A Tensor-Based Formulation of Hetero-functional Graph Theory
Amro M. Farid, Dakota Thompson, Prabhat Hegde, Wester Schoonenberg
SSUBMITTED FOR PUBLICATION: (DOI) 1
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 water10) 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. (cid:4)
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, USA. [email protected] a r X i v : . [ c s . A I] J a n UBMITTED FOR PUBLICATION: (DOI) 2
TABLE I
A Classification of Engineering Systems by Function and Operand [3]
Function/Operand LivingOrganisms Matter Energy Information MoneyTransform
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 NationalHighwayTra ffi c 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-layernetworks 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
UBMITTED FOR PUBLICATION: (DOI) 3 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 di ff erences in as precise a manneras possible. . . . [The multi-layer network community] has produced an equally immense explosion of disparateterminology, 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
Legend:
House with EV ChargingHouse without ParkingWater Treatment Facility with EV Chargers Office with EV ChargerWater Storage FacilityPower Plant SubstationIntersectionRoad with Wireless Charging Conventional RoadPower LineWater Pipe Line
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
UBMITTED FOR PUBLICATION: (DOI) 4 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 di ff erences 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 mind that constitute an abstraction 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 ModelAbstraction Model Abstraction Model
Fig. 2. Graphical Representation of Four Ontological Properties As Mapping Between Abstraction and Model: a Soundness, b Completeness, c Lucidity, and d Laconicity [95].
The ontological strength of hetero-functional graph theory comes from the “systems thinking” foun- It 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).
UBMITTED FOR PUBLICATION: (DOI) 5 dations in the model-based systems engineering literature [97]. In e ff ect, 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 di ff erentiationbetween elements related to transformation and transportation. In that regard, it takes great care to not overload 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 di ff erences 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 bu ff ersas 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 o ff ers 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 A S 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 = J S (cid:12) R (1) UBMITTED FOR PUBLICATION: (DOI) 6 where R is set of system resources, P is the set of system processes, J S is the system knowledge base,and (cid:12) is matrix Boolean multiplication (Defn. 47). Definition 2 – System Resource: [4] An asset or object r v ∈ R that is utilized during the execution of aprocess. (cid:4) Definition 3 – System Process [4], [105]:
An activity p ∈ P that transforms a predefined set of inputoperands into a predefined set of outputs. (cid:4) Definition 4 – System Operand: [4] An asset or object l i ∈ L that is operated on or consumed duringthe execution of a process. (cid:4) Definition 5 – System Knowledge Base [9]–[13], [94]:
A binary matrix J S of size σ ( P ) × σ ( R ) whoseelement J S ( w, v ) ∈ { , } is equal to one when action e wv ∈ E S (in the SysML sense) exists as a systemprocess p w ∈ P being executed by a resource r v ∈ R . The σ () notation gives the size of a set. (cid:4) 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 di ff erences 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 bu ff ers (Defn. 29), and H is the set of transportationresources (Defn. 30). Furthermore, the set of bu ff ers B S = M ∪ B (Defn. 31) is introduced for laterdiscussion. Similarly, P = P µ ∪ P ¯ η where P µ is the set of transformation processes (Defn. 32) and P ¯ η is theset of refined transportation processes (Defn. 33). The latter, in turn, is determined from the Cartesianproduct ( (cid:14) ) (Defn. 43) of the set of transportation processes P η (Defn. 34) and the set of holding processes P γ (Defn. 35). P ¯ η = P γ (cid:14) P η (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-ResourceArchitecture package
Model [ ] +Transform Operand() operations
Transformation Resources M operations +Transport Operand()+Hold Operand()
Resources RTransportation 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 P ¯ η = P γ (cid:14) P η [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. UBMITTED FOR PUBLICATION: (DOI) 7
Consequently, subsets of the design equation 1 can be written to emphasize the relationships betweenthe constitutent classes of processes and resources [9]–[13]. P µ = J M (cid:12) M (3) P γ = J γ (cid:12) R (4) P η = J H (cid:12) R (5) P ¯ η = J ¯ H (cid:12) R (6)where J M is the transformation knowledge base, J γ is the holding knowledge base, J H is the transportationknowledge base, and J ¯ H is the refined transportation knowledge base [10], [13], [106]–[109].The transportation knowledge base J H is best understood as a matricized rd -order tensor J H wherethe element J H ( y , y , v ) = 1 when the transportation process p u ∈ P η defined by the origin b s y ∈ B S andthe destination b s y ∈ B S is executed by the resource r v ∈ R . J H = F M ( J H , [2 , , [3]) (7) J H = F − M ( J H , [ σ ( B S ) , σ ( B S ) , σ ( R )] , [2 , , [3]) (8)where F M and F − M are the matricization and tensorization functions (Defns. 55 and 56) respectively. Here, F M () serves to vectorize the dimensions of the origin and destination bu ff ers into the single dimensionof transportation processes.The J H tensor reveals that the transportation knowledge base is closely tied to the classical under-standing of a graph A B S where point elements of form called nodes, herein taken to be the set of bu ff ers B S , 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 . A B S ( y , y ) = σ ( R ) (cid:95) v J H ( y , y , v ) = σ ( R ) (cid:95) v J H ( u, v ) ∀ y , y ∈ { , . . . , σ ( B S ) } , u = σ ( B S )( y −
1) + y , v ∈ { , . . . , σ ( R ) } (9) A B S = J H (cid:12) σ ( R ) = vec − (cid:16) J H (cid:12) σ ( R ) , [ σ ( B S ) , σ ( B S )] (cid:17) T (10) A T VB S = (cid:16) J H (cid:12) σ ( R ) (cid:17) T V = J H (cid:12) σ ( R ) (11)where the (cid:87) notation is the Boolean analogue of the (cid:80) notation (Defn. 44), (cid:12) n is the n-mode Booleanmatrix product (Defn. 61), vec − () is inverse vectorization (Defn. 58) and () V is shorthand for vectorization(Defn. 57). Furthermore, the notation n is used to indicate a ones-vector of length n. The transportationsystem knowledge base J H replaces the edges of the formal graph A B S with an explicit description offunction in the transportation processes P η . The multi-column nature of the transportation knowledgebase J H contains more information than the formal graph A B S and allows potentially many resourcesto execute any given transportation process. Consequently, the OR operation across the rows of J H (orthe third dimension of J H ) is su ffi cient to reconstruct the formal graph A B S . In short, a single columntransportation knowledge base is mathematically equivalent to a vectorized formal graph A B S .Similarly, the refined transportation knowledge base is best understood as a matricized 4 th order tensor J ¯ H where the element J H ( g, y , y , v ) = 1 when the refined transportation process p ϕ ∈ P ¯ η defined by theholding process p γg ∈ P γ , the origin b s y ∈ B S and the destination b s y ∈ B S is executed by the resource r v ∈ R . J ¯ H = F M ( J ¯ H , [3 , , , [4]) (12) J ¯ H = F − M (cid:16) J ¯ H , [ σ ( P γ ) , σ ( B S ) , σ ( B S ) , σ ( R )] , [3 , , , [4] (cid:17) (13)The J ¯ H tensor reveals that the refined transportation knowledge base is closely tied to the classicalunderstanding of a multi-commodity flow network A LB S [110]–[112]. Mathematically, it is a 3 rd -ordertensor whose element A LB S ( i, y , y ) = 1 when operand l i ∈ L is transported from bu ff er b s y to b s y . Again, UBMITTED FOR PUBLICATION: (DOI) 8 the multi-commodity flow network A LB S is purely a description of system form and any statement offunction is entirely implicit . In the special case of a system where the set of operands L maps 1-to-1 theset of holding processes P γ (i.e. i = g ): A LB S ( i, y , y ) = σ ( R ) (cid:95) v J ¯ H ( g, y , y , v ) ∀ g ∈ { , . . . , σ ( P γ ) } , y , y ∈ { , . . . , σ ( B S ) } , v ∈ { , . . . , σ ( R ) } (14)= σ ( R ) (cid:95) v J ¯ H ( ϕ, v ) ∀ ϕ = σ ( B S )( g −
1) + σ ( B S )( y −
1) + y , v ∈ { , . . . , σ ( R ) } (15) A LB S = J ¯ H (cid:12) σ ( R ) = vec − (cid:16) J ¯ H (cid:12) σ ( R ) , (cid:104) σ ( B S ) , σ ( B S ) , σ ( P γ ) (cid:105)(cid:17) T (16) A T VLB S = (cid:16) J ¯ H (cid:12) σ ( R ) (cid:17) T V = J ¯ H (cid:12) σ ( R ) (17)The refined transportation system knowledge base J ¯ H replaces the operands and edges of the multi-commodity flow network A MP with an explicit description of function in the holding processes P γ andtransportation processes P η . The multi-column nature of the refined transportation knowledge base J ¯ H contains more information than the multi-commodity flow network A LB S and allows potentially manyresources to execute any given refined transportation process. Consequently, the OR operation acrossthe rows of J ¯ H (or the fourth dimension of J ¯ H ) is su ffi cient to reconstruct the multi-commodity flownetwork A LB S . 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 ( J M , J γ , J H and J ¯ H ) readily serve to reconstruct the system knowledge base J S . First, the refined transportation knowledgebase is the Khatri-Rao product of the holding and transportation knowledge bases. J ¯ H ( σ ( P η )( g −
1) + u, v ) = J γ ( g, v ) · J H ( u, v ) ∀ g ∈ { , . . . , σ ( P γ ) } , u = σ ( B S )( y −
1) + y , v ∈ { , . . . , σ ( R ) } (18) J ¯ H = J γ (cid:126) J H (19)= (cid:104) J γ ⊗ σ ( P η ) (cid:105) · (cid:104) σ ( P γ ) ⊗ J H (cid:105) (20)where · is the Hadamard (or scalar) product (Defn. 50), (cid:126) is the Khatri-Rao product (Defn. 53) and ⊗ is the Kronecker product (Defn. 52). From this point, the system knowledge base J S is straightforwardlyreconstructed [9]–[13]: J S = (cid:34) J M | J ¯ H (cid:35) (21)Hetero-functional graph theory also di ff erentiates 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 K S of size σ ( P ) × σ ( R ) whoseelement K S ( w, v ) ∈ { , } is equal to one when a constraint eliminates event e wv from the event set. (cid:4) The system constraints matrix is constructed analogously to the system knowledge base [9]–[13]. K S = (cid:34) K M | K ¯ H (cid:35) (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 J S and the system constraints matrix K S have been constructed, thesystem concept A S follows straightforwardly. By Defn. 35, holding processes are distinguished by three criteria: 1.) di ff erent 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. UBMITTED FOR PUBLICATION: (DOI) 9
Definition 7 – System Concept [9]–[13], [94]:
A binary matrix A S of size σ ( P ) × σ ( R ) whose element A S ( w, v ) ∈ { , } is equal to one when action e wv ∈ E S (in the SysML sense) is available as a system process p w ∈ P being executed by a resource r v ∈ R . A S = J S (cid:9) K S = J S · ¯ K S (23)where (cid:9) is Boolean subtraction (Defn. 48) and ¯ K S = N OT ( K S ). (cid:4) Every filled element of the system concept indicates a system capability of the form: “Resource r v doesprocess p w ”. The system constraints matrix limits the availability of capabilities in the system knowledgebase to create the system concept A S . The system capabilities are quantified by the structural degrees offreedom. Definition 8 – Structural Degrees of Freedom [9]–[13], [94]:
The set of independent actions E S thatcompletely defines the available processes in a large flexible engineering system. Their number is givenby: DOF S = σ ( E S ) = σ ( P ) (cid:88) w σ ( R ) (cid:88) v [ J S (cid:9) K S ] ( w, v ) (24)= σ ( P ) (cid:88) w σ ( R ) (cid:88) v A S ( w, v ) (25)= (cid:104) J S , ¯ K S (cid:105) F (26) (cid:4) 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
DOF M and the refined transportation degrees of freedom DOF H are calculated similarly [9]–[12]: DOF M = σ ( P µ ) (cid:88) j σ ( M ) (cid:88) k [ J M (cid:9) K M ] ( j, k ) (27) DOF H = σ ( P ¯ η ) (cid:88) ϕ σ ( R ) (cid:88) v [ J ¯ H (cid:9) K ¯ H ] ( 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 A ρ 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 A ρ of size σ ( R ) σ ( P ) × σ ( R ) σ ( P ) whose element J ρ ( χ , χ ) ∈ { , } is equal to one when string z χ ,χ = e w v e w v ∈Z is available and exists, where index χ i ∈ [1 , . . . , σ ( R ) σ ( P )]. (cid:4) In other words, the hetero-functional adjacency matrix corresponds to a hetero-functional graph G = {E S , Z} with structural degrees of freedom (i.e. capabilities) E S as nodes and feasible sequences Z asedges.Much like the system concept A S , the hetero-functional adjacency matrix A ρ arises from a Booleandi ff erence [13], [108], [117]–[119]. A ρ = J ρ (cid:9) K ρ (29)where J ρ is the system sequence knowledge base and K ρ is the system sequence constraints matrix. Definition 10 – System Sequence Knowledge Base [13], [108], [117]–[119]:
A square binary matrix J ρ ofsize σ ( R ) σ ( P ) × σ ( R ) σ ( P ) whose element J ρ ( χ , χ ) ∈ { , } is equal to one when string z χ ,χ = e w v e w v ∈ Z exists, where index χ i ∈ [1 , . . . , σ ( R ) σ ( P )]. (cid:4) UBMITTED FOR PUBLICATION: (DOI) 10
Definition 11 – System Sequence Constraints Matrix [13], [108], [117]–[119]:
A square binary con-straints matrix K ρ of size σ ( R ) σ ( P ) × σ ( R ) σ ( P ) whose elements K ( χ , χ ) ∈ { , } are equal to one whenstring z χ χ = e w v e w v ∈ Z is eliminated. (cid:4) The definitions of the system sequence knowledge base J ρ and the system sequence constraints matrix K ρ feature a translation of indices from e w v e w v to z χ χ . This fact suggests that these matrices havetheir associated 4 th order tensors J ρ , K ρ and A ρ . J ρ = F M (cid:16) J ρ , [1 , , [3 , (cid:17) (30) K ρ = F M (cid:16) K ρ , [1 , , [3 , (cid:17) (31) A ρ = F M (cid:16) A ρ , [1 , , [3 , (cid:17) (32)Each of these are discussed in turn. J ρ and its tensor-equivalent J ρ create all the potential sequencesof the capabilities in A S . J ρ ( w , v , w , v ) = A S ( w , v ) · A S ( w , v ) ∀ w , w ∈ { . . . σ ( P ) } , v , v ∈ { . . . σ ( R ) } (33) J ρ ( χ , χ ) = A VS ( χ ) · A VS ( χ ) ∀ χ , χ ∈ { . . . σ ( R ) σ ( P ) } (34) J ρ = A VS A V TS (35) J ρ = (cid:104) J S · ¯ K S (cid:105) V (cid:104) J S · ¯ K S (cid:105) V T (36) J ρ = F − M (cid:16) J ρ , [ σ ( P ) , σ ( R ) , σ ( P ) , σ ( R )] , [1 , , [3 , (cid:17) (37)Of these potential sequences of capabilities, the system sequence constraints matrix K ρ serves toeliminate the infeasible pairs. The feasibility arises from five types of constraints:I: P µ P µ . Two transformation processes that follow each other must occur at the same transformationresource. m = m .II: P µ P ¯ η . A refined transportation process that follows a transformation process must have an originequivalent to the transformation resource at which the transformation process was executed. m − u − /σ ( B S ) where / indicates integer division.III: P ¯ η P µ . A refined transportation process that precedes a transformation process must have a destinationequivalent to the transformation resource at which the transformation process was executed. m − u − σ ( B S ) where % indicates the modulus.IV: P ¯ η P ¯ η . A refined transportation process that follows another must have an origin equivalent to thedestination of the other. ( u − σ ( B S ) = ( u − /σ ( B S )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 P w P w is feasible if and only if A P ( w , w ) = 1 where A P 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 K ρ was calcu-lated straightforwardly using for FOR loops to loop over the indices χ and χ 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, ¯ K ρ = N OT ( K ρ ) 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 and the last constraint be satisfied. ¯ K ρ = (cid:16) ¯ K ρI ⊕ ¯ K ρII ⊕ ¯ K ρIII ⊕ ¯ K ρIV (cid:17) · ¯ K ρV (38)where ⊕ is Boolean addition (Defn. 45) and ¯ K ρI , ¯ K ρII , ¯ K ρIII , ¯ K ρIV , ¯ K ρV are the matrix implementationsof the five types of feasibility constraints identified above. Their calculation is most readily achievedthrough their associated 4 th -order tensors. UBMITTED FOR PUBLICATION: (DOI) 11
For the Type I constraint, ¯ K ρI is constructed from a sum of 4 th -order outer products (Defn. 54) ofelementary basis vectors. ¯ K ρI = σ ( P µ ) (cid:88) w =1 σ ( M ) (cid:88) v =1 σ ( P µ ) (cid:88) w =1 v (cid:88) v = v e σ ( P ) w ◦ e σ ( R ) v ◦ e σ ( P ) w ◦ e σ ( R ) v (39)where the e ni notation places the value 1 on the i th element of a vector of length n . ¯ K ρI is calculatedstraightforwardly by matricizing both sides and evaluating the sums.¯ K ρI = σ ( P µ ) (cid:88) w =1 σ ( M ) (cid:88) v =1 σ ( P µ ) (cid:88) w =1 v (cid:88) v = v (cid:18) e σ ( R ) v ⊗ e σ ( P ) w (cid:19) ⊗ (cid:18) e σ ( R ) v ⊗ e σ ( P ) w (cid:19) T (40)¯ K ρI = σ ( M ) (cid:88) v =1 (cid:32) e σ ( R ) v ⊗ (cid:34) σ ( P µ ) σ ( P ¯ η ) (cid:35)(cid:33) (cid:32) e σ ( R ) v ⊗ (cid:34) σ ( P µ ) σ ( P ¯ η ) (cid:35)(cid:33) T (41)Similarly, for the Type II constraint:¯ K ρII = σ ( P µ ) (cid:88) w =1 σ ( M ) (cid:88) v =1 v (cid:88) y = v σ ( R ) (cid:88) v =1 e σ ( P ) w ◦ e σ ( R ) v ◦ σ ( P µ ) X σ ( P ¯ η ) y ◦ e σ ( R ) v (42)Here, the X σ ( P ¯ H ) y vector has a value of 1 wherever a refined transportation process p w originates at thetransformation resource m v . Drawing on the discussion of the 3rd-order tensor J ¯ H in Section II, X σ ( P ¯ H ) y ,itself, is expressed as a vectorized sum of 3 rd -order outer products. X σ ( P ¯ H ) y = σ ( P γ ) (cid:88) g =1 σ ( B S ) (cid:88) y =1 (cid:18) e σ ( P γ ) g ◦ e σ ( B S ) y ◦ e σ ( B S ) y (cid:19) T V = (cid:18) σ ( P γ ) ⊗ e σ ( B S ) y ⊗ σ ( B S ) (cid:19) (43)¯ K ρII is then calculated straightforwardly by matricizing both sides of Eq. 42 and evaluating the sums.¯ K ρII = σ ( P µ ) (cid:88) w =1 σ ( M ) (cid:88) v =1 v (cid:88) y = v σ ( R ) (cid:88) v =1 (cid:18) e σ ( R ) v ⊗ e σ ( P ) w (cid:19) ⊗ e σ ( R ) v ⊗ σ ( P µ ) X σ ( P ¯ H ) y T (44)¯ K ρII = σ ( M ) (cid:88) v =1 (cid:32) e σ ( R ) v ⊗ (cid:34) σ ( P µ ) P ¯ η (cid:35)(cid:33) σ ( R ) ⊗ σ ( P µ ) σ ( P γ ) ⊗ e σ ( B S ) v ⊗ σ ( B S ) T (45)Similarly, for the Type III constraint:¯ K ρIII = v (cid:88) y = v σ ( R ) (cid:88) v =1 σ ( P µ ) (cid:88) w =1 σ ( M ) (cid:88) v =1 σ ( P µ ) X σ ( P ¯ η ) y ◦ e σ ( R ) v ◦ e σ ( P ) w ◦ e σ ( R ) v (46)Here, the X σ ( P ¯ H ) y vector has a value of 1 wherever a refined transportation process p w terminates at thetransformation resource m v . X σ ( P ¯ H ) y , itself, is expressed as a vectorized sum of 3 rd -order outer products. X σ ( P ¯ H ) y = σ ( P γ ) (cid:88) g =1 σ ( B S ) (cid:88) y =1 (cid:18) e σ ( P γ ) g ◦ e σ ( B S ) y ◦ e σ ( B S ) y (cid:19) T V = (cid:18) σ ( P γ ) ⊗ σ ( B S ) ⊗ e σ ( B S ) y (cid:19) (47)¯ K ρIII is then calculated straightforwardly by matricizing both sides of Eq. 46 and evaluating the sums.¯ K ρIII = v (cid:88) y = v σ ( R ) (cid:88) v =1 σ ( P µ ) (cid:88) w =1 σ ( M ) (cid:88) v =1 e σ ( R ) v ⊗ σ ( P µ ) X σ ( P ¯ H ) y ⊗ (cid:18) e σ ( R ) v ⊗ e σ ( P ) w (cid:19) T (48)¯ K ρIII = σ ( M ) (cid:88) v =1 σ ( R ) ⊗ σ ( P µ ) σ ( P γ ) ⊗ σ ( B S ) ⊗ e σ ( B S ) v (cid:32) e σ ( R ) v ⊗ (cid:34) σ ( P µ ) P ¯ η (cid:35)(cid:33) T (49) UBMITTED FOR PUBLICATION: (DOI) 12
Similarly, for the Type IV constraint:¯ K ρIV = σ ( B S ) (cid:88) y =1 σ ( R ) (cid:88) v =1 y (cid:88) y = y σ ( R ) (cid:88) v =1 σ ( P µ ) X σ ( P ¯ H ) y ◦ e σ ( R ) v ◦ σ ( P µ ) X σ ( P ¯ H ) y ◦ e σ ( R ) v (50)¯ K ρIV is then calculated straightforwardly by matricizing both sides of Eq. 50 and evaluating the sums.¯ K ρIV = σ ( B S ) (cid:88) y =1 σ ( R ) (cid:88) v =1 y (cid:88) y = y σ ( R ) (cid:88) v =1 e σ ( R ) v ⊗ σ ( P µ ) X σ ( P ¯ H ) y ⊗ e σ ( R ) v ⊗ σ ( P µ ) X σ ( P ¯ H ) y T (51)¯ K ρIV = σ ( B S ) (cid:88) y =1 σ ( R ) ⊗ σ ( P µ ) σ ( P γ ) ⊗ σ ( B S ) ⊗ e σ ( B S ) y σ ( R ) ⊗ σ ( P µ ) σ ( P γ ) ⊗ e σ ( B S ) y ⊗ σ ( B S ) T (52)The Type V constraint must make use of the functional graph adjacency matrix A P . Consequently, thefourth-order tensor K ρV is calculated first on a scalar basis using the Kronecker delta function δ i (Defn.49) and then is matricized to K ρV .¯ K ρV ( w , v , w , v ) = δ v v · δ v v · A P ( w , w ) ∀ w , w ∈ { , . . . , σ ( P ) } , v , v ∈ { , . . . , σ ( R ) } (53)¯ K ρV = σ ( R ) (cid:88) v =1 σ ( R ) (cid:88) v =1 (cid:18) e σ ( R ) v ⊗ e σ ( R ) Tv (cid:19) ⊗ A P (54)¯ K ρV = (cid:16) σ ( R ) ⊗ σ ( R ) T (cid:17) ⊗ A P (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 z χ χ = e w v e w v ∈ Z of length 2 that completely describe the system language.The number is given by: DOF ρ = σ ( Z ) = σ ( R ) σ ( P ) (cid:88) χ σ ( R ) σ ( P ) (cid:88) χ [ J ρ (cid:9) K ρ ]( χ , χ ) (56)= σ ( R ) σ ( P ) (cid:88) χ σ ( R ) σ ( P ) (cid:88) χ [ A ρ ]( χ , χ ) (57) (cid:4) 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 J VS and A ρ : P S J VS = σ ( E S ) (58) P S A ρ P TS = (cid:101) A ρ (59)where P S 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: DOF ρ = σ ( Z ) = σ ( E S ) (cid:88) ψ σ ( E S ) (cid:88) ψ [ (cid:101) A ρ ]( ψ , ψ ) (60)where ψ ∈ [1 , . . . , σ ( E S )]. UBMITTED FOR PUBLICATION: (DOI) 13
IV. Hetero-functional Incidence Tensor
To complement the concept of a hetero-functional adjacency matrix A ρ and its associated tensor A ρ ,the hetero-functional incidence tensor (cid:102) M ρ describes the structural relationships between the physicalcapabilities (i.e. structural degrees of freedom) E S , the system operands L , and the system bu ff ers B S . (cid:102) M ρ = (cid:102) M + ρ − (cid:102) M − ρ (61) Definition 13 – The Negative 3 rd Order Hetero-functional Incidence Tensor (cid:102) M − ρ : The negative hetero-functional incidence tensor (cid:103) M ρ − ∈ { , } σ ( L ) × σ ( B S ) × σ ( E S ) is a third-order tensor whose element (cid:102) M − ρ ( i, y, ψ ) = 1when the system capability (cid:15) ψ ∈ E S pulls operand l i ∈ L from bu ff er b s y ∈ B S . (cid:4) Definition 14 – The Positive 3 rd Order Hetero-functional Incidence Tensor (cid:102) M + ρ : The positive hetero-functional incidence tensor (cid:102) M + ρ ∈ { , } σ ( L ) × σ ( B S ) × σ ( E S ) is a third-order tensor whose element (cid:102) M + ρ ( i, y, ψ ) = 1when the system capability (cid:15) ψ ∈ E S injects operand l i ∈ L into bu ff er b s y ∈ B S . (cid:4) 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 matrix M − LP ∈ { , } σ ( L ) × σ ( P ) whose element M − LP ( i, w ) = 1 when the system process p w ∈ P pulls operand l i ∈ L asan input. It is further decomposed into the negative transformation process-operand incidence matrix M − LP µ (Defn. 36) and the negative refined transformation process-operand incidence matrix M − LP ¯ η (Defn.37) which by definition is in turn calculated from the negative holding process-operand incidence matrix M − LP γ (Defn. 38). M − LP = (cid:104) M − LP µ M − LP ¯ η (cid:105) = (cid:104) M − LP µ M − LP γ ⊗ σ ( P η ) T (cid:105) (62) (cid:4) Definition 16 – The Positive Process-Operand Incidence Matrix M + LP : A binary incidence matrix M + LP ∈{ , } σ ( L ) × σ ( P ) whose element M + LP ( i, w ) = 1 when the system process p w ∈ P injects operand l i ∈ L as anoutput. It is further decomposed into the positive transformation process-operand incidence matrix M + LP µ (Defn. 39) and the positive refined transformation process-operand incidence matrix M + LP ¯ η (Defn. 40)which, by definition, is, in turn, calculated from the positive holding process-operand incidence matrix M + LP γ (Defn. 41) M + LP = (cid:104) M + LP µ M + LP ¯ η (cid:105) = (cid:104) M + LP µ M + LP γ ⊗ σ ( P η ) T (cid:105) (63) (cid:4) With the definitions of these incidence matrices in place, the calculation of the negative and positivehetero-functional incidence tensors (cid:102) M − ρ and (cid:102) M + ρ follows straightforwardly as a third-order outer product.For (cid:102) M − ρ : (cid:102) M − ρ = σ ( L ) (cid:88) i =1 σ ( B S ) (cid:88) y =1 e σ ( L ) i ◦ e σ ( B S ) y ◦ P S (cid:18)(cid:16) X − iy (cid:17) V (cid:19) (64)where X − iy = M − TLP µ e σ ( L ) i e σ ( M ) Ty | M − TLP γ e σ ( L ) i ⊗ (cid:18) e σ ( B S ) y ⊗ σ ( B S ) (cid:19) ⊗ σ ( R ) T (65)The X − iy matrix is equivalent in size to the system concept A S . It has a value of one in all elements wherethe associated process both withdraws input operand l i and originates at the bu ff er b s y . Consequently,when X − iy is vectorized and then projected with P S , the result is a vector with a value of one only wherethe associated system capabilities meet these criteria. UBMITTED FOR PUBLICATION: (DOI) 14
For (cid:102) M + ρ : (cid:102) M + ρ = σ ( L ) (cid:88) i =1 σ ( B S ) (cid:88) y =1 e σ ( L ) i ◦ e σ ( B S ) y ◦ P S (cid:18)(cid:16) X + iy (cid:17) V (cid:19) (66)where X + iy = M + TLP µ e σ ( L ) i e σ ( M ) Ty | M + TLP γ e σ ( L ) i ⊗ (cid:18) σ ( B S ) ⊗ e σ ( B S ) y (cid:19) ⊗ σ ( R ) T (67)The X + iy matrix is equivalent in size to the system concept A S . It also has a value of one in all elementswhere the associated process both injects output operand l i and terminates at the bu ff er b s y . Consequently,when X + iy is vectorized and then projected with P S , the result is a vector with a value of one only wherethe associated system capabilities meet these criteria.It is important to note that the definitions of the 3 rd order hetero-functional incidence tensors (cid:102) M − ρ ,and (cid:102) M + ρ are provided in projected form as indicated by the presence of the projection operator P S inEquations 64 and 66 respectively. It is often useful to use the un-projected form of these tensors. M − ρ = σ ( L ) (cid:88) i =1 σ ( B S ) (cid:88) y =1 e σ ( L ) i ◦ e σ ( B S ) y ◦ (cid:16) X − iy (cid:17) V (68) M + ρ = σ ( L ) (cid:88) i =1 σ ( B S ) (cid:88) y =1 e σ ( L ) i ◦ e σ ( B S ) y ◦ (cid:16) X + iy (cid:17) V (69)The third dimension of these unprojected 3 rd order hetero-functional incidence tensors can then besplit into two dimensions to create 4 th order hetero-functional incidence tensors. M + P R = vec − (cid:16) M + ρ , [ σ ( P ) , σ ( R )] , (cid:17) (70) M − P R = vec − (cid:16) M − ρ , [ σ ( P ) , σ ( R )] , (cid:17) (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 bu ff ers B S from which these are operands are sent and the system bu ff ers B S towhich these operands are received. They are used in the following section as part of the discussion onlayers. M P R = M + P R − M − P R (72)
Definition 17 – The Negative 4 th Order Hetero-functional Incidence Tensor M − P R : The negative 4 th Order hetero-functional incidence tensor M − P R ∈ { , } σ ( L ) × σ ( B S ) × σ ( P ) × σ ( R ) has element M − P R ( i, y, w, v ) = 1when the system process p w ∈ P realized by resource r v ∈ R pulls operand l i ∈ L from bu ff er b s y ∈ B S . (cid:4) Definition 18 – The Positive 4 th Order Hetero-functional Incidence Tensor M − P R : The positive 4 th Order hetero-functional incidence tensor M − P R ∈ { , } σ ( L ) × σ ( B S ) × σ ( P ) × σ ( R ) has element M − P R ( i, y, w, v ) = 1when the system process p w ∈ P realized by resource r v ∈ R injects operand l i ∈ L into bu ff er b s y ∈ B S . (cid:4) Returning back to the hetero-functional incidence tensor (cid:101) M ρ , it and and its positive and negativecomponents (cid:101) M + ρ , (cid:101) M − ρ , can also be easily matricized. M ρ = F M (cid:16) M ρ , [1 , , [3] (cid:17) (73) M − ρ = F M (cid:16) M − ρ , [1 , , [3] (cid:17) (74) M + ρ = F M (cid:16) M + ρ , [1 , , [3] (cid:17) (75) UBMITTED FOR PUBLICATION: (DOI) 15
The resulting matrices have a size of σ ( L ) σ ( B S ) × σ ( E S ) which have a corresponding physical intuition.Each bu ff er b s y has σ ( L ) copies to reflect a place (i.e. bin) for each operand at that bu ff er. Each of theseplaces then forms a bipartite graph with the system’s physical capabilities. Consequently, and as expected,the hetero-functional adajacency matrix A ρ can be calculated as a matrix product of the positive andnegative hetero-functional incidence matrices M + ρ and M + ρ . A ρ = 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 Graphs
Definition 19 – Layer:
A layer G λ = {E Sλ , Z Sλ } of a hetero-functional graph G = {E S , Z S } is a subset ofa hetero-functional graph, G λ ⊆ 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. (cid:4) 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 Λ λ of size σ ( P ) × σ ( R ) whose element Λ λ ( w, v ) = 1when the capability e wv ⊂ E Sλ . (cid:4) From this definition, the calculation of a hetero-functional graph layer follows straightforwardly. First,a layer projection operator P λ is calculated: P λ Λ Vλ = σ ( E S ) (77)Next, the negative and positive hetero-functional incidence tensors (cid:102) M − ρλ and (cid:102) M + ρλ for a given layer λ arecalculated straightforwardly. (cid:102) M − ρλ = σ ( L ) (cid:88) i =1 σ ( B S ) (cid:88) y =1 e σ ( L ) i ◦ e σ ( B S ) y ◦ P λ (cid:18)(cid:16) X − iy (cid:17) V (cid:19) = (cid:102) M − ρ (cid:12) P λ (78) (cid:102) M + ρλ = σ ( L ) (cid:88) i =1 σ ( B S ) (cid:88) y =1 e σ ( L ) i ◦ e σ ( B S ) y ◦ P S (cid:18)(cid:16) X + iy (cid:17) V (cid:19) = (cid:102) M + ρ (cid:12) P λ (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 (cid:101) A ρλ follows straightforwardly. (cid:101) M + ρλ = F M ( (cid:102) M + ρλ , [1 , , [3]) (80) (cid:101) M − ρλ = F M ( (cid:102) M − ρλ , [1 , , [3]) (81) (cid:101) A ρλ = (cid:101) M + Tρλ (cid:101) M − ρλ (82) UBMITTED FOR PUBLICATION: (DOI) 16
This approach of separating a hetero-functional graph into its constituent layers is quite generic becausethe layer selector matrix Λ λ 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 (Sequence-dependent) Degrees of Freedom of Trimetrica Legend:
Degree of Freedom w/ operand “Electric Power at 132kV”Sequence-dependent Degree of Freedom
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 TopologyElectrified PotableWater TopologyElectrified PowerTopology Charging TopologyTopologyTransportation 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 L λ ⊆ L . (cid:4) 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 2 σ ( L ) possible layers. For completeness, an index λ D ∈ { , . . . , σ ( L ) } is used todenote a given layer. In reality, however, the vast majority of physical systems exhibit far fewer than2 σ ( 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) f λ back to the λ D index. f λ : λ → λ D (83) UBMITTED FOR PUBLICATION: (DOI) 17
The utility of the λ d index (stated as a base 10 number) becomes apparent when it is converted intoa binary (base 2) number λ v ∈ { , } σ ( L ) which may be used equivalently as a binary vector of the samelength. λ v = bin ( λ D ) (84)The resulting binary vector λ v has the useful property that λ v ( l i ) = 1, i ff operand l i ∈ L λ . Consequently,a given value of λ v serves to select from L the operands that pertain to layer λ . The associated layerselector matrix follows straightforwardly: Λ λ ( w, v ) = (cid:40) λ v = M − LP (: , w ) ∀ r v ∈ R Λ above is e ff ectively a third order tensor whosevalue Λ ( λ, w, v ) = 1 when the capability e wv 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 2 σ ( 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 L λ ⊆ L . (cid:4) Λ λ ( w, v ) = (cid:40) λ v = M + LP (: , w ) ∀ r v ∈ R ff erential, algebraic, or di ff erential-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 di ff erent places in an electrical system have the sameconstitutive (Ohm’s) law, but have di ff erent transportation processes because their origin and destinationsare di ff erent. 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. (cid:4)
In such a case, the layer selector matrix Λ λ straightforwardly maps capabilities to their layer and dynamicdevice model interchangeably. A su ffi cient number of layers need to be created to account for all of thedi ff erent 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 ( A MLN ) defined by De Dominico et. al. [14] is chosen tofacilitate the discussion. This fourth order tensor has elements A MLN ( α , α , β , β ) where the indices α , α denote “vertices” and β , β 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 A MLN and the hetero-functional
UBMITTED FOR PUBLICATION: (DOI) 18 adjacency tensor A ρ 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 A MLN ( α , α , β , β ) = A B S l ( y , y , i , i ) where the multi-layernetwork’s vertices are equated to the bu ff ers B S and the layers are equated to the operands L . Thisinterpretation would well describe the departure of an operand l i from bu ff er b sy and arriving as l i at b sy . The equivalence of vertices to bu ff ers is e ff ectively 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 A B S l is neither lucid nor complete with respect to the systemprocesses P (as an abstraction). (cid:4) Proof 1:
By contradiction. Assume that A B S l is both lucid and complete network model with respect tosystem processes P . Consider an operand l that departs b s , undergoes process p , and arrives as l at b s . Now consider the same operand l that departs b s , undergoes process p , and arrives as l at b s .Both of these scenarios would be denoted by A B S l (1 , , ,
1) = 1. Consequently, this modeling element isoverloaded and as such violates the ontological property of lucidity. Furthermore, because A B S l makesno mention of the concept of system processes, then it violates the completeness property as well. (cid:4) 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 interpretation A B S l of a multi-layer network is inadequate even for this very simple counter-example . Rs L Cs(t) +- Rss(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 A MLN ( α , α , β , β ) = A B S l ( y , y , w , w )where the multi-layer network’s vertices are equated to the bu ff ers B S and the layers are equated to theprocesses P . This interpretation would well describe the execution of a process p w that is realized bybu ff er b sy followed by a process p w that is realized by bu ff er b sy . The equivalence of layers to processeswarrants further attention as well. Theorem 2:
The mathematical model A B S l is neither lucid nor complete with respect to the system’stransportation resources H (as an abstraction). (cid:4) Although 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 A B S l is both a lucid and complete network model with respect tosystem’s transportation resources H . Consider transportation process p between a bu ff er b s and a distinctbu ff er b s . If such a transportation process were realized by any bu ff er b s ∈ B S , then by definition it wouldno longer be a bu ff er but rather a transportation resource. Consequently, A B S l is not complete with respectthe system’s transportation resources H . Now consider a process p that is realized by bu ff er b s followedby a process p that is realized by a distinct bu ff er b s . This is denoted by A B S l (1 , , ,
2) = 1. Given thedistinctness of b s and b s , a transportation process must have happened in between p and p althoughit is not explicitly stated by the mathematical statement A B S l (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. (cid:4)
It is worth noting that the first multi-layer network interpretation A B S l can be derived directly fromthe positive and negative hetero-functional incidence matrices. A B S l = F − M (cid:16) M − Tρ M + ρ , [ σ ( B S ) , σ ( B S ) , σ ( L ) , σ ( L )] , [1 , , [2 , (cid:17) (87)When M − Tρ and M + ρ are multiplied so that the capabilities E s are the inner dimension, the result isan adjacency matrix that when tensorized becomes A B S l . In e ff ect, A B S l (in matricized form) is thedual adjacency matrix [123] of the hetero-functional adjacency matrix A ρ . 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 A B S l ( y , y , i , i ). 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 bu ff ers and their functional feasibilityin terms of intermediate operands remains intact. In other words, given the sequence of capabilities e w v e w v , one can immediately deduce the exchanged operands in L λ ⊆ L and the intermediate bu ff er b s . 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 bu ff er, onechecks if either or both of the resources are bu ff ers. If not, then two transportation processes followedone another and the intermediate bu ff er 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 Descriptors
In light of the commonalities and di ff erences between hetero-functional graphs and (formal) multilayernetworks, 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 di ff erent 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 Coe ffi cients: Clustering coe ffi cients 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 di ff erently. Fagiolo discussed this taxonomy andaccompanying clustering coe ffi cients [125]. These clustering coe ffi cients 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 coe ffi cients within and across layers for di ff erent types of systems. When investigating asystem, the clustering coe ffi cient 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 e ffi ciency.
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 di ff erence in definition between the conventional graph theoretic application and the hetero-functional graph theoretic application is the result of the di ff erence 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 di ff erences between thesemathematical models in as precise a manner as possible. UBMITTED FOR PUBLICATION: (DOI) 21
Appendix
A. Ontological Science Definitions
The formal definitions of soundness, completeness, lucidity, and laconicity rely on “Ullman’s Triangle”in Figure 6. a b s t r ac t s r e p r e s e n t s r e p r e s e n t s instantiation a b s t r ac t s refers toDomain Conceptualization: 𝒞 Abstraction: 𝒜 Physical Object/SystemModel: ℳ Real Domain: 𝒟 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 i ff 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.) (cid:4) Definition 25 – Completeness [96]:
A language L is complete w.r.t. a domain conceptualization C i ff every 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.) (cid:4) Definition 26 – Lucidity [96]:
A language L is lucid w.r.t. a domain conceptualization C i ff 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.) (cid:4) Definition 27 – Laconicity [96]:
A language L is laconic w.r.t. a domain conceptualization C i ff 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.) (cid:4) 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. p i ∈ P . • The i th elementary basis vector of size n is denoted by e ni . • A matrix of ones of size m × n is denoted by m × n . • A matrix of zeros of size m × n is denoted by m × n . • With the exception of elementary basis vectors, all vectors and matrices are indicated with a capitalletter. e.g. J H . • All tensors are indicated with capital letters in calligraphic script. e.g. J H . UBMITTED FOR PUBLICATION: (DOI) 22 • All elements in vectors, matrices, and tensors are indicated with indices within parentheses. e.g. J S ( w, v ). • A(:,i) denotes the i th column of A or equivalently the i th mode-1 fiber. The : indicates all elementsof the vector. • A(i,:) denotes the i th row of A or equivalently the i th 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 useof Einstein’s (shorthand) tensor notation where the sigma-notation (cid:80) is eliminated.
C. Hetero-functional Graph Theory Definitions
Definition 28 – Transformation Resource [7]:
A resource r ∈ R is a transformation resource m ∈ M i ff it is capable of one or more transformation processes on one or more operands and it exists at a uniquelocation in space. (cid:4) Definition 29 – Independent Bu ff er [7]: A resource r ∈ R is an independent bu ff er b ∈ B i ff 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. (cid:4) Definition 30 – Transportation Resource [7]:
A resource r ∈ R is a transportation resource h ∈ H i ff itis capable of transporting one or more operands between an origin and a distinct destination, withouttransforming these operands. (cid:4) Definition 31 – Bu ff er [7]: A resource r ∈ R is a bu ff er b s ∈ B S i ff it is capable of storing one or moreoperands at a unique location in space. B S = M ∪ B . (cid:4) Definition 32 – Transformation Process [7]:
A process is a transformation process p µj ∈ P µ i ff 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) (cid:4) Definition 33 – Refined Transportation Process [7]:
A process is a refined transportation process p ¯ ηϕ ∈ P ¯ η i ff it is capable of transporting one or more operands between an origin bu ff er b sy ∈ B S to a destinationbu ff er b sy ∈ B S while it is realizing holding process p γg ∈ P γ . It’s syntax is: { transport, operands, origin, destination,while transitive verb } →{ outputs, destination } (89) (cid:4) Definition 34 – Transportation Process [7]:
A process is a transportation process p ηu ∈ P η i ff it is capableof transporting one or more operands between an origin bu ff er b sy ∈ B S to a destination bu ff er b sy ∈ B S according to the following convention of indices [9]–[13] : u = σ ( B S )( y −
1) + y (90)It’s syntax is: { transport, operands,origin,destination } → { outputs,destination } (91) (cid:4) Definition 35 – Holding Process [7]:
A process is a holding process p γg ∈ P γ i ff it holds one or moreoperands during the transportation from one bu ff er 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 di ff erent operands, Note that a “storage process” is merely a transportation process with the same origin and destination.
UBMITTED FOR PUBLICATION: (DOI) 23 • Hold a given operand in a given way , or • Change the state of the operand. (cid:4)
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 withdi ff erent sets of inputs and outputs. (cid:4) Definition 36 – The Negative Transformation Process-Operand Incidence Matrix M − LP µ : A binaryincidence matrix M − LP µ ∈ { , } σ ( L ) × σ ( P µ ) whose element M − LP ( i, j ) = 1 when the transformation systemprocess p µ j ∈ P pulls operand l i ∈ L as an input. (cid:4) Definition 37 – The Negative Refined Transportation Process-Operand Incidence Matrix M − LP ¯ η : Abinary incidence matrix M − LP ¯ η ∈ { , } σ ( L ) × σ ( P ¯ η ) whose element M − LP ( i, ϕ ) = 1 when the refined transportationprocess p ϕ ∈ P ¯ η pulls operand l i ∈ L as an input. It is calculated directly from the negative holding process-operand incidence matrix M − LP γ . M − LP ¯ η ( i, ϕ ) = σ ( P η ) (cid:88) u =1 M − LP γ ( i, g ) · δ u ∀ i ∈ { , . . . σ ( L ) } , g ∈ { , . . . , σ ( P γ ) } , u ∈ { , . . . , σ ( P η ) } , ϕ = σ ( P η )( g −
1) + u (92) M − LP ¯ η ( i, ϕ ) = σ ( B S ) (cid:88) y =1 σ ( B S ) (cid:88) y =1 M − LP γ ( i, g ) · δ y · δ y ∀ y , y ∈ { , . . . , σ ( B S ) } , ϕ = σ ( B S )( g −
1) + σ ( B S )( y −
1) + y (93) M − LP ¯ η = σ ( P η ) (cid:88) u =1 M − LP γ ⊗ e σ ( P η ) Tu = σ ( B S ) (cid:88) y =1 σ ( B S ) (cid:88) y =1 M − LP γ ⊗ (cid:18) e σ ( B S ) y ⊗ e σ ( B S ) y (cid:19) T (94)= M − LP γ ⊗ σ ( P η ) T = M − LP γ ⊗ (cid:16) σ ( B S ) ⊗ σ ( B S ) (cid:17) T (95) (cid:4) Definition 38 – The Negative Holding Process-Operand Incidence Matrix M − LP γ : A binary incidencematrix M − LP γ ∈ { , } σ ( L ) × σ ( P γ ) whose element M − LP ( i, g ) = 1 when the holding process p g ∈ P γ pulls operand l i ∈ L as an input. (cid:4) Definition 39 – The Positive Transformation Process-Operand Incidence Matrix M + LP µ : A binary inci-dence matrix M + LP µ ∈ { , } σ ( L ) × σ ( P µ ) whose element M + LP ( i, j ) = 1 when the transformation system process p µ j ∈ P ejects operand l i ∈ L as an output. (cid:4) Definition 40 – The Positive Refined Transportation Process-Operand Incidence Matrix M + LP ¯ η : A binaryincidence matrix M + LP ¯ η ∈ { , } σ ( L ) × σ ( P ¯ η ) whose element M + LP ( i, ϕ ) = 1 when the refined transportationprocess p ϕ ∈ P ¯ η ejects operand l i ∈ L as an output. It is calculated directly from the negative holding UBMITTED FOR PUBLICATION: (DOI) 24 process-operand incidence matrix M + LP γ . M + LP ¯ η ( i, ϕ ) = σ ( P η ) (cid:88) u =1 M + LP γ ( i, g ) · δ u ∀ i ∈ { , . . . σ ( L ) } , g ∈ { , . . . , σ ( P γ ) } , u ∈ { , . . . , σ ( P η ) } , ϕ = σ ( P η )( g −
1) + u (96) M + LP ¯ η ( i, ϕ ) = σ ( B S ) (cid:88) y =1 σ ( B S ) (cid:88) y =1 M + LP γ ( i, g ) · δ y · δ y ∀ y , y ∈ { , . . . , σ ( B S ) } , ϕ = σ ( B S )( g −
1) + σ ( B S )( y −
1) + y (97) M + LP ¯ η = σ ( P η ) (cid:88) u =1 M + LP γ ⊗ e σ ( P η ) Tu = σ ( B S ) (cid:88) y =1 σ ( B S ) (cid:88) y =1 M + LP γ ⊗ (cid:18) e σ ( B S ) y ⊗ e σ ( B S ) y (cid:19) T (98)= M + LP γ ⊗ σ ( P η ) T = M + LP γ ⊗ (cid:16) σ ( B S ) ⊗ σ ( B S ) (cid:17) T (99) (cid:4) Definition 41 – The Positive Holding Process-Operand Incidence Matrix M + LP γ : A binary incidencematrix M + LP γ ∈ { , } σ ( L ) × σ ( P γ ) whose element M + LP ( i, g ) = 1 when the holding process p g ∈ P γ ejects operand l i ∈ L as an output. (cid:4) 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 ). (cid:4) Definition 43 – Cartesian Product (cid:14) [127]:
Given three sets, A, B, and C, A (cid:14) B = { ( a, b ) ∈ C ∀ a ∈ A and b ∈ B } (100) (cid:4) E. Definitions of Boolean Operations
The conventional symbols of ∧ , ∨ , and ¬ are used to indicate the AND, OR, and NOT operationsrespectively. Definition 44 – (cid:87)
Notation: (cid:87) notation indicates a Boolean OR over multiple binary elements a i . n (cid:95) i a i = a ∨ a ∨ . . . ∨ a n (101) (cid:4) Definition 45 – Matrix Boolean Addition ⊕ : Given Boolean matrices
A, B, C ∈ { , } m × n , C = A ⊕ B isequivalent to C ( i, j ) = A ( i, j ) ∨ B ( i, j ) ∀ i ∈ { . . . m } , j ∈ { . . . n } (102) (cid:4) Definition 46 – Matrix Boolean Scalar Multiplication ( · ): Given Boolean matrices
A, B, C ∈ { , } m × n , C = A · B is equivalent to C ( i, j ) = A ( i, j ) ∧ B ( i, j ) = A ( i, j ) · B ( i, j ) ∀ i ∈ { . . . m } , j ∈ { . . . n } (103) (cid:4) Definition 47 – Matrix Boolean Multiplication (cid:12) [10], [128]:
Given matrices A ∈ { , } m × n , B ∈ { , } n × p ,and C ∈ { , } m × p , C = A × B = AB is equivalent to C ( i, k ) = n (cid:95) i =1 A ( i, j ) ∧ B ( j, k ) = n (cid:95) i =1 A ( i, j ) · B ( j, k ) ∀ i ∈ { . . . m } , k ∈ { . . . p } (104) UBMITTED FOR PUBLICATION: (DOI) 25 (cid:4)
Definition 48 – Matrix Boolean Subtraction:
Given Boolean matrices
A, B, C ∈ { , } m × n , C = A (cid:9) B isequivalent to C ( i, k ) = A ( i, j ) ∧ ¬ B ( i, j )) = A ( i, j ) · ¬ B ( i, j ) ∀ i ∈ { . . . m } , j ∈ { . . . n } (105) (cid:4) F. Matrix Operations
Definition 49 – Kronecker Delta Function δ ij [129]: δ ij = (cid:40) i = j i (cid:44) j (106) (cid:4) Definition 50 – Hadamard Product [130]:
Given matrices
A, B, C ∈ R m × n , C = A · B is equivalent to C ( i, j ) = A ( i, j ) · B ( i, j ) ∀ i ∈ { . . . m } , j ∈ { . . . n } (107) (cid:4) Definition 51 – Matrix Product [130]:
Given matrices A ∈ R m × n , B ∈ R n × p , and C ∈ R m × p , C = A × B = AB is equivalent to C ( i, k ) = n (cid:88) i =1 A ( i, j ) · B ( j, k ) ∀ i ∈ { . . . m } , k ∈ { . . . p } (108) (cid:4) Definition 52 – Kronecker Product [131], [132]:
Given matrix A ∈ R m × n and B ∈ R p × q , the Kronecker(kron) product denoted by C = A ⊗ B is given by: C = A (1 , B A (1 , B . . . A (1 , n ) BA (2 , B A (2 , B . . . A (2 , n ) B... ... . . . ...A ( m, B A ( m, 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 ∈ { . . . m } , j ∈ { . . . n } , k ∈ { . . . p } , l ∈ { . . . q } (110) (cid:4) Definition 53 – Khatri-Rao Product [131], [132]:
The Khatri-Rao Product is the “column-wise Kroneckerproduct”. Given matrix A ∈ R m × n and B ∈ R p × n , the Khatri-Rao product denoted by C = A (cid:126) B is givenby: C = (cid:104) A (: , ⊗ B (: , A (: , ⊗ B (: , . . . A (: , n ) ⊗ B (: , n ) (cid:105) (111)= [ A ⊗ p ] · [ m ⊗ B ] (112)Alternatively, in scalar notation: C ( p ( i −
1) + k, j ) = a ( i, j ) · b ( k, j ) ∀ i ∈ { . . . m } , j ∈ { . . . n } , k ∈ { . . . p } (113)If A and B are column vectors, the Kronecker and Khatri-Rao products are identical. (cid:4) UBMITTED FOR PUBLICATION: (DOI) 26
G. Tensor Operations
Definition 54 – Outer Product of Vectors [131], [132]:
Given two vectors A ∈ R m and A ∈ R m , theirouter product B ∈ R m × m is denoted by B = A ◦ A = A A T (114) B ( i , i ) = A ( i ) · A ( i ) ∀ i ∈ { . . . m } , i ∈ { . . . m } (115)Given n vectors, A ∈ R m , A ∈ R m , . . . , A n ∈ R m n , their outer product B ∈ R m × m × ... × m n is denoted by B = A ◦ A ◦ . . . ◦ A n where B ( i , i , . . . , i n ) = A ( i ) · A ( i ) · . . . · A n ( i n ) ∀ i ∈ { . . . m } , i ∈ { . . . m } , . . . , i n = { . . . m n } (116) (cid:4) Definition 55 – Matricization F M ( [131], [132]): Given an n th order tensor A ∈ R p × p × ... × p n , and orderedsets R = { r , . . . , r L } and C = { c , . . . , c M } that are a partition of the n modes N = { , . . . , n } (i.e. R ∪ C = N , R ∩ C = ∅ ), the matricization function F M () outputs the matrix A ∈ R J × K A = F M ( A , R, C ) (117) A ( j, k ) = A ( i , i , . . . , i n ) ∀ i ∈ { , . . . , p } , i ∈ { , . . . , p } , . . . i n ∈ { , . . . , p n } (118)where j = 1 + L (cid:88) l =1 ( i r l − l − (cid:89) l (cid:48) =1 i r l (cid:48) , k = 1 + M (cid:88) m =1 ( i c m − m − (cid:89) m (cid:48) =1 i c m (cid:48) , J = (cid:89) q ∈ R p q K = (cid:89) q ∈ C p q (119)For the sake of clarity, F M () 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 (cid:4)
Definition 56 – Tensorization [131], [132]:
Given a matrix A ∈ R J × K , the dimensions P = [ p , p , . . . , p n ]of a target n th order tensor A ∈ R p × p × ... × p n , and ordered sets R = { r , . . . , r L } and C = { c , . . . , c M } that are apartition of the n modes N = { , . . . , n } (i.e. R ∪ C = N , R ∩ C = ∅ ), the tensorization function F − M () outputsthe n th order tensor A . A = F − M ( A, P , R, C ) (120) A ( i , i , . . . , i n ) = A ( j, k ) ∀ i ∈ { , . . . , p } , i ∈ { , . . . , p } , . . . i n ∈ { , . . . , p n } (121)where j = 1 + L (cid:88) l =1 ( i r l − l − (cid:89) l (cid:48) =1 i r l (cid:48) , k = 1 + M (cid:88) m =1 ( i c m − m − (cid:89) m (cid:48) =1 i c m (cid:48) , J = (cid:89) q ∈ R p q K = (cid:89) q ∈ C p q (122)For the sake of clarity, F − M () 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 (cid:4)
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 n th order UBMITTED FOR PUBLICATION: (DOI) 27 tensor
A ∈ R p × p × ... × p n and the dimensions P = [ p , p , . . . , p n ], the vectorization function vec () = () V outputsthe vector A ∈ R J A = vec ( A ) = A V (123) A ( j ) = A ( i , i , . . . , i n ) ∀ i ∈ { , . . . , p } , i ∈ { , . . . , p } , . . . i n ∈ { , . . . , p n } (124)where j = 1 + n (cid:88) l =1 ( i l − l − (cid:89) l (cid:48) =1 i l (cid:48) , J = n (cid:89) q =1 p q (125) (cid:4) Definition 58 – Inverse Vectorization [130]–[132]:
Inverse vectorization denoted by vec − () is a specialcase of tensorization when the input matrix is simply a vector. Formally, A ∈ R J and the dimensions P = [ p , p , . . . , p n ] of a target n th order tensor A ∈ R p × p × ... × p n , the inverse vectorization function vec − ()outputs the n th order tensor A . A = vec − ( A, P ) (126) A ( i , i , . . . , i n ) = A ( j ) ∀ i ∈ { , . . . , p } , i ∈ { , . . . , p } , . . . i n ∈ { , . . . , p n } (127)where j = 1 + n (cid:88) l =1 ( i l − l − (cid:89) l (cid:48) =1 i l (cid:48) , J = n (cid:89) q =1 p q (128)Furthermore, the above definition of inverse vectorization can be applied to a q th dimensional slice of atensor. In such a case, B = vec − ( A, P , r ) (129) B ( k , . . . , k r − , i , . . . , i n , k r +1 , . . . , k m ) = A ( k , . . . , k r − , j, k r +1 , . . . , k m ) (130)where index convention in Equation 128 applies. (cid:4) Definition 59 – Matrix and Tensor Transpose:
Given a matrix A ∈ R m × m , its matrix transpose A T ∈ R m × m is equivalent to: A T ( j, i ) = A ( i, j ) ∀ i ∈ { . . . m } , j ∈ { . . . m } (131)In this work, the generalization to tensors is a special case of the definition provided in [133]. Given atensor A ∈ R m × ... × m n , its tensor transpose A T ∈ R m n × ... × m is equivalent to: A T ( i n , . . . , i ) = A ( i , . . . i n ) ∀ i ∈ { . . . m } , . . . , i n ∈ { . . . m n } (132) (cid:4) Definition 60 – N-Mode Matrix Product × p [131], [132]: The N-mode matrix product is a generalizationof the matrix product. Given a tensor
A ∈ R m × m × ... × m p × ... × m n , matrix B ∈ R m p × q , and C ∈ R m × m × ... × q × ... × m n ,the n-mode matrix product denoted by C = A × p B is equivalent to: C ( i , i , . . . , i p − , j, i p +1 , . . . , i n ) = m p (cid:88) i p =1 A ( i , i , . . . , i n ) · B ( i p , j ) (133) ∀ i ∈ { , . . . , m } , . . . , i p − ∈ { , . . . , m p − } , i p +1 ∈ { , . . . , m p +1 } , . . . , i n ∈ { , . . . , m n } , j ∈ { , . . . , q } (cid:4) UBMITTED FOR PUBLICATION: (DOI) 28
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 ∈ { , } m × m × ... × m p × ... × m n , matrix B ∈ { , } m p × q , and C ∈ { , } m × m × ... × q × ... × m n , the n-mode matrix product denoted by C = A (cid:12) p B is equivalent to: C ( i , i , . . . , i p − , j, i p +1 , . . . , i n ) = m p (cid:95) i p =1 A ( i , i , . . . , i n ) · B ( i p , j ) (134) ∀ i ∈ { , . . . , m } , . . . , i p − ∈ { , . . . , m p − } , i p +1 ∈ { , . . . , m p +1 } , . . . , i n ∈ { , . . . , m n } , j ∈ { , . . . , q } (cid:4) References
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