1 Copyright © 2012 by ASME

Proceedings of the ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

IDETC/CIE 2012 August 12-15, 2012, Chicago, IL, USA

DETC2012-70096

ENDOGENOUS PROGRESS POTENTIAL    

Andy Dong, Somwrita Sarkar Design Lab

University of Sydney Sydney, NSW, Australia

Email: {andy.dong,somwrita.sarkar}@sydney.edu.au  

ABSTRACT This paper provides a graph-theoretic framework for

assessing the progress potential for a product or class of products based on the underlying knowledge structure. We characterize the knowledge structure based on singular value signatures and extend the characterization to multi-dimensional functional model representations of product knowledge. We show that the progress potential for a product is partially determined by graph properties of the knowledge structure, and whether the graph structure is modular, random, or hierarchically modular. The characterization is empirically tested on energy harvesting devices. Our model connects the knowledge underlying a product to its potential for progress, appropriately taking into account the nature of product knowledge and its ‘complexity’.

INTRODUCTION In his most highly cited paper on macroeconomics, Paul

Romer developed a model showing the surprising result that growth is driven by the accumulation of ideas that are ‘non-rival’ rather than the size of the population (of a nation) [1]. What makes Romer’s work particularly relevant to engineering design is that the “design for a new good” is a canonical example of a non-rival good, because design arises from significant intellectual activity, is a good that lives on, and can be put to multiple uses, indefinitely in Romer’s model.

The intellectual value of a design as embodied in a product and its influence on the potential for continued progress is of particular interest in this paper. What Romer’s work left off is the assumption that any design is as of equivalent intellectual value as another design, and further that the production of new designs, possibly from existing designs, is based only on the human capital devoted to research, rather than any intrinsic improvability of a design. In engineering design research, we

know that this is not true. The modularity of a product design is a key characteristic of a design, which would influence its potential for progress [2-5]. Products that are modular would be amenable to rapid modular innovation because a change in any one module would not necessarily require a cascade of changes to the product due to standardized interfaces between modules. Various modules of the product could improve at differing rates, which would result in a general improvement of the basic product. Regular upgrades to laptop computer memory, processor, and graphics cards and automotive platform strategy are perhaps the most well known examples of this principle in operation. Likewise, Nam Suh’s axiomatic design principles, and in particular the axiom on maximal independence, promote the hypothesis that functionally modular designs should have the maximal progress potential [6]. However, there is likely to be a limit to progress potential based on the modularity of product architecture. While fixing the modularity of product architecture might unlock its progress potential, product architecture alone is not likely to be a sufficient determinant of progress potential.

If we could model the intrinsic improvability of a design with respect to a growing stock of knowledge, having this knowledge would allow us to forecast a product’s rate of improvement and, possibly, to re-design a product (but not necessarily its product architecture alone) so that it will sustainably improve. At the moment, industries rely on historical data on innovation through so-called ‘progress functions’ or ‘learning curves’ derived from cumulative volume of production and investment; yet, these have been shown to be unreliable to predict future progress except in few instances [7]. This relationship is empirically observed in the form of a power law y = ax-b, where y is the input cost for the xth component, a is the input cost for the first unit, and b is the rate of progress [7]. Recently, McNerney et al. showed that the progress potential of a technology is driven by a power law with

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exponent b = 1/(γd*), where γ is the intrinsic difficulty of finding a better component and d* is the maximum design complexity of the product [8]. The complexity of the product is determined by the component that has the most influence on other components, such that it is not possible to alter that component without simultaneously altering dependent components.

McNerney et al. took a product architecture approach, which itself is not fully resolved, since their measure of design complexity is based upon the number of dependencies between components. The issue of structural modularity in engineering systems is an open issue in engineering design research, with various metrics for modularity [9-13] and methods for finding modularity [14]. There is still no firm agreement as to an ‘optimal’ degree of modularity, or whether there can even be a single modularity number [15]. In summary, there is yet to be a uniform way to assess the architectural complexity of a product.

Instead, this paper takes the view that the limiting factor in the progress potential of a product is not the complexity of the product architecture, but rather the complexity of the underlying knowledge structure for the product. Researchers in other fields have taken this perspective on the importance of knowledge structure and the progress of fields [16] and especially hierarchical knowledge structures that progress through integrative and subsumptive development. What is the knowledge structure underlying a product, though, and how can this structure be assessed for progress potential? This paper presents a graph theoretic approach to assess the progress potential for a class of products. It will show how the modeling and analysis technique is appropriate for two-dimensional and multi-dimensional representations of product knowledge.

GRAPH THEORETIC APPROACH The analysis of the progress potential for a class of

products begins with Nam Suh’s axiomatic approach to complex systems design. To characterize a product’s design, Suh models the relation between the functional requirements (FR) and the design parameters (DP) by a matrix A, which he calls the design matrix. In other words, the matrix A could be considered to represent the structure of the ‘knowledge’ about the design, because the matrix A describes how FR and DP are integrated and linked together into a coherent whole. Axiom 1, the Independence Axiom, states that the functional requirements of the product should be independent [6, p. 16]. No one function should affect another function. At times, functional independence could mean physical independence, that is, structural modularity, but this is not a necessary condition. For a design to satisfy Axiom 1 fully, the matrix A must be diagonal. However, the matter of whether there exists a truly decoupled design or not is not directly related to the problem at hand, other than that we hold that a product wherein A is nearly diagonal is preferred.

Let us suppose that the set of n design parameters for a product are represented in a graph G. In graph theoretic terms, we can define an adjacency matrix A, where

Aij =1 if an edge exists between nodes i and j0 otherwise

!"#

$#

Each type-1 node is a functional requirement and each

type-2 node is a design parameter. For the type of complex systems that Suh was envisioning, the size of A is very large, and simply finding which functions are independent and which are not independent is computationally complex. In graph theory, this problem of finding nodes that belong together because they share edges is the problem of community detection [17]. By definition, a community has more intra-community edges than inter-community edges. If there is community structure, than the number of communities will always be much lower than n dimensions. In contrast, if there is no community, and there is perfect independence in the design matrix, then there will be exactly n communities. If there are communities, what this means is that there is mathematical redundancy in the design matrix, because there are linearly dependent rows or columns, and we could find a basis for those vectors. We will return to this problem of finding community structure after extending Suh’s ideas to the problem of progress potential.

All of this prior axiomatic reasoning by Suh provides the foundation for the theory espoused by this paper. Let us suppose that the design matrix does not represent a single product, but rather a class of products or the set of technologies that achieve necessary functional requirements under a set of design parameters. The definition of the adjacency matrix A is slightly modified, where

Aij =d number of edges between nodes i and j0 otherwise

!"#

$#

As before, each type-1 node is a functional requirement

and each type-2 node is a design parameter, but now the number of edges (d) between each node counts the number of products that have a map between a functional requirement and a design parameter. That is, knowledge exists to map a design parameter to a functional requirement and this knowledge is embodied in a set of products. If we had a perfectly diagonal matrix, then what this implies is that any functional requirement in this class of products can be progressed independently without influencing another functional requirement. In other words, we could state that the total knowledge κ required to design the product is simply the sum of the knowledge to design each individual aspect of the product, as modeled in McNerney et al. [8]:

κ = K1 +K2 +Kn (1)

In general, however, this matrix A is not likely to be perfectly diagonal, and thus the total knowledge required to design the product involves a complex set of interactions between the knowledge associated with the mapping between the functional requirements and the design parameters. To the

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extent that a design parameter or a functional requirement can be replaced (e.g., by replacing a design parameter associated with analog technology with a design parameter associated with digital technology while satisfying the same functional requirement) without altering the mapping, this is progress that refines or extends an existing product, but progress that does not alter the underlying structure of knowledge about the product. In contrast, if we aim to modify the underlying knowledge structure, which Henderson and Clark characterized as architectural innovation [18] but at the level of interaction between components, then progress potential may be limited by the complex coupling between the functional requirements and the design parameters.

In short, the structure of knowledge underlying a product will have a bearing on the cost of producing knowledge. To analytically relate the structure of knowledge to the cost of knowledge production, we will model the cost of the production of knowledge. Central to this model is the cost of producing (advancing) a unit of knowledge about a functional requirement or a design parameter and the cost of dependencies (links) across multiple functional requirements and design parameters. An example of advancing knowledge about a functional requirement could include increasing the performance envelope whereas advancing knowledge about a design parameter could be achieved by improving the fit tolerance. An example of coupled advances between functional requirements and design parameters would include the number of knowledge advances associated with modifying the skin of commercial aircraft into a composite material instead of aluminum, which reduces the risk of corrosion due to humidity, which allowed an increase in the cabin pressure, humidity, and, eventually, the size of windows.

We characterize the total cost z of producing knowledge for a product as the sum of the cost c for producing a unit of knowledge for each functional requirement or design parameter in the product or technology and the cost of producing knowledge associated with the interactions between functional requirements and design parameters. In other words, there is a cost associated with the synthesis of new knowledge into the existing knowledge structure, due to the complex “web” of linkages in the existing knowledge structure. The cost of integrating knowledge is a function based on a variable amount of investment B (dollars) put into the integration. We characterize z such that

z = c+ B

d∑

n∑

(2)

where n is the number of functional requirements and design parameters in the product or technology and d is the degree of knowledge dependency between functional requirements and design parameters, which we will model through the knowledge structure. This characterization is similar to Cohen and Levinthal’s model for the absorptive capability of companies, which models the extent to which companies can make use of new, publicly available knowledge in their firm and the amount of investment the firm makes to absorb the new knowledge [19]. To show how this equation can take into account the

structure of knowledge, we consider the following simplifying assumptions, although the conclusion will not rest on all of these being necessarily true. First, we assume that the cost to produce any unit of knowledge for all functional requirements and design parameters is constant. Clearly, this is not uniform, since, for cell phones, producing knowledge for a smaller battery with a high storage capacity is much more costly than producing knowledge related to a higher resolution screen. However, these are scalar differences unrelated to the structure of knowledge of the product. Second, we assume that there are q independent sub-domains of knowledge within the knowledge structure for the product such that knowledge within each sub-domain is dependent upon each other, but knowledge within each sub-domain, and by definition each unit of knowledge within the sub-domain, is fully independent of other sub-domains. In other words, there is a knowledge structure A=G(p,q) with p units of knowledge and q perfectly disconnected sub-domains of equal size s, wherein s=p/q, with each unit of knowledge having a degree of connectivity k to other knowledge with k=s-1. Finally, we assume that the investment B is constant for each sub-domain of knowledge. The cost of producing knowledge for this product would be characterized by

z = nc+ skqB (3)

where the term sk accounts for the number of links within a sub-domain and the term q accounts for the number of disconnected sub-domains. Making the appropriate substitutions and algebraic manipulations gives the cost of producing knowledge in terms of the knowledge structure:

z = nc+ pq

pq−1

"

#$

%

&'qB

z = nc+ p pq−1

"

#$

%

&'B

(4) Since it is not possible for there to be fewer knowledge

units than sub-domains, i.e., p cannot be less than q, we can consider the two cases of p=q and p>q. If p=q, then we have the situation of perfect knowledge modularity, and thus the cost to improve knowledge is simply the total cost nc of improving the knowledge for all functional requirements and design parameters, which is what we would expect. If we have the situation where there are more knowledge units than sub-domains, then the cost of improvement scales as p2/q. In other words, the structure of the dependency of knowledge has a direct effect on the cost of producing knowledge to advance the product. The more modular sub-domains of knowledge there are, the more slowly the cost of knowledge integration increases. Nonetheless, the number of knowledge units will dominate the cost of knowledge integration.

Between the extremes of a fully diagonal matrix A and a fully connected graph, in other words, an edge between every node, what do these suggest about the progress potential of the class of products? To address this question, we return to the

4 Copyright © 2012 by ASME

problem of redundancy in the matrix A. If there is redundancy, then there are common functional requirement-design parameter mappings. However, if there is a group of common functional-requirement-design parameter mappings, that is, a community of them, then this implies that a set of functional requirements and design parameters might be best thought of as a single functional requirement-design parameter mapping. Then, the strategy would be to progress knowledge about these as a set. At the other extreme, if there is full redundancy, then it is not possible to think of this class of products as independent functional requirement-design parameters mappings, which means that the progress potential is low. We would have to advance collective knowledge of the mappings between functional requirements and design parameters simultaneously. The question then becomes, what is the dependency of knowledge about the product? In other words, we become interested in knowing the degree of knowledge modularity in a class of products, and that this degree of knowledge modularity may indicate the endogenous progress potential for a class of products. To address this question, we discuss the issue of finding modularity in multi-dimensional product representations and what the modularity signatures imply about progress potential. We then show a way to model product knowledge that allows us to assess the progress potential for a class of products.

KNOWLEDGE MODULARITY DETECTION The aim of this section is to present an approach to identify

the coupling between functional requirement-design parameter mappings, or more generally knowledge couplings. In a prior paper, we have shown how the singular value decomposition (SVD) provides an efficient way to do this and to detect the modularity and hierarchical modularity of complex systems architecture [15]. We review some of the main concepts of that paper herein so as to build towards similar concepts in higher orders. SVD is a matrix factorization technique that decomposes any rectangular matrix A into an orthonormal basis and a diagonal matrix of singular values, which contain scaling information. The factorization of A of dimension m × n is given by

A =USVT (5)

where U is an orthonormal basis for ℜm and V is an orthonormal basis for ℜn. The original correlated information between functional requirement-design parameter mappings is now represented in terms of uncorrelated independent vectors and a scaling factor. The diagonal matrix S describes how much scaling a vector undergoes as it is transformed from to ℜm to ℜn, and the singular values in this matrix are ordered in descending order. The number of singular values is equal to the rank of S, and thus, excluding the null space, S is r × r, U is m × r, and V is n × r.

The value of each mapping can now be calculated as linear combinations of the derived orthonormal bases and singular values. The US or SVT products provide a new way of describing each of the m type-1 functional requirement nodes

and n type-2 design parameter nodes as a linear combination. The i-th type-1 node is

ui1s11 +ui2s22 ++uirsrr, i =1 to m (6)

and the j-th type-2 node is

s11v1 j + s22v2 j ++ srrvrj, j =1 to n (7) We can now return to the question of minimum

redundancy in the knowledge about the product. If a set of functional requirement and design parameter nodes shares the same set of nodes, then they by definition belong in the same community. If we were to calculate each i-th type-1 node and each j-th type-2 node by ignoring the least significant singular values k to r, 1 < k < r, some of the nodes will still lie in same vicinity of the other nodes since each subsequent term in Eqn. (6) and Eqn. (7) contributes less to the node representation.

This rank reduction calculation has important implications on the minimum redundancy question. If we take the two extreme cases, a fully diagonal and a fully connected matrix A, we can show that in the case of the fully connected matrix, all singular values are needed to calculate the nodes whereas in a fully connected matrix, only one singular value is needed to recalculate the nodes. Suppose that A is a diagonal 4 × 4 matrix.

A =

1 0 0 00 1 0 00 0 1 00 0 0 1

!

"

####

$

%

&&&&

The singular value decomposition of A is

U =

1 0 0 00 1 0 00 0 1 00 0 0 1

!

"

####

$

%

&&&&

S =

1 0 0 00 1 0 00 0 1 00 0 0 1

!

"

####

$

%

&&&&

V =

1 0 0 00 1 0 00 0 1 00 0 0 1

!

"

####

$

%

&&&&

Every singular value is needed to calculate the nodes. Now suppose that A is a fully connected matrix.

A =

1 1 1 11 1 1 11 1 1 11 1 1 1

!

"

####

$

%

&&&&

The singular value decomposition of A is

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U =

−0.5 −0.5 −0.5 −0.5−0.5 −0.1667 −0.1667 0.8333−0.5 0.8333 −0.1667 −0.1667−0.5 −0.1667 0.8333 −0.1667

"

#

$$$$

%

&

''''

S =

4 0 0 00 0 0 00 0 0 00 0 0 0

"

#

$$$$

%

&

''''

V =

−0.5 0 0 0.866−0.5 −0.5774 −0.5774 −0.2887−0.5 0.7887 −0.2113 −0.2887−0.5 −0.2113 0.7887 −0.2887

"

#

$$$$

%

&

''''

Only one singular value is needed to calculate the nodes.

What this implies is that it is not possible to disaggregate the knowledge about the class of products into definable modules. In addition, it further implies that a significant amount of systems integration was associated in integrating knowledge to produce a product in this class.

In between these two extremes, we are thus interested in the extent to which all products either ‘look alike’ or that the generalized knowledge underlying all of them is mostly modular – and how modular. Let us suppose that we have a matrix A of dimension 7×7 such that all of the functional requirement-design parameters in the upper left block are connected only to nodes in the upper left block and all of the functional requirement-design parameters in the lower right block are connected only to those nodes in the lower right block, but there are no other interconnections other than with one shared node. The representation of matrix A is

A =

1 1 0 1 0 0 00 1 1 0 0 0 00 1 1 1 0 0 01 0 1 1 1 0 10 0 0 1 1 1 00 0 0 0 1 1 10 0 0 1 0 1 1

!

"

########

$

%

&&&&&&&&

The SVD of A has only two singular values, one to

calculate the location of the nodes in the upper left block, and one to calculate the location of the nodes in the lower right block. Thus, the number of singular values gives an indication of the degree of modularity of the knowledge structure.

EXTENSION TO MULTI DIMENSIONAL PRODUCT KNOWLEDGE REPRESENTATION

Up to this point, we have been using Suh’s design matrix representation to represent product knowledge, but there are other product knowledge representation formats which are more ‘natural’, complete, and, most important, component-independent, including the function-behavior-structure (FBS) ontology, functional modeling, [20, 21] and so-called high-definition design structure matrices [22]. Including the product dimension, the use of these product representation formats would result in matrices of order 4 for FBS and order 3 for the functional basis. Matrices of order greater than 2 are known as tensors. All of the concepts discussed to this point apply to tensors of arbitrary order using the higher-order singular value decomposition.

Consistent with accepted notation, tensors will be represented with Euler capitals, e.g., A . In order to handle

multi-dimensional psychological data, Tucker [23] proposed the decomposition of a tensor into derivational modes (equivalent to the left and right singular vectors in SVD) and a core mode to show the level of interaction between the components in the derivational modes. Tucker solved this composition for an order 3 tensor, and the solution was recently extended to N modes by De Lathauwer et al. [24], who showed that the Tucker decomposition is a generalization of the singular value decomposition. In so doing, they named the Tucker decomposition in N modes as higher-order singular value decomposition (HOSVD). Like SVD, the HOSVD decomposes a tensor A into a core tensor G (equivalent to the matrix of singular values S) and a set of matrices B (equivalent to the left and right singular vectors U and V) along each mode

of A . For a tensor A ∈ℜI1×I2×I3×IN of order N, having ℜn

elements along each mode, the HOSVD is given by [25]: A = G ×1 U(1) ×2 U(2) ×3 U(3) ×4 U(4) … ×N U(N) (8) Computing the HOSVD of an order N tensor is equivalent

to the computation of N different matrix SVDs, one for each n-mode matrix unfolding of the tensor G [24]. An n-mode matrix folding of a tensor is the process of reordering the elements of a tensor into a two-dimensional matrix [25]. Each element of A is thus calculated as:

ai1i2i3in= gr1r2r3rN ui1r1(1)ui2r2

(2)ui3r3(3)uiNrN

(N )

rN=1

RN

∑r2=1

R2

∑r1=1

R1

∑

for in =1,…, IN ,n =1,…,N. (9) Whereas the singular matrix S is ordered by decreasing

values of singular values in the two-dimensional SVD, in the HOSVD, the core tensor’s matrices in each mode are ordered in decreasing Frobenius-norm [24].

Each n-mode matrix of the core tensor is thus calculated from the n-mode matrix folding of the tensor A . The consequence of this homomorphism between SVD and HOSVD is that concepts about identifying the modularity of complex systems based upon pattern of successive singular [15] values apply equivalently to systems described as two-dimensional matrices/graphs and order N tensors.

We apply this homomorphism to consider the singular value signatures of tensors that could represent ideal or canonical knowledge structures. We first consider the situation of a class of products such that there is full knowledge independence component. In other words, the knowledge to progress any one component is independent of knowledge for another component progressing at the same time. This could be represented by an order 3 tensor of size 16×16×4 (‘cube’) with all the nodes being fully connected in blocks of 4 for each 1-mode ‘frontal’ (front view) slice of the tensor as shown in Figure 1(a) and each 2- and 3-mode matrix unfolding in Figure 1(b) and (c). This structure has small groups of clustered knowledge, which may relate to one or more components, but each cluster of knowledge is fully independent from another cluster.

6 Copyright © 2012 by ASME

(a)

(b)

(c)

Figure 1. MATRIX UNFOLDING OF (A) 16×16×4 TENSOR (B) MODE-1 AND MODE-2 UNFOLDING

16×64 AND (C) MODE-3 UNFOLDING 4×256.

To recalculate this tensor from its HOSVD decomposition, we would need all 4 singular values from the frontal and lateral slices, but only 1 singular value for the horizontal slice, as shown by the singular matrices for each n-mode singular matrix, because all of the nodes are fully connected ‘in a row’ from a ‘top-down’ view of the cube.

mode 1 (frontal) and mode 2 (lateral):

S =

8 0 8 0

8 08 0

00 0

!

"

#######

$

%

&&&&&&&

mode 3 (horizontal): S =

16 0 0 00 0 0 00 0 0 00 0 0 0

!

"

####

$

%

&&&&

Next, consider a hierarchically modular network with

modules of the same size. Such a network structure models the situation wherein there are modules of independent knowledge structures and that some modules of knowledge ‘sit’ within another or is subsumed by another knowledge structure. In the hierarchical matrix, we start with a perfectly modular network and then connect this network to another network. Consider an order 3 tensor of size 24×24×24 comprised of 4 blocks of size 6×6 such that within each block of 6×6 there is an embedded fully connected modular network of size 4×4 which is only

connected to the outer network at the ‘corners’, as shown in the following figure, Figure 2.

Figure 2. HIERARCHICAL MODULAR NETWORK. The singular value spectrum for this type of network,

shown in Figure 3, reveals an interesting signature in that it is stepped. The mode-1 and mode-2 singular value spectrums show 2 distinct steps each, one for each hierarchical module, that is, the ‘outer’ module circumscribed by the corner blocks and the inner module.

Figure 3. SINGULAR VALUE SPECTRUM FOR

HIERARCHICAL NETWORK. As with the previous example, the mode-3 unfolding

requires only one singular value to calculate the structure, with the remaining singular values being zero, whereas the mode-1 and mode-2 unfolding require 8 singular values, 4 for each ‘outer’ module and 4 for each ‘inner’ module. We can confirm this visual finding by calculating the singular values for each of the modes, as shown in Figure 4. This result is a generalization into tensors of a finding of hierarchical modularity that we found in very large-scale bench test networks [14].

24.7334 0 00 24.7334 0 0 24.7334 0

0 24.7334 00 7.7628 0

0 7.7628 00 7.7628 0

0 7.76280 0

0 0

!

"

##############

$

%

&&&&&&&&&&&&&&

(a) mode 1 (frontal) and mode 2 (lateral) singular values 24×24

. . .

6121314151617

12

2418

0 5 10 15 20 250

10

20

30

40

50

60Singular Values of Tensor

Index of Singular Values

Mag

nitu

de o

f Sin

gula

r Val

ue

mode−1mode−2mode−3

7 Copyright © 2012 by ASME

51.8459 0 00 0 0 0

!

"

####

$

%

&&&&

(b) mode 3 (horizontal) singular values 24×24 Figure 4. SINGULAR VALUES FOR HIERARCHICAL

MODULAR NETWORK. Finally, consider a random network with a random

probability p of having an edge between two nodes. When p = 0, the network is a perfectly disconnected network and when p = 1, it is a fully connected network. For small values of p, we can produce a small world network with random connections, wherein the distance between any two nodes in the network is much smaller than the total size of the network. This network models the situation wherein the knowledge connectivity between components is relatively unknown, and it is unknown if there is any knowledge modularity or ‘regular’ knowledge structure underlying the class of products.

Figure 5 presents a plot of the singular values for each n-mode of an order 3 tensor with a probability of an edge of p = {0.1, 0.3, 0.5, 0.7, 0.9}. We can see a few trends from this figure. First, as the probability of an edge p increases, the magnitude of the first singular value increases. Thus, the magnitude of this first singular value provides us some indication of the connectedness of the graph, and that one singular value is sufficient to calculate an approximation of the graph. Second, in the random network, there is only a single step from the first eigenvalue to the second eigenvalue; in other words, there is no modular structure other than the entire network itself in a random network. If the underlying knowledge for a product has a singular value spectrum similar to a random network, then this implies that there is no regular, underlying knowledge structure. In such a situation, we would conjecture that the progress potential is very low, since we do not know the knowledge dependence between components in the product.

Figure 5. N-MODE SINGULAR VALUES FOR A

RANDOM NETWORK.

In between a fully modular network and a fully random

network is a network having some modularity because each node is directly connected to exactly a given number of nodes. Due to limitations in space, we do not show that the singular value signature for networks of these types will follow the empirical trends shown in these examples, which we have demonstrated empirically for matrices [15], e.g., tensors of order 2. At this point, we thus have a graph-theoretic model and a set of indicators based on singular value spectra of tensors, which we will show provide a ‘natural’ way to model knowledge underlying products as the basis for predicting the progress potential.

These empirical trends lead us to the following hypotheses: Given the knowledge underlying a class of products

represented as a higher-order tensor A ∈ℜI1×I2×I3×IN:

H1: A class of products having the number of significant singular values approaching N for each mode has a higher progress potential than a class of products having the number of significant singular values approaching 1. (Modular and regular network)

H2: A class of products having a singular value spectrum similar to that of a random network has no generalizable knowledge and is not likely to progress in a regular manner. (Random network)

H3: A class of products having a singular value spectrum wherein there is only one significant singular value is likely to have a highly connected underlying knowledge structure and thus has a lower potential for progress. (Random network)

EMPIRICAL TEST WITH ENERGY HARVESTING DEVICES

To test whether this approach can identify the modularity in a knowledge structure underlying a set of products or class of technologies as a means for predicting the progress potential, we tested the approach on a data set of energy harvesting devices produced by Weaver et al. [26]. Weaver produced functional models of energy harvesting devices to investigate the innovation potential of these products, specifically, how ‘concepts’ from one product might be incorporated into other products. Originally, the functional models were represented in a two-dimensional matrix, with the rows (type-1 nodes) being products (i=39) and the columns (type-2 nodes) representing both function (k=21) and flow (j=16). This representation was converted into a three-dimensional order-3 tensor, where I1 is product, I2 is flow and I3 is function. The value xijk = 1 if product i uses flow j in function k. Some sample data is shown in Table 1.

First, we generated a plot of the singular values for each mode of the tensor representing the class of products, as shown in Figure 6. There is a large gap between the first and second singular values along each mode, suggesting that this class of products has an underlying knowledge structure that is rather fully connected. However, there is underlying modular structure since there is a gradual, stepped decrease in the magnitudes of the singular values, rather than the pattern seen in random networks. Thus, according to H2 and H3, there is progress potential for these products. Noting that the number of

0 2 4 6 8 10 12 14 160

20

40

60mode−1 Singular Values

Index

Sing

ular

Val

ue

p=0.1p=0.3p=0.5p=0.7p=0.9

0 2 4 6 8 10 12 14 160

20

40

60mode−2 Singular Values

Index

Sing

ular

Val

ue

p=0.1p=0.3p=0.5p=0.7p=0.9

0 2 4 6 8 10 12 14 160

20

40

60mode−3 Singular Values

Index

Sing

ular

Val

ue

p=0.1p=0.3p=0.5p=0.7p=0.9

8 Copyright © 2012 by ASME

significant singular values for mode-2 and mode-3 is less than the number of singular values, according to H1, we would hypothesize that the progress potential will be limited by the flow and function modes. In other words, while there is very little knowledge dependency across these products, there is a high level of knowledge dependency across the functions performed by these products.

Table 1. SAMPLE DATA FOR ENERGY HARVESTER

DEVICE. import transfer Product human

E rot ME

trans ME

human E

rot ME

trans ME

Perpetuum 0 0 0 0 0 0 Nova Energy Turbine

0 1 1 0 1 1

Wing Wave Generator

0 1 0 0 1 0

Micropelt STM-PEM

0 0 0 0 0 0

Figure 6. SINGULAR VALUE SPECTRUM OF ENERGY

HARVESTER DEVICES. To examine the structure of the knowledge about this class

of products, we first take truncated approximations of the original product tensor and then identify the knowledge sub-domains (modules) by clustering nodes that share edges. The interpretation of a cluster along an n-mode is that there are knowledge dependencies in the given mode. For example, if mode-1 is the product dimension, then modules along this mode characterize groupings of similar or dependent products. To identify the modules in the network along an n-mode of the tensor, we follow the methodology outlined by Sarkar and Dong [14] to calculate the cosine similarity along each n-mode of the tensor. The cosine calculation for tensors X ,Y is the inner product of two tensors divided by the product of the norm of the tensors:

X,YX Y

=

i2=1

I2∑i1=1

I1∑ xi1i2iN yi1i2iNiN=1

IN∑

i2=1

I2∑i1=1

I1∑ x2i1i2iNiN=1

IN∑ i2=1

I2∑i1=1

I1∑ y2i1i2iNiN=1

IN∑

Figure 7(a)(b)(c) shows the modularity of the knowledge structure for the product mode, flow mode, and function mode, respectively, where the vector r is the number of singular values retained in the truncated approximation along each mode. Each square in Figure 7(a)(b)(c) represents one unit of knowledge, that is, one product, flow, or function. Squares colored in red indicate a connection between units of knowledge; thus, there is always a diagonal of 1×1 red squares since a unit of knowledge always shares a (perfect) relationship with itself. Large blocks of red indicate a module. Where there are darker modules of red within a module of red, this indicates a module within a module. As the color trends toward blue, the level of knowledge dependency decreases.

(a)

(b)

0 10 20 30 400

10

20

30

40

50

60Singular Values of Tensor

Index of Singular Values

Mag

nitu

de o

f Sin

gula

r Val

ue

mode−1mode−2mode−3

Energy Harvesters r=[5 5 5]

10 20 30

5

10

15

20

25

30

35

Energy Harvesters r=[10 10 10]

10 20 30

5

10

15

20

25

30

35

Energy Harvesters r=[15 15 15]

10 20 30

5

10

15

20

25

30

35

Energy Harvesters r=[35 15 13]

10 20 30

5

10

15

20

25

30

35

Energy Harvesters r=[14 10 10]

10 20 30

5

10

15

20

25

30

35

Energy Harvesters r=[7 7 4]

10 20 30

5

10

15

20

25

30

35

Energy Harvesters r=[5 5 5]

5 10 15

5

10

15

Energy Harvesters r=[10 10 10]

5 10 15

5

10

15

Energy Harvesters r=[15 15 15]

5 10 15

5

10

15

Energy Harvesters r=[35 15 13]

5 10 15

5

10

15

Energy Harvesters r=[14 10 10]

5 10 15

5

10

15

Energy Harvesters r=[7 7 4]

5 10 15

5

10

15

9 Copyright © 2012 by ASME

(c)

Figure 7. MODULAR KNOWLEDGE STRUCTURE OF ENERGY HARVESTERS ALONG (A) PRODUCT MODE

(B) FLOW MODE (C) FUNCTION MODE. The number of singular values retained in the tensor

truncation to generate the first row of images in Figure 7(a)(b)(c) was chosen arbitrarily to explore the modularity of the knowledge structure; the number of singular values retained in the second row of images was chosen by selecting the index value when the magnitude of the singular value (Frobenius norm) dropped below 1, 5 and 10, respectively.

The images show the following trends. First, the modularity of the knowledge structure varies significantly between each mode. Whereas the modularity along the product mode is quite high, with many distinct modules, the modularity for the function mode is quite low. This is consistent with the graph for the singular values, in which the function mode (mode-3) had the highest singular value and the product mode (mode-1) had the lowest value. The implication is that the progress potential for this class of products is limited by generating knowledge about functions. Referring to Eqn. (4), the value of q is much lower for the function mode than for the product mode, although both have the similar number of units of knowledge. As such, the cost of advancing knowledge in the function mode will dominate and would limit the progress potential. Our cost model does not currently take into account the fraction of shared knowledge units in overlapping and hierarchically embedded modules. The degree of hierarchical embedding of knowledge could further limit progress potential by increasing the degree of connectivity higher than the size of the module, which we previously modeled as the size of the module less 1 in Eqn. (3), without taking into account that a unit of knowledge can be part of more than 1 module.

The lack of modules in the function mode relative to the high modularity in the product mode and the very high initial value of the first singular value in the function modes relative to the product mode further confirm the observation that there exists many types of products possible for the same set of functions. The figures show that functions can be modularized in products but functions are not necessarily themselves straightforward to modularize, due to latent interdependencies between them, at least in this data set. That is, the choice of a

function (and its associated flow) necessarily requires (has dependencies on) other functions. This result is a plausible interpretation of the data, given the investments in the underlying science of the functionality of energy-related technologies (e.g., ARPA-E). As we will see more clearly in Figure 10, the important implication is that functions depend upon other functions; in order to support the function of Import electrical energy, it is necessary to Sense and in order to Regulate electrical energy is also necessary to Sense. To Supply electrical energy, it is necessary to Store electrical energy and so on. The progress potential is thus limited to improving the stock of knowledge in all dependent functions, and this approach allows us to understand the degree of dependency. Whether the functional dependency is designed in or ‘natural’ is the next level consideration, since if the dependency was ‘designed’ into the system, intentionally or otherwise, it could then be designed out.

We next show which products, flows, and functions became clustered together in a module to check the validity of the modules. As exemplars, we chose the modules that revealed a high degree of hierarchical modularity, which are cases r=[7 7 4], r=[14 10 10] and r=[14 10 10] for product, flow and function, respectively.

Figure 8. PRODUCT MODULES.

Energy Harvesters r=[5 5 5]

5 10 15 20

5

10

15

20

Energy Harvesters r=[10 10 10]

5 10 15 20

5

10

15

20

Energy Harvesters r=[15 15 15]

5 10 15 20

5

10

15

20

Energy Harvesters r=[35 15 13]

5 10 15 20

5

10

15

20

Energy Harvesters r=[14 10 10]

5 10 15 20

5

10

15

20

Energy Harvesters r=[7 7 4]

5 10 15 20

5

10

15

20

Perpetuum FSH/C

Michigan U PFIG

Bistable Buckling Harvester

Innowattech Road/Rail

MIDE Volture

UTexas Inductive Vib

UTexas Piezoelectric Vib

Clarkson U Inductive Vib

AA Battery Harvester

Kinetic Flashlight

Seiko Kinetic Watch

Seiko Solar Watch

Seiko Thermic Watch

Hymini Solar/Wind/Crank

Kinesis Wind/Solar

Solar/Wind Streetlamp

Solar Powered Sterling Engine

Leviathan

Four Seasons

Enviro Energies

WindTamer

Enocean Eco 100

Columbia Power Manta Buoy

Soccket

Heel-Impact Shoe Harvester

Piezo Backpack Straps

Big Belly Trash Compactor

Tracking System

Solar Heat Engine With Mirrors

Inflatable Mat

Enocean ECT 310 Perpetuum

Micropelt STM-PEM

Micropelt TE-Power Ring

Micropelt TE-Power Probe

Vibration

Solar & Solar Hybrid

Wind & Wind Hybrid

Kinetic

Solar

Thermal

10 Copyright © 2012 by ASME

The modules shown in Figure 8 are accompanied by the group name assigned by Weaver [26] and a new group named Kinetic to describe the products in that module. The Vibration module incorporates both the Inductive Vibration and Piezoelectric Vibration groups determined by Weaver. The product modules are consistent with their original groupings, although not all products grouped by Weaver appear in a module. The accuracy of the modules ranges from a minimum of 60% for Vibration to a maximum of 81% for Solar, Wind and Hybrid combined. Only two products appear in modules not originally assigned by Weaver: Kinetic Flashlight and Seiko Kinetic Watch (originally in the group Inductive Vibration); however, it is likely that they appeared in the Solar category because of functions operating on light energy (i.e., solar). What are particularly interesting are the overlaps between some of the modules, showing how some of the products belong in multiple modules, which is again consistent with the terminology in the groups determined by Weaver.

The flow modules of Figure 9 group together common flows. The modules are physically plausible. The strong independence of the flows, combined with the overlaps in the function modules of Figure 10, suggest that multiple functions operate on what are otherwise independent flows. This is entirely plausible, since these energy conversion devices take in a flow of one type and convert it into a flow of another type, but the input and output flows are otherwise independent of each other. The only flows that have some partial dependency relate to indicating status and controlling the device.

Figure 9. FLOW MODULES.

As previously discussed, the high degree of overlap

between functional modules, as shown in Figure 10, characterizes how multiple functions operate simultaneously for this class of products. In other words, in order to have a product that is an energy harvesting device, the empirical evidence is that the product must implement functions of Import (e.g., Human Energy) and then Convert and Change (e.g., condition) it (e.g., into Electrical Energy) and Senses how much energy has been produced. That the functions Position, Guide, and Secure appear in a hierarchical module with Import and Convert correctly describes a key challenge for these devices: controlling the flow as it is brought into the device so that it is continuous and having sufficient ‘energy’ so that the

energy can converted into a useable energy form (electricity), and then exported continuously.

Figure 10. FUNCTION MODULES.

In summary, we were able to derive the correct modules

identifying known relations between units of knowledge in this set of products. Usefully, our approach is able to quantify the degree of knowledge modularity, which allows us to estimate the progress potential for these products and to demonstrate that it would be limited by producing new knowledge about functions.

CONCLUSIONS This paper presented a nascent framework for

understanding the progress potential of a product of class of products and the role of knowledge in progress potential. Whereas prior studies have sought to relate progress potential to the modularity of product architecture, this paper takes the approach that it is the structure and complexity of knowledge that is relevant. Such a perspective is consonant with the intuition that the progress potential for products is difficult because there is a cost associated with advancing knowledge related to the product. Knowing the structure of knowledge may give an indication as to the level of investment necessary to make advances.

The paper provides key building blocks for characterizing progress potential based on graph theoretic measures using the singular value signatures of matrices and tensors representing product knowledge. Other graph theoretic measures of knowledge structure such as degree distribution, overlap size between modules, and number of links between knowledge sub-domains could be considered in Eqn. (2) for a more sophisticated approach to the prediction of progress potential. The extensibility of the approach to multi-dimensional representations of product knowledge, which is fundamental to engineering design, based on tensors makes it possible to explore the properties of knowledge for a product in a manner that takes into account the complementarity and dependency of knowledge.

Solid

Electrical Energy

Status

Light Energy

Control

Gas

Pneumatic Energy

Human

Human Energy

Hydraulic Energy

Liquid

ExportTransferConvertSeparateImportChangePositionGuideSecureSenseRegulateActuateIndicateSupplyStore

11 Copyright © 2012 by ASME

ACKNOWLEDGEMENTS Andy Dong is the recipient of an Australian Research

Council Future Fellowship (project number FT100100376). The authors thank Prof. Kristin L. Wood and Dr. Jason Weaver for providing the dataset and Prof. Wood for discussions on the meaning of the results.

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