation process,although well-defined by its origin and destination could have been realized by any one of a numberof transportation resources. Consequently, the modeling element is overloaded and as such violates theproperty of lucidity. The lack of an explicit description of transportation processes or resources limitsthe utility of this type of multi-layer network model. �
It is worth noting that the first multi-layer network interpretation ABS l1 can be derived directly fromthe positive and negative hetero-functional incidence matrices.
ABS l1 = F −1M
(M−Tρ M+
ρ , [σ (BS ),σ (BS ),σ (L),σ (L)], [1,3], [2,4])
(87)
When M−Tρ and M+ρ are multiplied so that the capabilities Es are the inner dimension, the result is
an adjacency matrix that when tensorized becomes ABS l1 . In effect, ABS l1 (in matricized form) is thedual adjacency matrix [123] of the hetero-functional adjacency matrix Aρ. The presence of this matrixmultiplication obfuscates (i.e. creates a lack of lucidity) as to whether one capability or another occurredwhen expressing the adjacency tensor element ABS l1(y1, y2, i1, i2). In contrast, the matrix multiplicationin Eq. 76 does not cause the same problem. When two capabilities succeed one another, the informationassociated with their physical feasibility in terms of intermediate buffers and their functional feasibilityin terms of intermediate operands remains intact. In other words, given the sequence of capabilitiesew1v1
ew2v2, one can immediately deduce the exchanged operands in Lλ ⊆ L and the intermediate buffer
bs. In the case of the exchanged operands, one simply needs to intersect the output-operand set of thefirst process with the input operand set of the second process. In the case of the intermediate buffer, onechecks if either or both of the resources are buffers. If not, then two transportation processes followedone another and the intermediate buffer is deduced by Eq. 90. In short, the hetero-functional adjacencymatrix (or tensor) unambigously describes the sequence of two subject+verb+operand sentences whereasneither of the above interpretations of a multi-layer network do.
C. Network DescriptorsIn light of the commonalities and differences between hetero-functional graphs and (formal) multilayer
networks, this section discusses the meaning of network descriptors in the context of hetero-functionalgraphs. In this regard, the hetero-functional adjacency matrix is an adjacency matrix like any other.Consequently, network descriptors can be calculated straightforwardly. Furthermore, network descriptorscan be applied to subsets of the graph so as to conduct a layer-by-layer analysis. Nevertheless, that thenodes in a hetero-functional graph represent whole-sentence-capabilities means that network descriptorshave the potential to provide new found meanings over formal graphs based on exclusively formalelements.
1) Degree Centrality: Degree centrality measures the number of edges attached to a vertex. Since ahetero-functional graph is a directed graph, there is a need to distinguish between the in-degree centrality,which measures the number of edges going into vertex, and the out-degree centrality, which measuresthe number of edges going out of a vertex [100]. In the context of hetero-functional graph theory, the in-degree centrality of a vertex calculates the number of capabilities that potentially proceed the capabilityrelated to the vertex. The out-degree centrality calculates the number of capabilities that potentiallysucceed the vertex’s capability. The higher the degree centrality of a capability, the more connected thatcapability is to the other capabilities in the hetero-functional graph. It is important to recognize thatbecause transportation capabilities receive nodes in a hetero-functional graph, they can become the mostcentral node. In contrast, the degree centrality of a formal graph could not reach such a conclusionbecause the function of transportation is tied to formal edges rather than formal nodes.
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2) Closeness Centrality: Closeness centrality measures the average shortest path from one vertex to everyother reachable vertex in the graph. In a hetero-functional graph, the meaning of closeness centralityshows how a disruption has the potential to propagate through the graph across all different types ofoperands [100]. This metric is especially valuable for the resilience studies of interdependent systems,where the propagation of disruption across multiple disciplines is often poorly understood.
3) Eigenvector Centrality: Eigenvector centrality calculates the importance of a node relative to the othernodes in the network [124]. It also includes the eigenvector centrality of the node’s direct neighbors [14].The eigenvector centrality is specifically designed for the weighting of the in-degree of nodes in a directednetwork. The Katz centrality, on the other hand, provides an approach to study the relative importanceof nodes based on the out-degree [14].
4) Clustering Coefficients: Clustering coefficients describe how strongly the nodes of a network clustertogether. This is performed by searching for “triangles” or “circles” of nodes in a network. In a directednetwork, these circles can appear in multiple distinct combinations of directed connections. Each ofthese combinations needs to be measured and counted differently. Fagiolo discussed this taxonomy andaccompanying clustering coefficients [125]. These clustering coefficients for directed networks can bedirectly applied to hetero-functional graphs and show which capabilities are strongly clustered together.The definition of layers in hetero-functional graphs allows for a consistent definition and calculationof clustering coefficients within and across layers for different types of systems. When investigating asystem, the clustering coefficient may show clusters of capabilities that were not yet recognized as heavilyinterdependent. Such information can be used to revise control structures such that clusters of capabilitiesare controlled by the same entity for efficiency.
5) Modularity: Modularity serves as a measure to study if a network can be decomposed in disjointsets. In the hetero-functional graph theory literature, much has been published about modularity as itwas a prime motivation towards the inception of the theory [10], [126]. Hetero-functional graph theoryintroduces the concept of the Degree-of-Freedom-based Design Structure Matrix (or: the capability DSM)that does not only encompass the hetero-functional adjacency matrix, but extends the concept to theother elements of hetero-functional graph theory: the service model and the control model. The hetero-functional graph design structure matrix has the ability to visualize the couplings between the subsystemsof an engineering system and to classify those interfaces. Note that the capability DSM can also be appliedto just the hetero-functional adjacency matrix. Furthermore, the capability DSM applies to the conceptof layers in a hetero-functional graph. To study the interfaces between layers, the capability DSM canadopt layers as subsystems and classify the interfaces between the layers as mentioned previously. Inconclusion, hetero-functional graphs are described by flat adjacency matrices, regardless of the numberof layers in the analysis. Consequently, conventional graph theoretic network descriptors can be applied.The main difference in definition between the conventional graph theoretic application and the hetero-functional graph theoretic application is the result of the difference in the definition of the fundamentalmodeling elements, the nodes and edges, in a hetero-functional graph.
VI. Conclusions and Future Work
This paper has provided a tensor-based formulation of several of the most important parts of hetero-functional graph theory. More specifically, it discussed the system concept showing it as a generalizationof formal graphs and multi-commodity networks. It also discussed the hetero-functional adjacency matrixand its tensor-based closed form calculation. It also discussed the hetero-functional incidence tensor andrelated it back to the hetero-functional adjacency matrix. The tensor-based formulation described inthis work makes a stronger tie between HFGT and its ontological foundations in MBSE. Finally, thetensor-based formulation facilitates an understanding of the relationships between HFGT and multi-layer networks “despite its disparate terminology and lack of consensus”. In so doing, this tensor-basedtreatment is likely to advance Kivela et. al’s goal to discern the similarities and differences between thesemathematical models in as precise a manner as possible.
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Appendix
A. Ontological Science Definitions
The formal definitions of soundness, completeness, lucidity, and laconicity rely on “Ullman’s Triangle”in Figure 6.
abstracts
repr
esen
ts
repr
esen
ts
instantiation
abstracts
refers to
Domain Conceptualization: ! Abstraction: "
Physical Object/System
Model: ℳ
Real Domain: #
Language: ℒ
refers to
Fig. 6. Two Versions of Ullman’s Triangle. On the left is the relationship between reality, the understanding of reality, and thedescription of reality. On the right the instantiated version of the ontological definition [95].
Definition 24 – Soundness [96]: A language L is sound w.r.t. a domain conceptualization C iff everymodeling primitive in the language (M) has an interpretation in the domain abstraction A. (The absenceof soundness results in the excess of modeling primitives w.r.t. the domain abstractions as shown inFigure 2.c on lucidity.) �
Definition 25 – Completeness [96]: A language L is complete w.r.t. a domain conceptualization C iffevery concept in the domain abstraction A of that domain is represented in a modeling primitive of thatlanguage. (The absence of completeness results in one or more concepts in the domain abstraction notbeing represented by a modeling primitive, as shown in Figure 2.d on laconicity.) �
Definition 26 – Lucidity [96]: A language L is lucid w.r.t. a domain conceptualization C iff everymodeling primitive in the language represents at most one domain concept in abstraction A. (The absenceof lucidity results in the overload of a modeling primitive w.r.t. two or more domain concepts as shownin Figure 2.a on soundness.) �
Definition 27 – Laconicity [96]: A language L is laconic w.r.t. a domain conceptualization C iff everyconcept in the abstraction A of that domain is represented at most once in the model of that language.(The absence of laconicity results in the redundancy of modeling primitives w.r.t the domain abstractionsas shown in Figure 2.b on completeness.) �
B. Notation Conventions
Several notation conventions are used throughout this work:• All sets are indicated by a capital letter. e.g. P – the set of processes..• All elements within a set are indicated by a lower case letter. e.g. p ∈ P .• A subscript number indicates the position in an ordered set. e.g. pi ∈ P .• The ith elementary basis vector of size n is denoted by eni .• A matrix of ones of size m×n is denoted by 1m×n.• A matrix of zeros of size m×n is denoted by 0m×n.• With the exception of elementary basis vectors, all vectors and matrices are indicated with a capital
letter. e.g. JH .• All tensors are indicated with capital letters in calligraphic script. e.g. JH.
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• All elements in vectors, matrices, and tensors are indicated with indices within parentheses. e.g.JS(w,v).
• A(:,i) denotes the ith column of A or equivalently the ith mode-1 fiber. The : indicates all elementsof the vector.
• A(i,:) denotes the ith row of A or equivalently the ith mode-2 fiber.• A(i,j,:) denotes the i, j mode-3 fiber of A.• Given the presence of Booleans, real numbers and their operators, this work refrains from the use
of Einstein’s (shorthand) tensor notation where the sigma-notation∑
is eliminated.
C. Hetero-functional Graph Theory Definitions
Definition 28 – Transformation Resource [7]: A resource r ∈ R is a transformation resource m ∈M iffit is capable of one or more transformation processes on one or more operands and it exists at a uniquelocation in space. �
Definition 29 – Independent Buffer [7]: A resource r ∈ R is an independent buffer b ∈ B iff it is capableof storing one or more operands and is not able to transform them or transport them to another locationand it exists at a unique location in space. �
Definition 30 – Transportation Resource [7]: A resource r ∈ R is a transportation resource h ∈ H iff itis capable of transporting one or more operands between an origin and a distinct destination, withouttransforming these operands. �
Definition 31 – Buffer [7]: A resource r ∈ R is a buffer bs ∈ BS iff it is capable of storing one or moreoperands at a unique location in space. BS =M ∪B. �
Definition 32 – Transformation Process [7]: A process is a transformation process pµj ∈ Pµ iff it is capableof transforming one or more properties of a set of operands into a distinct set of output properties inplace. It’s syntax is:
{transitive verb, operands} → {outputs} (88)
�
Definition 33 – Refined Transportation Process [7]: A process is a refined transportation process pηϕ ∈Pη iff it is capable of transporting one or more operands between an origin buffer bsy1
∈ BS to a destinationbuffer bsy2
∈ BS while it is realizing holding process pγg ∈ Pγ . It’s syntax is:
{transport, operands, origin, destination,while transitive verb} →{outputs, destination} (89)
�
Definition 34 – Transportation Process [7]: A process is a transportation process pηu ∈ Pη iff it is capableof transporting one or more operands between an origin buffer bsy1
∈ BS to a destination buffer bsy2∈ BS
according to the following convention of indices [9]–[13]4:
u = σ (BS )(y1 − 1) + y2 (90)
It’s syntax is:
{transport, operands,origin,destination} → {outputs,destination} (91)
�
Definition 35 – Holding Process [7]: A process is a holding process pγg ∈ Pγ iff it holds one or moreoperands during the transportation from one buffer to another. In order to maintain the independenceaxiom and the mutual exclusivity of the system processes (Theorem 3), holding processes are specifiedso as to distinguish between transportation processes that:• Have different operands,
4Note that a “storage process” is merely a transportation process with the same origin and destination.
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• Hold a given operand in a given way, or• Change the state of the operand.
�
Theorem 3 – Mutual Exclusivity of System Processes [7]: A lucid representation of system processesas a domain conceptualization distinguishes between two system processes as modeling primitives withdifferent sets of inputs and outputs. �
Definition 36 – The Negative Transformation Process-Operand Incidence Matrix M−LPµ : A binary
incidence matrix M−LPµ ∈ {0,1}σ (L)×σ (Pµ) whose element M−LP (i, j) = 1 when the transformation system
process pµj ∈ P pulls operand li ∈ L as an input.�
Definition 37 – The Negative Refined Transportation Process-Operand Incidence Matrix M−LPη : A
binary incidence matrix M−LPη ∈ {0,1}σ (L)×σ (Pη ) whose element M−LP (i,ϕ) = 1 when the refined transportation
process pϕ ∈ Pη pulls operand li ∈ L as an input. It is calculated directly from the negative holding process-operand incidence matrix M−LPγ .
M−LPη (i,ϕ) =σ (Pη )∑u=1
M−LPγ (i,g) · δu ∀i ∈ {1, . . .σ (L)}, g ∈ {1, . . . ,σ (Pγ )},u ∈ {1, . . . ,σ (Pη)},ϕ = σ (Pη)(g − 1) +u
(92)
M−LPη (i,ϕ) =σ (BS )∑y1=1
σ (BS )∑y2=1
M−LPγ (i,g) · δy1· δy2
∀y1, y2 ∈ {1, . . . ,σ (BS )},ϕ = σ2(BS )(g − 1) + σ (BS )(y1 − 1) + y2 (93)
M−LPη =σ (Pη )∑u=1
M−LPγ ⊗ eσ (Pη )Tu =
σ (BS )∑y1=1
σ (BS )∑y2=1
M−LPγ ⊗(eσ (BS )y1 ⊗ eσ (BS )
y2
)T(94)
=M−LPγ ⊗1σ (Pη )T =M−LPγ ⊗
(1σ (BS ) ⊗1σ (BS )
)T(95)
�
Definition 38 – The Negative Holding Process-Operand Incidence Matrix M−LPγ : A binary incidence
matrix M−LPγ ∈ {0,1}σ (L)×σ (Pγ ) whose element M−LP (i,g) = 1 when the holding process pg ∈ Pγ pulls operand
li ∈ L as an input.�
Definition 39 – The Positive Transformation Process-Operand Incidence Matrix M+LPµ
: A binary inci-
dence matrix M+LPµ∈ {0,1}σ (L)×σ (Pµ) whose element M+
LP (i, j) = 1 when the transformation system processpµj ∈ P ejects operand li ∈ L as an output.
�
Definition 40 – The Positive Refined Transportation Process-Operand Incidence Matrix M+LPη
: A binary
incidence matrix M+LPη∈ {0,1}σ (L)×σ (Pη ) whose element M+
LP (i,ϕ) = 1 when the refined transportationprocess pϕ ∈ Pη ejects operand li ∈ L as an output. It is calculated directly from the negative holding
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process-operand incidence matrix M+LPγ
.
M+LPη
(i,ϕ) =σ (Pη )∑u=1
M+LPγ
(i,g) · δu ∀i ∈ {1, . . .σ (L)}, g ∈ {1, . . . ,σ (Pγ )},u ∈ {1, . . . ,σ (Pη)},ϕ = σ (Pη)(g − 1) +u
(96)
M+LPη
(i,ϕ) =σ (BS )∑y1=1
σ (BS )∑y2=1
M+LPγ
(i,g) · δy1· δy2
∀y1, y2 ∈ {1, . . . ,σ (BS )},ϕ = σ2(BS )(g − 1) + σ (BS )(y1 − 1) + y2 (97)
M+LPη
=σ (Pη )∑u=1
M+LPγ⊗ e
σ (Pη )Tu =
σ (BS )∑y1=1
σ (BS )∑y2=1
M+LPγ⊗(eσ (BS )y1 ⊗ eσ (BS )
y2
)T(98)
=M+LPγ⊗1σ (Pη )T =M+
LPγ⊗(1σ (BS ) ⊗1σ (BS )
)T(99)
�
Definition 41 – The Positive Holding Process-Operand Incidence Matrix M+LPγ
: A binary incidence
matrix M+LPγ∈ {0,1}σ (L)×σ (Pγ ) whose element M+
LP (i,g) = 1 when the holding process pg ∈ Pγ ejects operandli ∈ L as an output.
�
D. Definitions of Set Operations
Definition 42 – σ() Notation [7]: returns the size of the set. Given a set S with n elements, n = σ (S). �
Definition 43 – Cartesian Product � [127]: Given three sets, A, B, and C,
A�B = {(a,b) ∈ C ∀a ∈ A and b ∈ B} (100)
�
E. Definitions of Boolean Operations
The conventional symbols of ∧, ∨, and ¬ are used to indicate the AND, OR, and NOT operationsrespectively.
Definition 44 –∨
Notation:∨
notation indicates a Boolean OR over multiple binary elements ai .n∨i
ai = a1 ∨ a2 ∨ . . .∨ an (101)
�
Definition 45 – Matrix Boolean Addition ⊕: Given Boolean matrices A,B,C ∈ {0,1}m×n, C = A ⊕ B isequivalent to
C(i, j) = A(i, j)∨B(i, j) ∀i ∈ {1 . . .m}, j ∈ {1 . . .n} (102)
�
Definition 46 – Matrix Boolean Scalar Multiplication (·): Given Boolean matrices A,B,C ∈ {0,1}m×n,C = A ·B is equivalent to
C(i, j) = A(i, j)∧B(i, j) = A(i, j) ·B(i, j) ∀i ∈ {1 . . .m}, j ∈ {1 . . .n} (103)
�
Definition 47 – Matrix Boolean Multiplication � [10], [128]: Given matrices A ∈ {0,1}m×n, B ∈ {0,1}n×p,and C ∈ {0,1}m×p, C = A×B = AB is equivalent to
C(i,k) =n∨i=1
A(i, j)∧B(j,k) =n∨i=1
A(i, j) ·B(j,k) ∀i ∈ {1 . . .m}, k ∈ {1 . . .p} (104)
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�
Definition 48 – Matrix Boolean Subtraction: Given Boolean matrices A,B,C ∈ {0,1}m×n, C = A B isequivalent to
C(i,k) = A(i, j)∧¬B(i, j)) = A(i, j) · ¬B(i, j) ∀i ∈ {1 . . .m}, j ∈ {1 . . .n} (105)
�
F. Matrix Operations
Definition 49 – Kronecker Delta Function δij [129]:
δij ={
1 if i = j0 if i , j
(106)
�
Definition 50 – Hadamard Product [130]: Given matrices A,B,C ∈Rm×n, C = A ·B is equivalent to
C(i, j) = A(i, j) ·B(i, j) ∀i ∈ {1 . . .m}, j ∈ {1 . . .n} (107)
�
Definition 51 – Matrix Product [130]: Given matrices A ∈Rm×n, B ∈Rn×p, and C ∈Rm×p, C = A×B = ABis equivalent to
C(i,k) =n∑i=1
A(i, j) ·B(j,k) ∀i ∈ {1 . . .m}, k ∈ {1 . . .p} (108)
�
Definition 52 – Kronecker Product [131], [132]: Given matrix A ∈Rm×n and B ∈Rp×q, the Kronecker(kron) product denoted by C = A⊗B is given by:
C =
A(1,1)B A(1,2)B . . . A(1,n)BA(2,1)B A(2,2)B . . . A(2,n)B
......
. . ....
A(m,1)B A(m,2)B . . . A(m,n)B
(109)
Alternatively, in scalar notation:
C(p(i − 1) + k,q(j − 1) + l) = a(i, j) · b(k, l) ∀i ∈ {1 . . .m}, j ∈ {1 . . .n}, k ∈ {1 . . .p}, l ∈ {1 . . .q} (110)
�
Definition 53 – Khatri-Rao Product [131], [132]: The Khatri-Rao Product is the “column-wise Kroneckerproduct”. Given matrix A ∈Rm×n and B ∈Rp×n, the Khatri-Rao product denoted by C = A~B is givenby:
C =[A(:,1)⊗B(:,1) A(:,2)⊗B(:,2) . . . A(:,n)⊗B(:,n)
](111)
= [A⊗1p] · [1m ⊗B] (112)
Alternatively, in scalar notation:
C(p(i − 1) + k, j) = a(i, j) · b(k, j) ∀i ∈ {1 . . .m}, j ∈ {1 . . .n}, k ∈ {1 . . .p} (113)
If A and B are column vectors, the Kronecker and Khatri-Rao products are identical. �
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G. Tensor Operations
Definition 54 – Outer Product of Vectors [131], [132]: Given two vectors A1 ∈Rm1 and A2 ∈Rm2 , theirouter product B ∈Rm1×m2 is denoted by
B = A1 ◦A2 = A1AT2 (114)
B(i1, i2) = A1(i1) ·A2(i2) ∀i1 ∈ {1 . . .m1}, i2 ∈ {1 . . .m2} (115)
Given n vectors, A1 ∈ Rm1 ,A2 ∈ Rm2 , . . . ,An ∈ Rmn , their outer product B ∈ Rm1×m2×...×mn is denoted byB = A1 ◦A2 ◦ . . . ◦An where
B(i1, i2, . . . , in) = A1(i1) ·A2(i2) · . . . ·An(in) ∀i1 ∈ {1 . . .m1}, i2 ∈ {1 . . .m2}, . . . , in = {1 . . .mn} (116)
�
Definition 55 – Matricization FM( [131], [132]): Given an nth order tensor A ∈Rp1×p2×...×pn , and orderedsets R = {r1, . . . , rL} and C = {c1, . . . , cM} that are a partition of the n modes N = {1, . . . ,n} (i.e. R∪C =N,R∩C =∅), the matricization function FM() outputs the matrix A ∈RJ×K
A = FM(A,R,C) (117)
A(j,k) =A(i1, i2, . . . , in) ∀i1 ∈ {1, . . . ,p1}, i2 ∈ {1, . . . ,p2}, . . . in ∈ {1, . . . ,pn} (118)
where
j = 1 +L∑l=1
(irl − 1)l−1∏l′=1
irl′
, k = 1 +M∑m=1
(icm − 1)m−1∏m′=1
icm′
, J =∏q∈R
pq K =∏q∈C
pq (119)
For the sake of clarity, FM() is implemented in MATLAB code:
ATensor = rand(4,7,5,3); R = [4 1]; C = [2 3];
function AMatrix=matricize(ATensor,R,C);
P = size(ATensor); J = prod(P(R)); K = prod(P(C));
AMatrix = reshape(permute(ATensor,[R C]),J,K); % Matricize
�
Definition 56 – Tensorization [131], [132]: Given a matrix A ∈RJ×K , the dimensions P = [p1,p2, . . . ,pn]of a target nth order tensor A ∈Rp1×p2×...×pn , and ordered sets R = {r1, . . . , rL} and C = {c1, . . . , cM} that are apartition of the n modes N = {1, . . . ,n} (i.e. R∪C =N,R∩C = ∅), the tensorization function F −1
M () outputsthe nth order tensor A.
A = F −1M (A,P ,R,C) (120)
A(i1, i2, . . . , in) = A(j,k) ∀i1 ∈ {1, . . . ,p1}, i2 ∈ {1, . . . ,p2}, . . . in ∈ {1, . . . ,pn} (121)
where
j = 1 +L∑l=1
(irl − 1)l−1∏l′=1
irl′
, k = 1 +M∑m=1
(icm − 1)m−1∏m′=1
icm′
, J =∏q∈R
pq K =∏q∈C
pq (122)
For the sake of clarity, F −1M () is implemented in MATLAB code:
AMatrix = rand(12,35); P=[4,7,5,3]; R = [4 1]; C = [2 3];
function ATensor=tensorize(AMatrix,P,R,C);
ATensor = ipermute(reshape(AMatrix,[P(R) P(C)]),[R C]); % Tensorize
�
Definition 57 – Vectorization [130]–[132]: Vectorization denoted by vec() or ()V as a shorthand is aspecial case of matricization when the resulting matrix is simply a vector. Formally, given an nth order
SUBMITTED FOR PUBLICATION: (DOI) 27
tensorA ∈Rp1×p2×...×pn and the dimensions P = [p1,p2, . . . ,pn], the vectorization function vec() = ()V outputsthe vector A ∈RJ
A = vec(A) =AV (123)
A(j) =A(i1, i2, . . . , in) ∀i1 ∈ {1, . . . ,p1}, i2 ∈ {1, . . . ,p2}, . . . in ∈ {1, . . . ,pn} (124)
where
j = 1 +n∑l=1
(il − 1)l−1∏l′=1
il′
, J =n∏q=1
pq (125)
�
Definition 58 – Inverse Vectorization [130]–[132]: Inverse vectorization denoted by vec−1() is a specialcase of tensorization when the input matrix is simply a vector. Formally, A ∈ RJ and the dimensionsP = [p1,p2, . . . ,pn] of a target nth order tensor A ∈ Rp1×p2×...×pn , the inverse vectorization function vec−1()outputs the nth order tensor A.
A = vec−1(A,P ) (126)
A(i1, i2, . . . , in) = A(j) ∀i1 ∈ {1, . . . ,p1}, i2 ∈ {1, . . . ,p2}, . . . in ∈ {1, . . . ,pn} (127)
where
j = 1 +n∑l=1
(il − 1)l−1∏l′=1
il′
, J =n∏q=1
pq (128)
Furthermore, the above definition of inverse vectorization can be applied to a qth dimensional slice of atensor. In such a case,
B = vec−1(A,P , r) (129)
B(k1, . . . , kr−1, i1, . . . , in, kr+1, . . . , km) =A(k1, . . . , kr−1, j,kr+1, . . . , km) (130)
where index convention in Equation 128 applies. �
Definition 59 – Matrix and Tensor Transpose: Given a matrix A ∈ Rm1×m2 , its matrix transpose AT ∈Rm2×m1 is equivalent to:
AT (j, i) = A(i, j) ∀i ∈ {1 . . .m1}, j ∈ {1 . . .m2} (131)
In this work, the generalization to tensors is a special case of the definition provided in [133]. Given atensor A ∈Rm1×...×mn , its tensor transpose AT ∈Rmn×...×m1 is equivalent to:
AT (in, . . . , i1) =A(i1, . . . in) ∀i1 ∈ {1 . . .m1}, . . . , in ∈ {1 . . .mn} (132)
�
Definition 60 – N-Mode Matrix Product ×p [131], [132]: The N-mode matrix product is a generalizationof the matrix product. Given a tensor A ∈Rm1×m2×...×mp×...×mn , matrix B ∈Rmp×q, and C ∈Rm1×m2×...×q×...×mn ,the n-mode matrix product denoted by C =A×p B is equivalent to:
C(i1, i2, . . . , ip−1, j, ip+1, . . . , in) =mp∑ip=1
A(i1, i2, . . . , in) ·B(ip, j) (133)
∀i1 ∈ {1, . . . ,m1}, . . . , ip−1 ∈ {1, . . . ,mp−1}, ip+1 ∈ {1, . . . ,mp+1}, . . . , in ∈ {1, . . . ,mn}, j ∈ {1, . . . , q}
�
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Definition 61 – N-Mode Boolean Matrix Product: The N-mode Boolean matrix product is a generaliza-tion of the Boolean matrix product. Given a tensor A ∈ {0,1}m1×m2×...×mp×...×mn , matrix B ∈ {0,1}mp×q, andC ∈ {0,1}m1×m2×...×q×...×mn , the n-mode matrix product denoted by C =A�p B is equivalent to:
C(i1, i2, . . . , ip−1, j, ip+1, . . . , in) =mp∨ip=1
A(i1, i2, . . . , in) ·B(ip, j) (134)
∀i1 ∈ {1, . . . ,m1}, . . . , ip−1 ∈ {1, . . . ,mp−1}, ip+1 ∈ {1, . . . ,mp+1}, . . . , in ∈ {1, . . . ,mn}, j ∈ {1, . . . ,q}
�
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