Avoidance, Adjacency, and Association in Distributed Systems Design
IInterplay of Logical and Physical Architecture in Distributed System Design
Andrei A. Klishin,
1, 2, 3
David J. Singer, and Greg van Anders
1, 2, 5 Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Department of Physics, Engineering Physics, and Astronomy,Queen’s University, Kingston, Ontario, K7L 3N6, Canada ∗ (Dated: October 2, 2020)Patterns of avoidance, adjacency, and association in complex systems design emerge from the system’s un-derlying logical architecture (functional relationships among components) and physical architecture (componentphysical properties and spatial location). Understanding the physical–logical architecture interplay that givesrise to patterns of arrangement requires a quantitative approach that bridges both descriptions. Here, we showthat statistical physics reveals patterns of avoidance, adjacency, and association across sets of complex, dis-tributed system design solutions. Using an example arrangement problem and tensor network methods, weidentify several phenomena in complex systems design, including placement symmetry breaking, propagatingcorrelation, and emergent localization. Our approach generalizes straightforwardly to a broad range of complexsystems design settings where it can provide a platform for investigating basic design phenomena. A fundamental question in the design of complex, mul-ticomponent systems is how the components of the sys-tem are arranged.[1–3] Arrangement problems, generically,present the challenge of anticipating or identifying “primereal estate”,[4, 5] i.e. sectors of the system’s architecture thathave premium or priority because of the mutual avoidance,adjacency, or association between system components (seeFig. 1). Determining or anticipating components’ patternsof avoidance, adjacency, and association is important in so-called “greenfield” settings, i.e. before design aspects havebeen specified, and in “brownfield” settings, i.e. when one ormore system design aspects have been determined.[6–8] Inboth greenfield and brownfield settings, determining the pat-terns of arrangement and identifying system factors drivingthose behaviors is crucial for managing, mitigating, or adapt-ing to likely design outcomes.[9]Managing likely design outcomes by identifying patternsof arrangement depends crucially on both a system’s logicalarchitecture, i.e. the set of functional connections betweencomponents, and on the system’s physical architecture, i.e.the physical properties of the components and their arrange-ment in space.[10] A system’s logical architecture is essen-tially topological, and can be treated using network theorytechniques.[11] In contrast, describing the physical architec-ture of a system is typically done by disciplinary engineeringapproaches that rest on known physical principles. Treatingproblems in arrangements that arise from an interplay of a sys-tem’s logical and physical architecture requires a frameworkthat bridges a system’s network-theory-level description andits physical/spatial description. Whereas approaches exist atthe network theory and at the physical spatial levels, how theycan be bridged is an open question.Here, we show that topological and physical descriptionsof complex systems design can be bridged using statistical ∗ [email protected] – θ + Avoidance Adjacency Association ? ? ?
FIG. 1. Complex system design raises the question of identifying ar-rangement patterns of avoidance, adjacency, and association. Avoid-ance patterns (left) can be probed by testing the “cost” of creating avoid in the design. Adjacency patterns (center) describe arrangementmotifs found in the design, e.g. angles between the placement of de-sign elements. Association patterns (right) relate to the preferencefor proximity between design elements, e.g. measures “preferred”locations in adding design elements. physics. Using statistical physics we demonstrate a frame-work that reveals patterns of avoidance, adjacency, and asso-ciation in arrangement problems. We use an example arrange-ment problem to concretely demonstrate how our frameworkcan identify patterns of arrangement and how those patternsare driven by the design’s logical and physical architecture, inboth greenfield and brownfield settings. a r X i v : . [ phy s i c s . s o c - ph ] S e p I. SYSTEMS PHYSICS FRAMEWORKA. Motivation: Design Challenges Lurk Between Logical andPhysical Architectures
Complex systems are typically comprised by several inter-acting entities. The interactions among the entities are oftendescribed at two different levels: the description of what-is-connected-to-what, which is mathematically a graph-theoreticdescription, and by the description of how the entities phys-ically interact with one another in space, which is describedby physics. Taken separately, both levels of description giveuseful but incomplete insights into design.The logical architecture’s graph-theoretic description ofa complex system design is valuable because it isolatesthe connections between system elements that underlyfunctionality.[10] Functionality in the logical architectureis reduced to the topology of connections, and this con-nection topology can be analyzed with network theorytechniques.[12–14] Network theory approaches to analyzinglogical architecture are powerful because they abstract out thesystem’s physical realization.[15] However, realizing the log-ical architecture physically can produce emergent functionalconnections that are lost when logical architecture is analyzedalone.The physical architecture describes the realization of acomplex system design in terms of physical entities with phys-ical properties. Whereas entities in the logical architecturehave abstract interactions that are encoded topologically, inthe physical architecture interactions interact mechanically,thermodynamically, electromagnetically, etc., depending onphysical factors such as energy consumption and proximity inspace. By retaining this level of detail, the physical architec-ture provides an intimate picture of the performance of designelements. However, this intimate portrait of performance typ-ically describes a single physical architecture instance. Whatthat single instance means for the space of possible designsmore generally is often unclear.Though they can provide key insight into single designinstances, both physical and logical architecture descrip-tions restrict our ability to understand general characteris-tics of design. This restriction exists because general de-sign characteristics are properties of design problem spacesrather than of design instances. The focus on design in-stances has been described previously as design organizedaround “product structures”, i.e., around a particular outcomeof the design process.[16] Contrasting with product struc-tures are “knowledge structures” that organize the design pro-cess around relationships between design elements that persistacross instances.[17]Searching across instances is key for identifying patterns ofavoidance, adjacency, and association that are generic featuresof design problem spaces. Achieving this requires a differentapproach. To formulate this approach, the key challenge isin framing knowledge that emerges from collections of pos-sible instances of a system. The problem of many instancesthat give rise to collective behavior is the underlying principlethat motivated the development of statistical mechanics.[18] The fact that an analogous problem emerges in design, i.e. theneed to formulate knowledge structures to identify patterns indesign space, suggests that statistical physics could serve asthe foundation for a similar approach. Fig. 2 illustrates thisstrategy of attack.
B. Statistical Physics Approach
The need to address the problem of identifying patternsof avoidance, adjacency, and association that persist acrossspaces of designs points to statistical physics as a framework.To construct this framework there are two key challenges: for-mulating the design problem as a statistical mechanics model,and extracting from the model the knowledge structures thatencode design space properties.To construct statistical physics models of design, we needtwo things: the space of states and some metric on this space.For design problems that are studied with optimization tech-niques like simulated annealing, these two things are alreadyat hand. A generic approach for constructing a statisticalphysics design framework given a space of possible designsand a set of design objectives was developed in Ref. [19].We denote the design space as the set { α } consisting ofindividual design solutions α . Each solution α can be quan-titatively evaluated with a design objective O ( α ) . Instead ofexclusively focusing on the design that minimizes O , we con-sider a probability distribution over designs.[19] The maximalentropy distribution driven exclusively by the design objectiveis given by:[20] p α = 1 Z e − λ O ( α ) ; Z = (cid:88) α e − λ O ( α ) , (1)where λ is the design pressure , or the relative importance ofthe corresponding design objective in driving the distribution.The normalization Z is known as the partition function andcontains a wealth of information on the properties of the whole design space.With this formulation of design problems as statistical me-chanics models, the next challenge is to extract collectiveproperties that encode information about the structure of thedesign problem and the space it lives in. Doing this typi-cally requires computing sums over a combinatorially largeset of α , which relies on various problem-specific mathemat-ical techniques. For simple scenarios involving only one ortwo nodes with integrable interactions, prior work has shownthat this can be done by coarse-graining to extract effective,so-called Landau, free energies.[21] We expect that effectivefree energy approaches will, as they do in the ordinary statis-tical mechanics of particles, provide the means to gain insightinto the collective properties of more complex design spaces.However for more complex design spaces, again as is the casein the ordinary statistical mechanics of particles, some math-ematical techniques are required to study systems that lackclosed-form, integrable interactions.To meet the challenge of extracting information about de-sign spaces with complex forms of interaction it is useful to ProductStructure KnowledgeStructureLogical ArchitecturePhysical ArchitectureEngineeringDesignNetworkTheorySystemsPhysics fifi SpaceTopology ΣΣ StatisticalPhysics
FIG. 2. The need to grow from “product structure” approaches, the focus on single design instances, to “knowledge structure” approaches,the patterns of design outcomes or challenges that persist across collections of instances, suggests the applying the framework statisticalphysics. Statistical physics collectively sums the topological-level description of the logical architecture that expresses the system’s underlyingfunctionality and the physical/spatial description of design instances. Connecting the logical and physical descriptions of design in this way,the resulting “systems physics” picture that emerges bridges between traditional network theory and engineering design approaches. take cues from the structure of the problem. For complexproblems the advantage of the logical architecture is that it re-duces the complexity of interactions among elements to sim-ple, binary, yes/no connections. The disadvantage of this sim-plicity is that it loses the richness and specificity of the under-lying design problem. This suggests a more complete treat-ment of the design problem would be to “decorate” the topo-logical description of the logical architecture with informationabout the “topography” of the underlying design space and itsphysical architecture. This topographic decoration can be car-ried out by encoding the design space as a tensor network.Tensor networks were originally introduced as a graphicnotation for geometric tensors,[22] but over the last 25years have grown into powerful computational tools for stor-ing and manipulating high-rank data. The tensor networkcomputations are especially efficient when the connectionsare sparse. This property spurred the popularity of ten-sor networks in a broad range of applications, from encod-ing entangled wavefunctions in quantum condensed mattersystems,[23–25] to performing precision quantum chemistrycalculations,[26] renormalizing lattice models,[27, 28] solv-ing constraint counting problems,[29, 30] accelerating numer-ical linear algebra,[31–33] and learning multilinear classifiersin machine learning.[34] Across these applications, tensornetworks serve as an information structure that contains anexhaustive but raw description of the system.We use tensor networks to bridge the logical and physicaldescriptions of a design problem space. Network nodes en-code the design elements’ topography in the physical spaceof their placement and their properties. Network connections encode the functional connection topology and topographyof the physical interaction of design elements based on theirspatial location and physical properties. The topography–topology connection that the tensor network encodes has theuseful side effects that it provides a simple graphical repre-sentation of the interaction of design elements, and that well-developed methods exist for extracting information from ten-sor networks.Tensor networks encode information about the designspace, knowledge about patterns of avoidance, adjacency,and association among design elements, has to be extracted.Extracting this information requires adding specifically for-mulated pieces to it that represent key design questions (inphysics language, i.e., observables or order parameters). Weformulate design questions about avoidance, adjacency, andassociation among design elements by acting on the tensornetwork with a combination of elementary “moves”. Themoves yield patterns of the placement over sets of solutions indesign space, that are computed via contraction of tensor net-works. See Appendix A for a detailed description of movesand contraction.
C. Example Model System
Functional Units and Connections —To demonstrate theSystems Physics analysis and tensor network computationson a concrete example of a design problem, we first definethe specific logical and physical architectures for the prob-lem. We use a problem from Naval Engineering,[5] in which
Logical Architecture Physical ArchitectureTensor Network A ij ' ~x i “ a b c FIG. 3. Tensor network bridges the logical and physical descriptions of the design space for an example Naval Engineering arrangementproblem. (a) The logical architecture is represented graphically by a network of seven functional units (red circles) and their specific patternof functional connections (red lines), and algebraically with the adjacency matrix A ij . The dashed gray lines represent the non-links in thenetwork, which do not directly drive the arrangement but can be investigated. (b) The structure of the whole design space is contained in aninformation structure in form of a tensor network. Logical architecture determines the network pattern in which the tensors are connected,while physical architecture determines the contents of both site and coupling tensors. (c) The physical architecture is represented graphicallyby a square grid within a complex hull shape, and algebraically by the set of possible unit locations { (cid:126)x i } . A particular arrangement consists ofthe placement of all seven functional units within the hull and the routing of all functional connections between them (blue circles and lines). the functional units of a shipboard system need to be arrangedwithin the hull of a naval vessel, while respecting their func-tional connections, such as pipes or cables. The network pat-tern of connections constitutes the logical architecture, alsorepresented algebraically as an adjacency matrix A ij . Wefind a wide variety of network motifs arise in networks of asfew as n = 7 functional units without any graph symmetries(Fig. 3a), which we will study in the remainder of this work. Ship Hull —We position units and connections within a shiphull that we represent, following [5], by a 2D square grid witha complex, but fixed boundary (Fig. 3c). We constrain all con-nections within the ship hull to always run along a shortestpath between the two functional units; we choose our hull tobe L -convex to ensure that at least one shortest path existsbetween any pair of cells. We find that hull models with afew tens of cells are sufficient to establish placement patterns;computations reported below are for hulls with Y = 78 dis-tinct cells for unit placement each labelled as (cid:126)x i . Design Objectives —Picking the locations of all units andthe routings of functional connections between them definesa design solution α = ( { (cid:126)x i } , routing ) . In early-stage design,design architectures are typically not fixed, therefore the fullcombinatorial design space needs to be considered. Each de-sign solution is quantitatively evaluated with a design objec-tive O ( α ) , here we model routing cost: λ O ( α ) = λ (cid:88) ij A ij CL ( (cid:126)x i , (cid:126)x j ) , (2)where L is the “Manhattan” distance between the two cells, C is the cost per unit distance, and λ is the design pressure.Given the placement of units, we consider all allowed shortestpaths between them. By definition, all shortest paths have thesame length, so the value of O doesn’t depend on the partic-ular routing chosen, yet the number of routings is important.To account for the redundancy of routings, we introduce an effective design objective : λ O eff ( { (cid:126)x i } ) = (cid:88) ij A ij f ( (cid:126)x i , (cid:126)x j ; T ) ; (3) f ( (cid:126)x i , (cid:126)x j ; T ) = CT L ( (cid:126)x i , (cid:126)x j ) − ln n rout ( (cid:126)x i , (cid:126)x j ) , (4)where n rout is the number of shortest routings between (cid:126)x i and (cid:126)x j within the ship hull, typically growing with dis-tance. The routing lengths L ( (cid:126)x i , (cid:126)x j ) and the number rout-ings n rout ( (cid:126)x i , (cid:126)x j ) are fully determined by the shape of the hulland can be precomputed, stored as matrices, and scaled by thedesign pressure as needed. Tensor Network —The representation of the design space inform of a tensor network depends on both logical and phys-ical architectures (Fig. 3b). Logical architecture in form ofthe network A ij determines the pattern in which the site andcoupling tensors are connected. Physical architecture deter-mines the set of available locations for all units { (cid:126)x i } that isused as index for all tensors. The effective design objective f ( (cid:126)x i , (cid:126)x j ; T ) determines the entries of the coupling tensor. SeeAppendix A for a detailed mathematical discussion. Greenfield/Brownfield Settings —In the above formulation,the design space { α } of the problem is the space of all possi-ble arrangements of each functional unit { (cid:126)x i } , and is it nec-essary to establish a means of distinguishing units with fixedand variable position. This distinction is necessary becausethe formulation needs to address arrangement before or af-ter some of the units have been placed. We refer to situa-tions in which no units have fixed placement as greenfieldsettings. Greenfield settings are generically associated withgreen color-coding in results figures that follow. Also, werefer to situations in which one or more units have fixed lo-cations as brownfield settings. Brownfield settings are gener-ically associated with brown color-coding in results figuresthat follow. In Results figures that describe brownfield set-tings that combine placed units with yet-to-be-placed unitswe make a visual distinction between the two by color-codingplaced units and their effects brown and yet-to-be-placed unitsgreen. Low Cost, High Flexibility, and Crossover Regimes —Theformulation of design problems in terms of spaces of solu-tions weighted by objectives of the form of Eq. (4) has beenstudied in Ref. [19]. As in Ref. [19] we expect that the choiceof the design pressure associated with each objective (gen-eral case: Eq. (1); this model: Eq. (2)) will have qualitativelydistinct effects on design outcomes.[19] To maximize gener-ality, we study design pressures that correspond to multiplebehavioral regimes. We do this by first expressing the designpressure via its inverse λ = 1 /T , where T is the cost toler-ance . Low cost tolerance means that minimizing the routingcost O is a strong driver of a design solution choice, whereashigh cost tolerance means that the choice among the designsolution is not driven by cost. Ref. [19] showed that the sys-tem driven by this design objective undergoes a large-scalerearrangement (akin to a phase transition, but at finite-size)around T crit = C/ ln 2 ≈ . C . We pick C = 1 to fix themeasurement units for T . T < T crit favors low cost and wetherefore expect units to organize into motifs that facilitateshort (cheap) connecting paths. We expect this setting to becharacterized by effective attraction.
T > T crit favors maxi-mal flexibility and we expect units to organize into motifs thatfacilitate maximizing routing degeneracy. We expect this set-ting to be characterized by effective repulsion. We expect thatfor T ≈ T crit where cost and flexibility drivers are competingon near-equal footing there will be a crossover in behavior. II. RESULTSA. Avoidance
Void Premium —The interplay of logical and physical con-straints among design elements induces a complex landscapefor element placement. Intervening in that landscape by re-serving space for future use could induce functional units tomake a complex, collective rearrangement to avoid the re-served space. We characterize the cost of avoidance by com-puting the void premium that must be paid to forbid any unitsto be placed in the reserved space.We implement reserved space mathematically by creating a void × cells in size. In the results below we introduce thevoid on the hull midline, though in general it could be placedanywhere. We vary the horizontal location of the void, x v ,from zero to the hull length L (see Fig. 4a). We compute thecost of the void via the tensor network approach by suppress-ing several rows and columns of the coupling tensor (see Ap-pendix A). Contracting the modified tensor network results ina modified partition function, which is smaller or equal to theoriginal one Z ( x v ; T ) ≤ Z ( T ) . The ratio of the two partitionfunctions defines the non-negative free energy: ∆ F ( x v ; T ) = − ln Z ( x v ; T ) Z ( T ) . (5)We take this void free energy as a measure of the void pre- mium, the effective “cost” of the avoidance of a specified re-gion in space. Void Design Stress —The magnitude of the void premium ∆ F corresponds to the placement opportunity cost for thefunctional units. To understand this opportunity cost, notethat functional units that are not yet placed form a green-field “cloud” of possibilities within the hull, the location anddensity of which depends on the cost tolerance T . Cuttinga void from a dense part of the cloud costs a lot of free en-ergy, whereas cutting a cloud from a sparse part of the cloudcosts almost nothing. In this way, scanning the void free en-ergy along the void coordinate x v gives a direct probe of themorphology of the cloud. Conversely, if we regard the unit po-sitions as fixed, and the void as moveable, the cloud of unitsdrives the void with an effective force σ = − ∆ F/ ∆ x v , whichwe call void design stress . The void free energy and designstress are then a concise description of the collective effect ofavoidance in functional unit placement.The void premium and void design stress give a descriptionof collective avoidance effects in placement. These effects canbe examined in greenfield and brownfield settings. Greenfield —We studied avoidance metrics in greenfieldsettings, i.e. before any unit data have been fixed, in three de-sign regimes specified by cost tolerance T . We plot resultsin Fig. 4c-e. At subcritical T = 1 . (low cost priority), thevoid free energy curve shows a clear single maximum in themiddle of the hull, and two minima on the ends of the hull.At near-critical T = 2 . (cost-flexibility tradeoff) the curvemaintains the same qualitative shape, but the maximum getsflatter. At supercritical T = 3 . (high flexibility priority), thecurve shape flips to have a local minimum at the center ofthe hull, surrounded by local maxima on two sides. In otherwords, at low T the void prefers to be at either of the twoends of the ship (but a choice needs to be made in favor ofone of them). In contrast, at high T the void prefers to be inthe center of the ship. Thus the change from designing forflexibility (high T ) to designing for cost (low T ) induces achange from one architecture class (central-void) to two ar-chitecture classes (bow-void and stern-void). This collectiveeffect is analogous to symmetry-breaking phase transitions inconventional physical systems.[35]To understand the origin of the symmetry breaking we notethat void free energy is a proxy for the morphology of theunit cloud. To illustrate the shape of the cloud in a differentway, we approximate the cloud density as a sum of one-unitdensities ρ ( (cid:126)x ) = (cid:80) i p i ( (cid:126)x ) and plot densities as heatmaps inFig. 5f-h. These heatmaps are approximate because, unlikethe void free energy curves, they ignore the correlations inunit placement. At low T (panel f), units attract each otherand thus preferentially form a cloud in the center of the hulland push the void to either side of the hull. At near-critical T (panel g), the distribution becomes more homogeneousthroughout the hull, flattening the curve. At high T (panelh), functional units strongly repel one another, concentratingnear the edges of the hull. This leaves the center nearly empty,resulting in a single void free energy minimum. Brownfield —Both the void free energy curve and the unitcloud morphology can, however, change dramatically in a ~σ f g hb x v L V o i d P r e m i u m ∆ F ( x v ; T ) c low cost T = x v L d intermediate T = x v L e high flexibility T = U n i t D e n s i t y FIG. 4. Void premium quantifies the cost of avoidance of reserved space across the whole design space in a greenfield scenario. (a) Schematicof the ship hull and square cells within (Physical Architecture). Pink square represents a void where unit placement is prohibited, driven bythe void design stress (cid:126)σ along the center line of the hull (pink dashed lines). (b) Tensor network used to compute the void premium, with eachcoupling tensor modified. (c-e) Graphs of void premium (void free energy ∆ F ( x v ; T ) ) against the void coordinate x v for three values of T (color coded). (f-g) Functional unit density in presence of the void. brownfield settings, e.g. if even one unit is fixed to a spe-cific location in space. We pick the location indicated with thebrown square in Fig. 5d-f and fix one unit there. We choosethree different units to fix: unit 3 (which has 1 functional con-nection), unit 5 (2 functional connections) and unit 1 (3 func-tional connections). We plot the resulting void free energycurves in panels a-c, and unit clouds in panels d-f.Consider first the low-cost regime T = 1 . (Fig. 5a,d). Inthe greenfield setting (Fig. 4f) units positions were determinedsolely by ship geometry, and formed a dense cloud in the mid-dle of the ship (Fig. 4f). Fixing a unit position places an ad-ditional constraint on unit positions, and forces the unit cloudto condense around it (Fig. 5d). Because of this condensation,the void free energy curve becomes simultaneously steeperand more focused around the fixed unit point (panel a), but de-cays faster close to the edges of the hull. The void free energycost also depends on the topological position of the anchoredunit: it is highest for the most-connected unit 1 (bottom curve)and lowest for the least-connected unit 3 (top curve).At the near-critical and supercritical T = 2 . , . the an-chored unit similarly creates a reference point for the cloud,but the units in the cloud repel from that point. When repul-sion and attraction are nearly balanced at T = 2 . , the cloudprofile becomes nearly uniform and is not strongly affected bythe fixed unit (compare Fig. 4g and Fig. 5e). Similarly, fixinga unit at supercritical T = 3 . creates a point of strong repul-sion, forcing the unit cloud to the opposite corners of the shiphull (Fig. 4h and Fig. 5f). At both values of T = 2 . , . , the cloud morphology is not affected strongly by the single fixedunit position, and thus the brownfield void premiums (Fig. 5b-c) closely resemble their greenfield couterparts (Fig. 4b-c).Discussion of the effects of avoidance on unit positions, i.e.backreactions on the cloud, can be found in Appendix B. Avoidance: Logical–Physical Architecture Interplay —Theabove avoidance analysis gives a case study of basic phe-nomenology of the interplay between design pressure (favor-ing low-cost vs high-flexibility) and the logical and physicalarchitecture. Shifting the design priority from low-cost tohigh-flexibility changed the interaction between pairs of func-tional units. However, unit interactions were modulated byconnection topology (i.e., logical architecture) and by the spa-tial domain (i.e., the physical architecture). We captured theeffect of these complex interactions on spatial avoidance theunit clouds in Figs. 4,5. However, within the unit clouds, theinterplay of design pressure with the logical and physical ar-chitectures also induces emergent coupling. This emergentcoupling within the cloud induces patterns of adjacency andassociation between units, which we turn to next.
B. Adjacency
Whereas our avoidance analysis derives from and illustratesthe basic morphology of the unit cloud within the ship hull,questions about unit adjacency derive from correlations withinthe cloud, and the emergent coupling between units that arise. d e f a low cost T = b intermediate T = c high flexibility T = V o i d P r e m i u m ∆ F ( x v ; T ) x v L x v L x v L U n i t D e n s i t y FIG. 5. A brownfield scenario, such as anchoring one functional unit, sharpens the void premium curve. (a-c) Graphs of void premium (voidfree energy ∆ F ( x v ; T ) ) against the void coordinate x v for three values of T (columns, color coded) and different choice of the anchoredunit (rows, anchor shown in the tensor network on the right). (d-e) Functional unit density in presence of the void and unit 1 anchored in theindicated cell (brown square). a , D = low cost T = b intermediate T = c high flexibility T = N u ll m o d e l d e , D = f g N u ll m o d e l h FIG. 6. Bond diagrams quantify the adjacency patterns for both directly and indirectly connected functional units. (a-b) Tensor networks usedfor computations of bond diagrams for two pairs of units, one connected directly (0 → D = 1 ), the other indirectly (0 → D = 2 ). The sitetensors in the measured pair have external legs (green). The unit 1 is anchored in the center of the hull (brown square). (c-h) Bond diagramsfor the angle between the two units of the pair, for two different pairs (rows) and three values of T (columns, color coded). Yellow curve showsthe null model of the bond diagram, identical for all graphs. Black rim of bond diagram axes indicated the topological distance D = 1 , grayrim indicates D = 2 . Bond-Diagram Measures of Adjacency —To determine howemergent coupling between units leads to arrangment motifs,we consider pairs of units and their relative positions. We ex-amine pairs of units in 2D space, and express motifs as thepolar angles θ ( i → j ) , which vary along with the positions ofthe units in the cloud. Across the cloud, the angle takes form of a probability distribution p i → j ( θ ) . This distribution is anal-ogous to the bond order that is used to describe structure incondensed matter.[36, 37] In condensed matter, bond anglesare whole-system aggregate measures of adjacency. Here, theheterogeneous connectivity of design elements yields “bond”diagrams that are specific to each pair of units i, j . Depend- c d e Original Logical Architecture a Modified Logical Architecture fb g FIG. 7. Small changes of Logical Architecture lead to large changes of adjacency patterns, visible in the difference of bond diagrams. Panelsa-c correspond to the original Logical Architecture, while panels d-g illustrate two different changes of Logical Architecture. (a-b) Bonddiagrams at T = 3 . for two pairs of units, 0 → D = 1 ) and 0 → D = 2 ). (c) Tensor network used for computations of bond diagramswith original Logical Architecture, external legs not shown. (d) Tensor network used for computation of the 0 → → D = 1 , gray rim indicates D = 2 . ing on whether the units i and j are directly connected or not( A ij = 1 or ), the bond diagrams illuminate the strength ofdirect or emergent adjacency patterns. Computational Approach —To compute the bond diagramsmathematically, we use tensor networks to compute the raw2-unit marginal distributions p ( (cid:126)x i , (cid:126)x j ) (Fig. 6a,e), and con-vert them into the angular distributions p i → j ( θ ) using KernelDensity Estimation to reduce the numerical artifacts (see Ap-pendix A). In order to demonstrate more sharply defined bonddiagrams, we assume that one unit has already been placed(anchored) in the center of the hull and all other units needto be placed with respect to it. The bond diagram p i → j ( θ ) ofany pair of units is not uniform with respect to the angle θ even if the units are not connected at all, directly or indirectly.This non-uniformity is driven by the shape of the hull, a man-ifestation of the physical architecture, and we account for itby computing null model p ( θ ) of the bond diagram (see Ap-pendix A). Differences between the null model and computedbond diagrams are indicators of interaction-driven adjacency.This interaction driven adjacency depends on unit connectiv-ity; we use a topological distance , D ( i, j ) metric. D ( i, j ) isthe minimal number of network hops to get from unit i to unit j . In our example problem, the minimal number of hops variesfrom 1 (e.g. units 0 →
1) to 5 (e.g. units 3 → Direct Adjacency —We show topological distance, bond di-agrams, and the null model for our model in Fig. 6 in formof polar plots for two example unit pairs (corresponding plotsfor all unit pairs are given in Appendix B). We start discussion with the bond diagrams for direct adjacency (0 → D = 1 ).At subcritical T = 1 . (panel b) most units are located veryclose to each other, either in cardinal or intercardinal direc-tions (orthogonally or diagonally), resulting in a bond diagramwith a strong eightfold signal. At near-critical T = 2 . (panelc) the orthogonal attraction is balanced with diagonal repul-sion, resulting in a bond diagram with smaller peaks. At su-percritical T = 3 . (panel d) the units are located relativelyfar from each other and prefer diagonal relative location (sincediagonal location allows them to maximize their routing en-tropy), resulting in a fourfold, X-shaped signal. The symme-try of the fourfold signal is further broken by anchoring theunit 1. This additional symmetry breaking is driven by thehigh density of units in top-left and bottom-right corners ofthe hull (see Fig. 4). Emergent Adjacency —Across the whole T range, the bonddiagrams for direct adjacency are significantly different fromthe null model. However, adjacency can also be induced forindirectly connected unit pairs. The indirectly connected unitpair 0 → T and the ex-plicit details of Logical Architecture. However, our resultsreveal that for emergent adjacency topological distance is agood predictor of strength but is not a good predictor of shape . Logical Architecture Modifications —To further test the in-terplay between Logical Archiecture and Adjacency, we in-vestigate what happens to bond diagrams when we modifythe Logical Architecture. We consider two types of modifica-tions: removing an existing functional connection (Fig. 7d), oradding a new one between two units (Fig. 7e). Instead of com-paring the resulting bond diagrams to the null model again, wefocus on the difference between bond diagrams before and af-ter modification (panels f-g).
Addition:
The effect of addingthe (0,3) connection is strong. We can anticipate that with theadded direct adjacency, the adjacency pattern should approachthat of other directly adjacent units. The fourfold signal thatresults (panel g) is a signal of this. Since neither of the units0 or 3 is explicitly anchored in space, at high T = 3 . theywant to be positioned at the opposite ends of the longest diag-onal available within the hull, in this case the diagonal fromtop-left to bottom-right corners, similarly to the original adja-cency 0 → Removal:
Like for addition the effect ofremoving the (0,2) connection is dramatic, however the resultis unexpected. Instead of fourfold diagonal direct adjacency,the two units now have twofold horizontal emergent adjacency(panel f). The reason for this is that with the (0,2) connectionremoved, the units 1-2-6-0 now form a rhombus. Unit 1 isfixed in space, and because of high T = 3 . all unit pairs pre-fer to have diagonal adjacency. In this case the units 0 and 2on the opposite corners of the rhombus will have an orthog-onal adjacency, of which only horizontal adjacency manifestsbecause the ship hull is larger in length than height. Adjacency: Changes and Constraints Drive Patterns —We showed that the irregular, complex-network nature of asystem’s Logical Architecture drives the patterns of directand emergent adjacency. We showed adjacency patterns canchange significantly with changes in Logical Architecture.One outcome of this approach was the ability to detect emer-gent adjacency. The emergent effects we observed were witha single fixed unit. Though having having few fixed units isa characteristic of early-stage design, later-stage design situa-tions will result in more fixed units. Fixing more units willinduce more constraints, and further constraints will com-plicate the interplay of the logical and physical architecture.A more complicated logical–physical architecture interplayshould induce more complex patterns of association betweenunits, which we will examine next.
C. Association
Analyzing association patterns extends our avoidance andadjacency investigations to situations in which multiple, pre-existing constraints restrict functional units, i.e. in brownfieldsettings. These settings model either of two situations: (i) ac-tual late-stage design in which multiple functional units havebeen fixed during preceding design stages, or (ii) an early- stage design investigation of hypothetical late-stage situationsunder different decision scenarios.
Constraints and Localization —In either case, the expecta-tion that multiple active constraints will drive complex formsof interaction suggests that identifying patterns of associationthat result will require different techniques than identifyingpatterns of avoidance and adjacency. In general we expectthat patterns of association arising from multiple constraintswill localize those patterns relative to fixed design elements.This suggests that metrics of association patterns should sig-nal a tendency toward (or away from) placement proximityrelative to fixed elements, either globally or locally. Here, fora global signal we adapt measures of emergent localization tocompute a scalar design freedom for unit locations. For a localsignal we compute the design stress associated with specified,hypothetical unit placement.To study the effect of added constraints, their interplay withthe logical and physical architecture and the resulting local-ization, we employ the same model system as in the avoid-ance and adjacency investigations. However we introduceconstraints that fix units 1,2, and 6 to specific locations. Weinvestigate the emergent localization of the other 4 units withtwo metrics via the global design freedom Φ and local designstress ∆ F . Results are shown in Fig. 8 and broken down be-low. Global Signal of Association: Design Freedom
Mathemati-cally, both global and local metrics of localization use a tensornetwork computation of the marginal probability distribution p ( (cid:126)x i ) . The conversion of the distribution into design free-dom is inspired by the metric of existence area, commonlyused in studies of the Anderson localization of wavefunc-tions in disordered media and the localization of vibrationaleigenmodes.[38, 39] We define design freedom as: Φ = 1 Y (cid:18)(cid:80) (cid:126)x p ( (cid:126)x ) (cid:19) (cid:80) (cid:126)x p ( (cid:126)x ) , (6)where the normalization Y is the total number of cells withinPhysical Architecture; in this example Y = 78 . Given thisnormalization, Φ takes a value between 0 and 1 and has themeaning the effective fraction of the total area available forunit placement, if the distribution was uniform. For a unit withuniform distribution p ( (cid:126)x i ) = const , Φ would be 1, whereasfor an anchored unit Φ would be /Y → .Because of the heterogeneous connectivity of the logical ar-chitecture, design freedom Φ varies between units. The varia-tion between units is in addition to variation with design pres-sure, via changing cost tolerance T . Fig. 8c plots Φ( T ) byunit, and shows that all units have design freedom peaks near T ≈ . . In the range of this near-critical T , cost (effectiveattraction) and flexibility (effective repulsion) drivers of unitinteractions balance and allow the units to explore the largestrange of placement. As well, we observe Φ( T ) to fall intothree groups according to how constrained each unit is. Unit4 is not directly connected to any of the anchored units andthus enjoys the largest design freedom, almost approaching Φ = 1 . Units 3 and 5 are each connected to one anchored0 P e n d i n g A n c h o r e d ∆ F ab ? d low cost T = Φ = intermediate T = Φ = high flexibility T = Φ = m o r e Φ a n c h o r s ? e Φ =
Φ =
Φ = a n c h o r ? f Φ =
Φ =
Φ = ? g Φ =
Φ =
Φ = l e ss Φ a n c h o r s Cost Tolerance T D e s i g n F r ee d o m Φ c a n c h o r s a n c h o r a n c h o r s FIG. 8. Early stage design decisions determine the association patterns and design freedom for subsequent ones. (a) Tensor network used forassociation computations. Three out of seven units have already been anchored to specific locations (brown), other four are pending placement(green). External legs not shown. (b) The units 1,2,6 are anchored at the indicated locations within the ship hull (brown squares). The units0,3,4,5 can still be placed in many locations, some demonstrated for an example (green circles). (c) Graph of the design freedom Φ for thefour units without anchors across a range of cost tolerance T . Brown brackets on the right indicate that units with more adjacent anchors haveless remaining design freedom. Vertical dotted lines indicate the T values investigated in more detail in panels (d-g), as well as in avoidanceand adjacency patterns. (d-g) Design stress ∆ F patterns for the placement of each pending unit (rows in order of decreasing design freedom Φ ) at three values of T (columns, color coded). Legend for design stress magnitude ∆ F is shown to the left of panel (d). unit and thus have intermediate Φ . Unit 0 is connected to allthree anchored units and thus has the lowest Φ which quicklydecays at both low and high T . Local Signal of Association: Design Stress —Whereas Φ serves as an global scalar metric of design freedom, it is alsoimportant to understand how global design freedom is dis-tributed locally. This local distribution is captured by designstress. Design stress is closely related to an effective (Landau)free energy (LFE), defined as follows: F ( (cid:126)x i ) = − ln p ( (cid:126)x i ) + C , (7)where C is an arbitrary additive constant. We chose a con-vention where C is such that the minimal value of F is zero.The LFE can be interpreted as an effective design objectivefor the chosen degree of freedom, given that other degrees offreedom have been fixed or integrated out. This interpretationis analogous to the void premium (Eq. 5) in our avoidanceinvestigation, but instead of the effects of unit placement onvoids, here we examine the effects of unit placements on oneanother. Similar to the definition of void design stress via aspatial difference, the difference of LFE between two hori-zontally or vertically adjacent cells is the design stress ∆ F .Design stress is then an effective “force” that pushes individ-ual functional units towards their preferred locations.[19, 21]Compared with global design freedom, design stress patternsgive a more detailed picture of effective localization.Fig. 8d-g presents the design stress patterns for all four pending units at three values of T . Design stress is repre-sented by brown arrows drawn across the boundary of twoadjacent cells and pointing from higher to lower LFE. In otherwords, to decrease LFE and reach lower values of its effectivedesign objective, a unit needs to follow the arrows towards abasin. As the basin gets smaller and its walls get steeper, thepending units become more closely associated with the an-chored ones and thus exhibit stronger emergent localization.The localization effect is strongest at lowest T , where all ofpending units are strongly attracted to the two anchored units1,6 in a single basin. The basin is steepest for the most con-strained unit 0, less steep for the units 3,5, and the shallowestfor the least constrained unit 4, consistent with our expecta-tion based on design freedom Φ . At higher T = 2 . , . , theconstrained unit 0 develops complex LFE and design stresslandscapes with multiple local minima, maxima, and ridges(panel g). Units 3,5, connected respectively to the anchoredunits 6 and 1 in the center of the hull, show an X-shaped pat-tern of LFE (panels e,f), similar to the bond diagrams for di-rectly connected unit pairs, e.g. Fig. 6d. Lastly, the unit 4 isnot connected to any of the anchored units, instead it is “dan-gling off” unit 5 and thus shows almost nonexistent designstress across the whole hull (panel d). Association: Interaction and Decision Drivers —Both thedesign stress and design freedom metrics show that the as-sociation patterns and the emergent localization phenomenonstrongly depend on the position of both the fixed and the pend-1 C oa r s e G r a i n P h ys i c a l A r c h i t e c t u r e Coarse Grain Logical Architecture Z none all units { ~ x i } , r ou t i ng { ~ x i } x v θ θ ( i → j ) FIG. 9. System exploration across levels of detail in physical and log-ical architectures via two orthogonal directions of coarse-graining.Horizontal axis represents the reduction in the number of functionalunits explicitly considered. Vertical axis represents the reduction inspatial detail considered. Black arrows represent the computationalpathway across the study of avoidance, adjacency, and associationpatterns. The full design objective O (Eqn. 2) depends on all unit lo-cations and routings. The effective design objective O eff (Eqn. 4) de-pends only on locations of all units. The association pattern considersdetailed location (cid:126)x i , but only for a single unit. The adjacency patternreduces the spatial detail to just the relative direction θ ( i → j ) be-tween two units. The avoidance pattern further reduces the spatialdetail to just the void position x v as a single collective coordinateof the unit cloud. Finally, the partition function Z loses all detail ofphysical and logical architecture and summarizes the properties ofthe whole design space. ing units within the logical architecture. The logical archi-tecture alone gives an interpretation of the emergent local-ization result by counting the anchored neighbors. However,fully predicting localization requires examining the logical–physical architecture interplay that arises from the systemsphysics analysis. Unlike the simplified unidirectional designstress discussed in Ref. [21], in this system the design stresspattern is emergent both from unit interactions and from pre-vious design decisions. Chaining design decisions into se-quences and achieving optimal control of emergent localiza-tion stands out as an important question for further study. III. DISCUSSION
In this paper we showed that questions of avoidance, adja-cency, and association among the elements of complex, dis-tributed systems hinge on the interaction between logical- andphysical-architecture description planes. We bridged these de-scriptive planes with statistical physics techniques and showedthat patterns of avoidance, adjacency, and association can be mapped for an example system.
Design Phenomena: Symmetry-Breaking, Emergent Adja-cency, Localization — Our mapping of avoidance gave a spacepremium landscape. We found this landscape to undergoea symmetry-breaking transition with a change from designpressure that prioritizes high flexibility to pressure that pri-oritizes low cost (Figs. 4,5). Our mapping of adjacency gavea description analogous to “bond” directions in matter sys-tems. From this bonding description we observed that in-directly connected design elements developed emergent ad-jacency (Fig. 6). We also found large downstream changesin adjacency from small changes in underlying connectiv-ity (Fig. 7). Our mapping of association patterns quantifiedchanges in global design freedom driven by fixing design ele-ments and changes in design pressure. Mapping these effectslocally showed the emergent localization of design elements(Fig. 8).
Coarse Graining for Other Design Contexts —Our map-pings of avoidance, adjacency, and association patterns weredone for a model system motivated by problems in Naval En-gineering. However, for other design contexts where ques-tions of avoidance, adjacency, and association patterns arise,our statistical physics approach opens new lines of attack. Inparticular, our approach can be summarized in two steps. Firstwe “decorated” the logical architecture with detail from thephysical architecture. Then, we systematically chose two setsof system details, one to examine in detail, and the other totreat in aggregate, in a coarse-grained way. The aggregateddetails induce effective patterns of interaction among the re-maining elements, that reveal underlying patterns of arrange-ment. We illustrate this strategy in Fig. 9. Fig. 9 casts thestrategy into two orthogonal forms of coarse-graining: onein the physical architecture, the other in the logical architec-ture. In this representation, in statistical physics language,microstates that retain complete detail of both the physicaland logical architecture sit in one corner, whereas the partitionfunction, which aggregates microstates into a single scalar sitsin the opposite corner. Though the specific locations that cor-respond to our investigations are given at specific points onthese axes, regardless of design context, answers to questionsabout patterns of avoidance, adjacency, and association lie atintermediate levels of detail between those extremes.
ACKNOWLEDGEMENTS
We thank C.X. Du for useful discussions, A.S. Jermyn forextensive help with the
PyTNR package, and N. Mackay forassistance in development of the
TenZ package. This workwas supported by the U.S. Office of Naval Research GrantNos. N00014-17-1-2491 and N00014-15-1-2752. GvA ac-knowledges the support of the Natural Sciences and Engineer-ing Research Council of Canada (NSERC).2
Appendix A: Methods1. Tensor Network Construction
The partition function of the system can be expressed interms of the effective design objective in the following factor-ized form: Z ( T ) = (cid:88) { (cid:126)x } (cid:89) i 2. Move 1: External Legs The first move adds extra legs to specific site tensors to con-trol whether specific design degrees of freedom are marginal-ized or not. If none of the degrees of freedom are marginal-ized, then carrying out the multiplication but not the summa-tion in the sum (A1) would result in an un-normalized jointprobability distribution p ( { (cid:126)x i } ) over all the units, which isa rank- N tensor of prohibitive size. However, following theusual probability theory calculus, in a joint probability distri-bution each of the entering variables can be in three states:joint, marginalized, or conditional. In the tensor network rep-resentation of Fig. 10a, every variable (cid:126)x , (cid:126)x , . . . is marginal-ized, resulting in the distribution normalization, i.e. the parti-tion function Z ( T ) .In this perspective, a special action needs to be taken to not marginalize some of the variables. We do this by addingexternal legs to the corresponding site tensors (green lines inFig. 10b). External legs are the site degrees of freedom that are not summed over, functioning as free indices for the sum (A1).The result of contracting the network in Fig. 10b is a rank-2tensor that represents the un-normalized joint probability dis-tribution p ( (cid:126)x , (cid:126)x ) . Since the original network contracted toyield the full Z , the normalized probability distribution can beexpressed as ˜ p = p/ Z . 3. Move 2: Anchors The second move adds anchor tensors to the network. Theanchors represent the design decisions already taken and wo-ven into the information structure, thus encoding the brown-field aspects of design. In tensor network language, this isequivalent to fixing some of the local degrees of freedom (cid:126)x i and thus summing over a restricted ensemble, conditional onthe fixed (cid:126)x i . We do this by creating an additional tensor thatwe call an “anchor”. An anchor is a rank-1 tensor (vector) thatis coupled to a site tensor and is illustrated as purple square inFig. 10c. The elements of an anchor vector are given by theKronecker delta δ ( (cid:126)x i , (cid:126)x a ) , where (cid:126)x i is the index connected tothe site and (cid:126)x a is the specific location to which the functionalunit is pinned as result of a design decision. Since we didn’tcreate any external legs, the tensor network in Fig. 10c alsocontracts to a scalar number of conditional partition function Z x that functions as a similar statistical summary of the sys-tem as the original partition function Z . 4. Move 3: Modified Coupling The third move modifies the coupling tensors M ( (cid:126)x i , (cid:126)x j ) and traces the effect of this modification on the partition func-tion. In our study, we use the modification to account for thevoid where no units can be placed. The modification consistsof suppressing the statistical weight of the void cells in the3 contractionsite tensorcoupling tensorexternal leganchormodified coupling tensor a bcd efg Original Network Z ( T ) = + External Legs p ( ~x , ~x ) = + One Anchor Z ~x ( T ) = + Modified Coupling Z ( x v ; T ) = AvoidanceAdjacencyAssociation FIG. 10. Tensor networks can be used as information structures as combinations of basic moves express complex questions about the designspace. Panel (a) shows the original network, panels (b-d) describe the three basic moves that modify its topology, panels (e-g) show how theelementary moves are recombined to study the emergent patterns. (bottom-left corner) Legend of tensor network elements. (a) The originalnetwork connects n = 7 site tensors (green circles) with coupling tensors (gray squares) in the same pattern as the Logical Architecture (Fig. 2top). Since this network has no outgoing legs, it contracts into a single scalar number equal to the partition function Z ( T ) . (b) Move 1adds extra outgoing legs on site tensors 0 and 3 (green lines), making the contraction result in the rank-2 tensor containing the joint marginalprobability distribution on the spatial positions of the two units p ( (cid:126)x , (cid:126)x ) . (d) Move 2 attaches an additional rank-1 anchor tensor (brownsquare) to site 1, fixing it to a specific spatial location. This network contracts to the conditional partition function Z (cid:126)x . (d) Move 3 modifiesall of the coupling tensors (shown as pink squares), for example to account for the voids. This network contracts to the modified partitionfunction Z ( x v ; T ) . (e) For the avoidance pattern, we use both anchors and modified couplings to compute the placement opportunity cost viaEqn. 5. (f) For the adjacency pattern, we fix node 1 with an anchor and compute the 2-unit marginal distributions p ( (cid:126)x i , (cid:126)x j ) for all possible pairsof external legs i, j , and further convert them into bond order diagrams. (g) For the association pattern, we encode the past design decisionswith anchors on units 1, 2, 6 and study the 1-unit marginal distributions on each of the other units. coupling tensor: M ∗ ( (cid:126)x i , (cid:126)x j ) = M ( (cid:126)x i , (cid:126)x j ) (cid:89) (cid:126)x v (1 − δ ( (cid:126)x i , (cid:126)x v ))(1 − δ ( (cid:126)x j , (cid:126)x v )) , (A2)where (cid:126)x v denotes the positions falling into the excluded void.In the network on Fig. 10d we modified each coupling tensorin this way (marked in pink). The network results in the mod-ified partition function Z ( x v ; T ) , from which we compute thevoid free energy via Eq. 5. 5. Bond Diagrams To compute bond diagrams, the raw two-unit distributions p ( (cid:126)x i , (cid:126)x j ) need to be converted into the angular distributions p i → j ( θ ) in post-processing. Since all functional units are placed in a discrete, finite, and fixed set of cells (cid:126)x i , we pre-compute the directions between any pair of cells θ ( (cid:126)x i , (cid:126)x j ) ,measured in radians from to π , ahead of time and storethem.Within the design ensemble, the locations of units (cid:126)x i arerandom, drawn from the joint distribution encoded in the ten-sor network. We compute a series of marginal two-units dis-tributions p ( (cid:126)x i , (cid:126)x j ) for all pairs i < j (see the tensor networkin Fig. 10f). Since the possible unit locations are discrete,the possible directions θ ( (cid:126)x i , (cid:126)x j ) form an artificially irregulardiscrete set. This numerical artifact would result in a jaggeddirection distribution p i → j ( θ ) . We smooth the distribution byusing a version of non-parameteric Kernel Density Estimation(KDE) [41, 42] with periodic boundary conditions in which4higher angular harmonics are suppressed: p i → j ( θ ) = 1 N (cid:88) k (cid:88) (cid:126)x i ,(cid:126)x j p ( (cid:126)x i , (cid:126)x j ) e − ( hk ) cos ( k ( θ − θ ( (cid:126)x i , (cid:126)x j ))) (A3)Here N is a normalization factor, h is the KDE smoothingfactor (bandwidth), k ∈ { , , , . . . } is the angular mode in-dex. We find that using smoothing factor of h = 0 . radiansand angular modes up to k max = 30 gives good results.The resulting distributions p i → j ( θ ) need to be comparedwith the null distribution induced by the Physical Architecture(ship hull shape). We compute the null distribution by evalu-ating the formula (A3) for p ( (cid:126)x i , (cid:126)x j ) = const . The identicalnull distribution is shown in every panel of Fig. 6. 6. Numerical Aspects of Computation Implementing the tensor networks described above on acomputer requires two different kinds of computational work:constructing the networks from Logical and Physical Archi-tecture and possible modifications with the three moves, andcontracting said networks numerically. We perform these twotasks in Python.Using code we developed, we create tensor networks byspecifying the network topology, the spatial domain geome-try, the design objective, and additional moves. These speci-fications are done via high-level commands, allowing for therapid generation of diverse networks.Tensor network contraction is handled by a Python pack-age. Existing tensor network packages use different meth-ods of executing a sequence of pairwise tensor contractions.The contraction result does not depend on the contraction se-quence, but the computational time and memory requirementsrise by orders of magnitude for suboptimal sequences. Op-timal sequences are known for certain frequently used net-works, wherease for others one can use exhaustive enumer-ation algorithms to find the optimal sequence and then exe-cute it repeatedly for the same network topology.[43] How-ever, almost all networks that we contract in the present workare subtly different, and therefore might require different con-traction sequences. We perform all contractions with the PyTNR package, an open-source general purpose tensor net-work contractor.[40, 44] The features of PyTNR include usingheuristics to automatically generate the contraction sequenceson the fly, and performing SVD approximations of controlledprecision to reduce the dimension of stored tensors.The features of PyTNR define the computational con-straints on the size of systems that our approach can han-dle. The size and structure of the Logical Architecture di-rectly change the number of units n in the network and thenumber of tensors n t (counting both site and coupling ten-sors). The size or resolution of the Physical Architecture do-main directly affect the tensor bond dimension D . A rigorous,though pessimistic upper bound on the time complexity ofcontraction stands at O (cid:16) D √ ∆ n t (cid:17) , where ∆ is the maximaltensor rank.[30] In contrast, PyTNR relies on a heuristic and stochatic generation of contraction sequences, which compli-cates even the empirical investigations of numerical scalingof complexity. For highly structured networks, such as d -dimensional hypercubic lattices, the time complexity scalesa power law O ( N γ ) , where the exponent γ is a bit larger thanthe space dimension d and depends on the nature of system’sboundary conditions (periodic or closed).[40]To provide more concrete numbers, each tensor networkcontraction in this paper takes less than 10 seconds on a lap-top computer (Intel Core i5-3360M @ 2.8GHz CPU, 8GbRAM) for our example system ( n = 7 units, n t = 15 tensors, D = 78 , total number of combinatorial states O (cid:0) (cid:1) ). Theexample system size was chosen to best illustrate the physicalphenomena at single unit resolution. In other investigationswe reliably contracted lattice networks of up to n t = O (cid:0) (cid:1) tensors accounting for O (cid:0) (cid:1) combinatorial states using PyTNR .[45] This result suggests that the current tensor net-work methods would remain tractable for systems even oneorder of magnitude larger, or perhaps even larger as the tensornetwork methods develop. Appendix B: Supplementary Results1. Avoidance: Excess Density In the main text of the paper we show that void premiumquantifies the cost of reserving space within the ship hull, withthe effect being further amplified by the presence of an anchor(Figs. 4-5). We associated high void premium with a largerearrangement of the functional units. We can quantify thedegree of rearrangement by computing the unit density pro-file without void ρ no void ( (cid:126)x ) , the unit density profile with void ρ void ( (cid:126)x ) , and their difference: ∆ ρ ( (cid:126)x ) = ρ void ( (cid:126)x ) − ρ no void ( (cid:126)x ) , (B1)which we term excess density that can be both positive andnegative. Since the total number of functional units does notchange upon addition of void, the positive and negative re-gions of excess density have to cancel each other. In thiscase large rearrangements of the unit cloud are characterizedby large contrast between the positive and negative regions,graphically visible in saturation of colors.We plot all three densities in Fig. 11, both without and withan anchor on unit 1. In low-cost case T = 1 . (panel a) theunits want to form a compact cloud, which can be locatedanywhere within the hull. Upon creation of a void slightly tothe left of center, the unit cloud relocates to the right of thevoid, as seen by large negative ∆ ρ in the left half of the hulland positive in the right half (visible as red and blue clouds).When unit 1 is anchored (panel d), this effect becomes evenmore pronounced since the unit cloud condenses around a ref-erence point. Creation of a void pushes the unit cloud to theright and above the anchor, but it cannot move far from the an-chor, resulting in high contrast of excess density (strong colorsaturation in the figure) and thus high void premium. At in-termediate and high values of T , both with and without an5 a low cost T = b intermediate T = c high flexibility T = d e f 11 1 11 1 1 U n i t D e n s i t y w i t h o u t V o i d U n i t D e n s i t y w i t h V o i d -0.10.00.1 E x ce ss D e n s i t y U n i t D e n s i t y w i t h o u t V o i d U n i t D e n s i t y w i t h V o i d -0.10.00.1 E x ce ss D e n s i t y FIG. 11. Reserving space in locations with large void premium causes large rearrangements of the unit cloud morphology. The rearrangementis shown via the density profile without the void, with the void, and their pointwise difference (rows) for three values of T (columns, colorcoded). Panels (a-c) show the rearrangement in a fully greenfield scenario (no anchors). Panels (d-f) show the rearrangement in a brownfieldscenario (one anchor indicated with a brown square). anchor (panels b,c,e,f) the rearrangements are much smaller,visible in much paler colors on excess density heatmaps. 2. Adjacency: All Bond Diagrams In the main text of the paper we show how to compute thebond diagram p i → j ( θ ) for any pair of units i, j . Since thedirections θ ( i → j ) and θ ( j → i ) only differ by a trivial rota-tion by angle π , and the direction from a node to itself is not defined, a system of n units would have n ( n − / indepen-dent bond diagrams. The Logical Architecture of the exampleproblem was deliberately chosen to not have any graph sym-metries, therefore the bond diagrams are not related to eachother via any symmetries.While in the main text we only show several representativebond diagrams (Figs. 6-7), Fig. 12 shows all of the bond dia-grams for the low-cost regime T = 1 . and the high-flexibilityregime T = 3 . . The diagrams are arranged as a lower-triangular and an upper-triangular matrix of polar plots so that6 Null Model T = T = i j D ( i, j ) j i j i j i j i j i j i j j j j j j i i i i i i FIG. 12. Adjacency patterns between the 7 functional units shown via bond diagrams p i → j ( θ ) . (top-left corner) Typical tensor networkused for bond diagram computations, with a brown anchor for unit 1 and green external legs at units i and j for each origin-destination pair i, j . (bottom-right corner) Legend for the opacity of axes boundaries representing the topological distance D ( i, j ) varying from 1 to 5 hops.(top-right triangle, red) Bond diagrams for T = 3 . . (bottom-left triangle, blue) Bond diagrams for T = 1 . . The origin and destination ofeach bond diagram are indicated on the outside boundaries of the triangle. Axes in positions symmetric with respects to the diagonal refer tothe same origin-destination pair and thus can be directly compared. The yellow curve in each axes shows the null model bond diagram p ( θ ) . the diagrams in positions symmetric with respect to the di-agonal refer to the same pair of functional units and are thusdirectly comparable. The full set of diagrams shows all ad-jacency features highlighted in the main text. For units that are directly connected, most of the diagrams at T = 1 . showeightfold signal (e.g. 2 → T = 3 . showX-shaped fourfold signal (e.g. 4 → T = 1 . and T = 3 . (e.g. 3 → [1] C. H. 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