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IEEE SYSTEMS JOURNAL 1

An Inﬁnite Regress Model of

Design Change Propagation in Complex Systems

Serhad Sarica, Jianxi Luo

Abstract—In complex systems, design changes of one compo-

nent or subsystem may require the redesign of other components

or be stimulated by the redesign of other components. Changes

propagate through the design dependency paths among the

components of a system and thus present challenges to the

design and management of the system. Therefore, an assessment

of the inﬂuence and susceptibility of components is useful for

system design decisions such as which components to standardize,

modularize, or embed ﬂexibility to address future design changes.

Finding the overall inﬂuence or susceptibility of individual

components throughout a complex system is an inﬁnite regress

problem, as both the sources and targets of the inﬂuences are

the same set of components. A component inﬂuences some other

components, which inﬂuence other components, and so on. Such

inﬂuences can be propagated back to the initiating components

through cycles of design dependencies among components. In this

paper, we model change propagation in complex systems as an

inﬁnite regress problem and derive eigenvector-based indicators

of component inﬂuence and susceptibility. We demonstrate our

method based on several case studies.

Index Terms—complex systems, change propagation, inﬂuence,

susceptibility, cyclic dependency.

I. INTRODUCTION

THE evolution of an engineering system is driven by

many design changes for innovation or to meet evolving

customer requirements over time. Design changes may affect

manufacturing, supply chains, project schedule and various

costs, and need to be properly analyzed and managed. In

a complex system, the design change of a component or

subsystem may emerge or be initiated by the change of other

components [1] and propagate through physical or design

dependencies among components and necessitate changes in

those components [2]. Therefore, the knowledge of critical

components, which are most inﬂuential on other components

and most susceptible to design changes of other components

is crucial for change management and system architecting.

However, it is difﬁcult to assess the overall inﬂuence1and

susceptibility2of individual components in complex systems

that have many components and intricate dependencies among

them [3].

In particular, in a complex system where there are cyclic

dependencies among components , components may propagate

S. Sarica and J. Luo are with Engineering Product Development Pillar of

Singapore University of Technology and Design, 487372, Singapore

Manuscript received May 16, 2018; revised December 2, 2018; accepted

February 12, 2019.

1The inﬂuence of a component in design change propagation concerns its

overall effects on the system (on other components, and itself) if a design

change occurs in that particular component.

2The susceptibility of a component in design change propagation concerns

the overall effects on itself received from the design changes in the system.

design changes to every other component in the same cycles,

then back to themselves, and then onward again to other

components (and themselves), ad inﬁnitum [4]–[7].When the

sources and targets of inﬂuences are the same set of com-

ponents, design changes may propagate in an inﬁnite regress

manner across the components. In theory, design change prop-

agation may be ampliﬁed in the inﬁnite regress process and

cause system instability. Therefore, it is naturally difﬁcult to

estimate the overall inﬂuences and susceptibility of individual

components by tracing and counting speciﬁc component-to-

component changes.

To address such challenges, we directly model design

change propagation in complex systems as an inﬁnite regress

problem for generality and derive the overall inﬂuence and

susceptibility scores of individual components without the

need to trace and calculate the inﬂuences from speciﬁc

components to components. Solving our model results in

the eigenvector-based indicators of component inﬂuence and

susceptibility. Herein, the inﬂuence of a component in design

change propagation is assessed by considering all possible and

inﬁnite propagation paths for its design change to impact any

component (including itself). Similarly the susceptibility of

a component is assessed by considering all possible paths

through a component which propagate a change initiated

by any component (including itself). In turn, the component

inﬂuence and susceptibility indicators can inform system de-

signers which components should be prioritized to redesign,

standardize, modularize, or embed ﬂexibility to address future

design changes.

II. LITERATURE REVIEW

Change propagation via interdependencies in complex sys-

tems has been studied in a variety of ﬁelds, such as software

design [6], mechanical-electrical system design [8], infras-

tructure [9], supply chain management [10], and innovation

ecosystems [11]. Particularly, prior studies in complex system

design have investigated the probability and impact of possible

design changes of an instigating component to other com-

ponents in a system through their direct and indirect design

dependencies [1], [6], [12]. In the following, we review related

prior works.

First of all, the dependencies between components have

been widely analyzed based on the design structure matrix

(DSM). For example, Cohen et al. [13] introduced one of

the ﬁrst studies that uses a matrix-form product information

model, which consists of entities, relations between entities

and attributes of entities, to predict and evaluate change prop-

agation and its consequences. Second, many studies developed

IEEE SYSTEMS JOURNAL 2

component indicators concerning design changes and their

propagations. For example, Martin and Ishii [14] focused

on direct dependencies among components and analyzed a

component-based design structure matrix (DSM) to understand

change dynamics and use the speciﬁcation ﬂow concept to

accomplish a robust product platform. They introduced the

Generational Variety Index (GVI), Coupling Index-Receive

(CI-R) and Coupling Index-Supply (CI-S). GVI represents

external drivers of change, CI-R represents internal drivers

of change, and CI-S represents the probability of change

propagation. They used the plot of CI-R vs. GVI to show

how likely the components in a product are to change.

Design ﬂexibility studies particularly adopted or developed

relevant metrics to support proactive mitigation of change

propagation. Suh et al. [15] introduced the change propaga-

tion index (CPI), which differentiated the changes instigated

from a component and changes propagated to that particular

component. Cardin [16] and De Lessio et al. [17] adopted

the CPI to identify the components that could be designed

with ﬂexibility to absorb propagated changes. Smaling and de

Weck [18] proposed the technology invasiveness index based

on a component-based change DSM and used it to numerically

identify the changes required to adopt a new technology into a

product. Suh et al. [19] used DSM to track the changes caused

by the infusion of a new technology to a system and calculate

the cost of the changes. Most of these indices only considered

direct dependencies to assess change propagation.

Clarkson et al. [1] accounted for indirect dependency paths

of change propagation between system components. They

speciﬁcally analyzed the likelihood (i.e., the probability of

change propagating from one component to another and,

respectively) and impact (the amount of rework needed on the

receiver component to adopt the propagated change) of change

propagation between components. Their change prediction

method (CPM) uses a depth-ﬁrst search algorithm to analyze

the direct likelihood and impact DSM as inputs, discarding

cyclic and self-dependencies, to reach a multi-step risk DSM

that contains the likelihood and impact indices for both directly

and indirectly related components. The likelihood and impact

indices are then used to generate a graphical representation

called risk plot, which indicates the risk associated with

components that might be affected by a change that has

emerged in a speciﬁc component.

A strand of publications using CPM-based change propaga-

tion analysis followed Clarkson et al. [1]. For example, Eckert

et al. [20] categorized components as constants, absorbers,

carriers and multipliers in terms of their reactions to change.

Ahmad et al. [21] extended the CPM by allowing concurrent

initiating changes. Koh et al. [22] introduced a parameter

called change propagation reachability to limit the probability

of the propagation of a change beyond a predeﬁned number

of steps, hence lowering the number of steps that must be

examined. Generally, the inﬂuence of a change is estimated

to be able to propagate at most 4 steps from initiating

components to other components, depending on the speciﬁc

cases [1]. Lastly, Ullah et al. [23] introduced a method based

on propagation probability and impact concepts to identify

the least-risky change propagation paths for aiding designers

in reducing the rework period caused by requirement changes.

Some studies considered the change propagation between

different types of component interactions. Rutka et al. [24]

searched for suitable propagation paths depending on the types

of changes and level of changes among components. Koh et al.

[25] integrated the house of quality and the CPM to quantify

the performance of various options of changes considering

their suitability to requirements using multi-domain matrices

[26] and building a feedback loop between change options

and requirements. Hamraz et al. [27] presented a linkage

model called function-behavior-structure (FBS) model, which

includes a multi-domain matrix that consists of function,

behavior and component DSMs and domain mapping matri-

ces between these domains. Ahmad et al. [21] introduced a

multi-domain method that consists of requirement, function,

component and process domains to better identify propagation

paths by analyzing cross-domain dependencies.

MacCormack et al. [28], Milev et al. [29], Morkos and

Summers [30] and Baldwin et al. [6] analyzed the matrix

powers of component DSMs to consider indirect change

propagations between components. MacCormack et al. [28]

proposed two metrics, namely, propagation cost and clustered

cost, to analyze the inﬂuence of product architecture (modular

vs. integral) on change cost in the midst of design change and

propagation. Milev et al. [29] added another metric, relative

clustered cost, to compare systems of different sizes. Morkos

and Summers [30] used a DSM to investigate requirement

change propagation based on component dependencies. Ham-

raz et al. [31] introduced a method for obtaining a risk matrix

using simple matrix operations that avoid cyclic propagation

paths.

Another strand of studies used network analysis measures,

such as degree, distance, centrality, reachability and cluster-

ing coefﬁcients, to analyze the tendency of a component to

propagate changes to other system components via direct and

indirect dependencies. For instance, Sosa et al. [8] used three

centrality-based metrics to evaluate component modularity and

suggested that more modular components are better redesign

candidates. Cheng and Chu [3] used network measures to form

a reachability-vs.-distance plot of components for change risk

assessment. Wang and Wang [32] applied network measures

to a source code requirement dependency network to predict

the number of bugs in software systems. Gifﬁn et al. [33]

analyzed change requests using graph theory measures and

searched for network motifs of changes connected by parent-

child or sibling relationships. They classiﬁed components as

acceptors of changes and reﬂectors of changes based on graph-

theoretical indices. Pasqual and de Weck [34] proposed using

network methods and metrics with a change prediction model

and a multi-domain structure composed of product, change

and organizational domains for the analysis and management

of change propagation.

A few recent studies have paid particular attention to inter-

component design dependency cycles and their impact on

system or component performances. For instance, Sosa et al.

[5] found empirical evidence of components becoming fault-

prone if they are involved in cyclical design dependency paths

in software systems. Baldwin et al. [6] visually revealed hidden

IEEE SYSTEMS JOURNAL 3

cyclical cores and multi-core structures in software systems by

using network analysis methods. Luo [7] used simulations to

show that inter-component design dependency cycles in system

architecture give rise to product evolvability, i.e., the ability

of a system design to further evolve for improved system

performances.

Likewise, dependency cycles also concern the system of

systems in different domains. For example, Chai et al. [9]

focused on the interdependency of infrastructure systems in a

cyclically dependent network structure to assess the resiliency

of the network in the case of the failure of a speciﬁc system.

Wang et al. [10] formulated a cyclic dependency structure

for the supply chain networks to assess the networks overall

health. Luo and Triulzi [11] found the participation of verti-

cally integrated ﬁrms in inter-ﬁrm transaction cycles gives rise

to their innovation performances.

In sum, prior studies have focused on quantifying the

changeability of components by analyzing the propagation

of changes through direct or a small number of indirect

inﬂuence steps among the components. The core of such

efforts is the development of indices, which may be used

to identify and classify the change propagation behaviors

of components, such as multipliers, absorbers, carriers and

constants [20]. However, the inﬁnite regress nature of change

propagation has not been formally modeled in the literature.

No methods or indices based on inﬁnite regress propagation

have been developed to inform and support system design.

This paper aims to address this gap by directly modeling the

inﬁnite regress phenomenon of inter-component design change

propagation to derive the indicators on the overall inﬂuences

and susceptibility of individual components to support system

design decisions.

III. INFI NI TE RE GR ES S MOD EL (IRM) OF CHANGE

PROPAG ATIO N

From a network perspective of inﬂuence, if a component

inﬂuences the components that are themselves inﬂuential, this

will make the component more inﬂuential than inﬂuencing less

inﬂuential components. Similarly, if a component is inﬂuenced

by components that are themselves more susceptible to the

design changes of other components, this will make the

component more susceptible than being inﬂuenced by less

susceptible components.

A. Inﬁnite regress model for the inﬂuence and susceptibility

of system components

To model the inﬂuences of components, we begin by esti-

mating the inﬂuences of individual components simply based

on the sum of direct inﬂuences and then use them as weights of

the inﬂuences from respective components, following the logic

that the inﬂuences from more inﬂuential components are more

inﬂuential. Then, such estimates are improved by iterating for

n runs if there are n components in total.

Suppose that Ais the design structure matrix (DSM). Its

element ai,j denotes the strength of design dependency of

component ion component j, and implies that one unit of

change in the design of component jmay require ai,j amount

of design change in component i. On this basis, assume that

riis the vector of inﬂuence indicators of the components of

a system and that the indicator vector ri+1 is updated as the

sum of initiating-to-affected connections (i.e., columns of the

DSM A) weighted by the normalized rifrom the previous

iteration, i.e., ATriwhere ATis the transpose of A. In this

manner, the direct inﬂuence from a component with a higher

inﬂuence score receives a larger weight than the inﬂuence from

a component with lower inﬂuence scores. For the inﬂuence

scores in rto be used as weighting factors, they are normalized

to have an average of 1 and thus take the form

ri+1 =ATri

eTATri/n (1)

where nis the total number of components and eis the

row summation vector (ei= 1 for all i). The denominator

represents the average inﬂuence score of all components.

Therefore, the estimation of component inﬂuence indicators

can be improved through iterations (of weighting) to inﬁnity

(i→ ∞). The indicator will converge to the eigenvector of

the transpose of the DSM A. That is, the inﬂuence indicator

vector can be calculated as np/(pTe)with

ATp=λp(2)

where pis the eigenvector corresponding to an eigenvalue of

matrix AT. According the Perron-Freobenious theorem, if we

wish pto be non-negative, λmust be the largest eigenvalue and

pis the corresponding eigenvector. Therefore, the elements

of np/(pTe)indicate the total inﬂuences of corresponding

components to the overall system of directly or indirectly

dependent components, including themselves.

Similar procedures can be applied for modeling and deriving

component susceptibility indicators. Assume that siis the

vector of susceptibility indicators of the components of a

system. The indicator vector si+1 is updated as the sum of

affected-by-initiating connections (i.e., rows of A) weighted by

the normalized sifrom the previous iteration, i.e., Asi. In this

manner, the direct inﬂuence from a component with a higher

susceptibility score receives a larger weight than the inﬂuence

from a component with a lower susceptibility score. For the

susceptibility scores in the vector sto be used as weighting

factors, they are normalized to have an average of 1 and thus

take the form

si+1 =Asi

eTAsi/n (3)

where nis the total number of components and eis the row

summation vector (ei= 1 for all i). The denominator is the

average susceptibility score of all components.

Therefore, the estimation of component susceptibility indi-

cators can be improved through iterations to inﬁnity (i→ ∞).

The indicator will converge to the eigenvector corresponding

to the dominant eigenvalue of A. That is, the susceptibility

indicator vector is calculated as nq/(qTe)with

Aq =λq(4)

where qrepresents the eigenvector corresponding to the dom-

inant eigenvalue of matrix A. Therefore, the elements of

IEEE SYSTEMS JOURNAL 4

nq/(qTe)indicate the total susceptibility of the corresponding

components from the inﬂuences of all inter-dependent compo-

nents, including themselves.

In brief, with modeling the inﬁnite regress of inﬂuences

throughout the inter-component design dependency network,

accounting for every possible path in the network including

cycles and self-dependencies, the overall inﬂuence and sus-

ceptibility of the components in a system can be indicated by

the elements of the eigenvectors corresponding to the dominant

eigenvalues of ATand A, respectively, where Ais the design

dependency matrix.

B. Alternative modeling and derivation of the eigenvector-

based indicators

Here, we provide an alternative formulation of the inﬁnite

regress of inter-component change propagation. Assume that

the inﬂuence score of a component can be calculated as a

weighted sum of the inﬂuence scores of those it inﬂuences.

The inﬂuence of each of those components, in turn, is a

weighted sum of the components that they further inﬂuence

ad inﬁnitum, following the network paths. For a network of n

components, the inﬂuence index of a component {p(i)}(i=

1,2, , n)can be formulated as follows:

λp(1) = a11p(1) + a21 p(2)+ · · · +ak1p(k) + · · · +an1p(n)

λp(2) = a12p(1) + a22 p(2)+ · · · +ak2p(k) + · · · +an2p(n)

. . .

λp(k) = a1kp(1) + a2kp(2)+ · · · +akk p(k) + · · · +ankp(n)

. . .

λp(n) = a1np(1) + a2np(2)+ · · · +akn p(k) + · · · +annp(n)

(5)

where λis a scaling constant and aki is the direct design

dependency of component kon component iand indicates the

direct design inﬂuence of component ion component k. The

constant is required such that the equations have a nonzero

solution [35]–[37]. The system of equations (5) above can be

rewritten as

ATp=λp(6)

where Ais the DSM and pis the vector of inﬂuence indices

of n components, [p(1), p(2), . . . , p(n)]. The solution for p,

the elements of which are component inﬂuence scores, is

the eigenvector corresponding to the dominant eigenvalue of

matrix AT, to ensure the non-negativity of p.

Similarly, the susceptibility score of a component can be

calculated as a weighted sum of the susceptibility scores of

the components that can inﬂuence it. For those components,

the susceptibility of each, in turn, is a weighted sum of

the susceptibility scores of the components that can further

inﬂuence them, ad inﬁnitum, following the network paths.

Therefore, for a network of ncomponents, the susceptibility

index of a component {q(i)}(i= 1,2, , n)can be formulated

as follows:

λq(1) = a11 q(1) + a12q(2)+ · · · +a1kq(k) + · · · +a1nq(n)

λq(2) = a21 q(1) + a22q(2)+ · · · +a2kq(k) + · · · +a2nq(n)

. . .

λq(k) = ak1q(1) + ak2q(2)+ · · · +akk q(k) + · · · +akn q(n)

. . .

λq(n) = an1q(1) + an2q(2)+ · · · +ankq(k) + · · · +ann q(n)

(7)

where λis a scaling constant and aki is the direct design

dependency of component kon component iand indicates

the direct design inﬂuence of component i on component k.

The system of (7) above can be rewritten as

Aq =λq(8)

where Ais the DSM and qis the vector of susceptibility scores

of ncomponents, [q(1), q(2), , q(n)]. The solution for q,the

elements of which are component susceptibility scores, is the

eigenvector corresponding to the dominant eigenvalue of A.

Taken together, the eigenvector corresponding to the dom-

inant eigenvalue of ATprovides the indicators of the to-

tal inﬁnite regress inﬂuences of respective components, and

the eigenvector corresponding to the dominant eigenvalue

of Aprovides the indicators of the overall inﬁnite regress

susceptibility of respective components. Therefore, we have

derived the same eigenvector-based indicators for inﬂuence

and susceptibility of components as those in section III-B. The

eigenvector-based indicators can be easily acquired by basic

linear algebra operations of the design dependency matrix.

C. Illustration of the use of the IRM

We use a randomly generated artiﬁcial DSM (Fig.1), rep-

resenting a complex system with 20 components and their

intricate dependencies, to demonstrate the IRM-based analysis.

Given this speciﬁc DSM, we follow the IRM to calculate the

eigenvector-based scores for the inﬁnite regress inﬂuence and

susceptibility of each of the 20 components. These scores

can be further used to rank the components in terms of

their inﬂuence and susceptibility throughout the system. The

component scores and rankings are reported in Table I.

Fig.2 presents a scatter plot of the 20 components according

to their inﬂuence and susceptibility rankings. Components

are denoted by the same IDs in Fig.2 and Table I. Such a

plot can be used to classify the components as absorbers,

carriers, constants, and multipliers deﬁned in the taxonomy

by Eckert et al. [20] for system design decision support

. For instance, the components in the upper left quadrant

are highly susceptible to changes but not inﬂuential. They

are absorbers. The upper right quadrant contains the carriers

that are both highly inﬂuential in introducing changes and

susceptible to changes propagated to them. Carriers are critical

for system design stability and demand special attention to

the designs of these components themselves as well as their

interfaces with other components. The bottom left quadrant

contains components with relatively low susceptibility and low

IEEE SYSTEMS JOURNAL 5

Fig. 1. An artiﬁcial system DSM.

TABLE I

EIGENVECTOR-BASED COMPONENT INFLUENCE AND SUSCEPTIBILITY

SCORES AND RANKINGS BY THE IRM

Component Inﬂuence

Score

Inﬂuence

Rank

Susceptibility

Score

Susceptibility

Rank

1 2.07 1 2.32 1

2 0.35 17 0.45 15

3 1.90 3 1.29 9

4 0.99 10 0.20 17

5 1.29 6 0.81 11

6 0.16 19 1.74 4

7 1.21 7 0 20

8 0.86 12 1.08 10

9 1.10 9 0.51 13

10 1.66 4 0.48 14

11 0.79 13 2.06 2

12 0.58 15 1.62 5

13 0.93 11 1.34 7

14 0 20 1.89 3

15 0.77 14 0.09 19

16 1.47 5 0.17 18

17 1.15 8 0.79 12

18 0.44 16 1.33 8

19 2.02 2 1.41 6

20 0.28 18 0.43 16

inﬂuence. These components are constants and stable midst

design changes. They are not generally affected by changes in

other components, and changes in these components are not

likely to propagate to other components as well. Therefore,

constants are good candidates for implementing changes for

further design improvements. Components in the lower right

quadrant are multipliers that are inﬂuential but not susceptible.

The design changes of these components propagate changes to

others, but they themselves are relatively resistant to changes

propagating in the remainder of the system. The multipliers

are relatively enduring against design changes propagating in

the system. However, when a change is propagated from the

multipliers, the systemic impact of the change in the system

is relatively high comparing to absorbers and constants.

On this basis, system designers may consider different

design strategies for different types of components, e.g.,

multipliers, carriers, absorbers and constants. For instance,

Fig. 2. Distribution of components by their susceptibility and inﬂuence

rankings in absorber, carrier; constant and multiplier quadrants

several studies using CPI [15] have suggested multipliers as

the potential candidates for embedding ﬂexibility since such

components have a higher potential of causing changes in the

rest of the system than other components [16], [17]. Likewise,

the absorbers should be designed with high resilience or

robustness as they are likely to be affected by design changes

propagated from other components in the systems. The system

designers may prioritize the redesigns of the constants that can

enhance system performance, since their designs are resistant

to external changes in the system, and their redesigns propa-

gate limited inﬂuences on other components. Such isolation

suggests inexpensive and rapid redesigns and autonomous

innovation opportunities in such components. In contrast, the

carriers with both high inﬂuence and susceptibility are the

riskiest components in change propagation and deserve careful

design considerations in terms of system architecture and inter-

component interfaces. Their designs are expected to be ﬂex-

ible, robust and resilient. Furthermore, system designers may

simplify or decouple the interfaces of multipliers, absorbers,

or carriers with the rest of the system, in order to limit design

change propagation from, to, or through them.

In the following, we illustrate two examples of applying the

inﬁnite regress model (IRM) to analyze component inﬂuence

and susceptibility based on empirical DSMs for hardware and

software systems.

IV. CAS E EXA MP LE S

A. Pratt & Whitney Jet Engine

The DSM of the Pratt & Whitney jet engine is from a prior

study published by Sosa et al. [8], [38]. The DSM reveals all

direct design dependencies between 54 major components in

8 subsystems, including fan, low-pressure compressor (LPC),

high-pressure compressor (HPC), combustion chamber (CC),

high-pressure turbine (HPT), low-pressure turbine (LPT),

mechanical components (MCH), and externals and controls

(EXT). The MCH and EXT are considered integrative sub-

systems because of the physically distributed and functionally

IEEE SYSTEMS JOURNAL 6

Fig. 3. Pratt & Whitney jet engine DSM

Fig. 4. Distribution of Pratt & Whitney jet engine components by their susceptibility and inﬂuence in the absorber, carrier, constants and multiplier quadrants.

IEEE SYSTEMS JOURNAL 7

integrative features of their components. The remaining 6 sub-

systems are relatively modular and the design dependencies

of their components of such subsystems are mostly within the

respective subsystems.

Sosa et al. [8], [38] conducted intensive interviews with the

engineers working on all components of the system to collect

the information on the type (e.g., spatial, structural, material,

energy, information) and criticality of design interface between

components. They further used such information to analyze

the strength of the nonzero design interfaces and determined

319 weak interfaces and 250 strong interfaces. We mapped

their design interface strength data into a design dependency

matrix (see Fig. 3), in which a nonzero off-diagonal entry is

given the value of 1 or 2 if the design dependency between

the corresponding pair of components is weak or strong

respectively.

Based on this DSM, we again calculated the IRM-based

inﬂuence and susceptibility scores and rankings of the 54

components and use the results to identify them as absorbers,

carriers, constants or multipliers in Fig. 4. The plot reveals

that most components of the jet engine are either carriers or

constants. On one hand, 22 components appear to be highly

susceptible and inﬂuential carriers. 9 out of 10 components in

the External and Controls subsystem are carriers, attributing

to the physically distributive but functionally integrative nature

of this subsystem. These components should be designed to be

ﬂexible to mitigate change costs. System designers may also

consider decoupling them from other components via interface

redesign and standardization, or rerouting energy, material or

information ﬂows to bypass these components.

On the other hand, 22 components are in the quadrant of

constants. Their designs are relatively resistant to changes

initiated in other components of the system, and their own

design changes also propagate limited inﬂuences on other

components. Due to the isolation, such components are good

candidates for inexpensive redesigns and modular innovations

in themselves. For example, Sosa et al. [8] reported 70% of

the total redesigns of the Pratt & Whitney engine took place

in the fan subsystem, in which 5 out of the total 7 components

are in the quadrant of constants. Redesign costs for constants

are lower than for the carriers and multipliers.

B. Linux Kernel

Fig. 5 shows the DSM for the historically ﬁrst release of the

Linux kernel. Linux kernel is an open source software system

developed by self-organized contributors around the world. Its

ﬁrst version was created and released by Linus Torvalds in

year 1991. In this case, the DSM is indeed the function call

graph of the 35 components, excluding the 10 isolated code

ﬁles. The matrix entry ai,j is 1 if code ﬁle i calls for the output

of code ﬁle jas its input, i.e., code ﬁle idepends on jand

jmay inﬂuence i. The code-to-code dependency data were

extracted using software architecture analysis tool Understand

and ﬁrst published by Luo and Magee [39]. In contrast to the

DSM for Pratt & Whitney jet engine based on interview data

and estimates, the design dependency data based on inter-code

function calls are objective and physical.

Fig. 5. DSM for Linux kernel version 0.01

Fig. 6. Distribution of Linux kernel components by their susceptibility and

inﬂuence in the absorber, carrier, constant and multiplier quadrants

Based on the DSM for Linux kernel, we again calculated

the inﬁnite regress inﬂuence and susceptibility scores of the 35

components and use the results to identify them as absorbers,

carriers, constants or multipliers in Fig. 6. Compared to the

foregoing jet engine case, the Linux kernel components are

more spread out in the whole susceptibility-inﬂuence space.

The plot reveals 8 absorbers, 9 carriers, 10 constants and 8

multipliers. For managing change propagation from the overall

system perspective, system designers may pay special attention

to the 9 highly inﬂuential and highly susceptible carriers for

system architecture design consideration.

For example, component 3 among the carriers can be

redesigned with increased ﬂexibility in its codes structure

IEEE SYSTEMS JOURNAL 8

to adapt easily (without signiﬁcant redesigns of its own) to

external changes. Given that components 24 and component

25 are highly inﬂuential multipliers and have direct interface

with the highly susceptible component 3, open programming

interfaces can be developed for component 3 to decouple the

redesign and development efforts of components 24 and 25

from its own. For components 24 or 25 and other multipliers,

system designers may consider decomposing them into more

than one component to reduce the probability and impact of

potentially harmful change propagation. Their designs are also

expected to be ﬂexible and thus reduce the needs for redesigns

that will propagate changes to the rest of the system. For the

highly susceptible absorbers, e.g., component 14, their designs

need to be robust to external changes. In contrast, system

designers may encourage redesigns of the 10 constants, e.g.,

component 35, which can upgrade the system performances or

functionality without incurring signiﬁcant system-wide change

costs.

Note that, some of the components of the Linux kernel are

not involved in cycles but in the middle of sequential depen-

dencies, the IRM may give inaccurate results. For example,

when component idepends on component jwhich depends

on component k, but components i,jand kare not in any

cycle and iis a sink component without outgoing inﬂuence

links, the IRM will give a zero score for the inﬂuence of

component j. This is not accurate because jis susceptible to

the design change of kand also propagates design change to i.

To overcome this problem, we create a dummy component that

has a mutual change propagation relationship with every actual

component that is not involved in any cycle. The cyclic links of

the dummy component are extremely weak (e.g., a value close

to 0) to assure that the change propagation behavior of the

overall system is not affected. This method is also applicable

to other systems that have components not involved in cyclic

dependencies.

V. DISCUSSION

The foregoing case examples ﬁrst suggest the data require-

ment ﬂexibility and generality of the IRM and eigenvector-

based indicators. In these cases, DSM entry values are ratios,

integers and binary, respectively. The DSMs include cyclic

dependencies and non-cyclic ones. These DSM data-forms

together can represent a variety of complex system contexts. In

other words, IRM is applicable to a wide spectrum of complex

systems (with theoretically possible inﬁnite regress inﬂuences

among non-linearly coupled components or subsystems), and

contrasts with other eigenvalue-eigenvector analyses that have

speciﬁc restrictions on the matrix and entry values [4] and the

change propagation methods that only concern the propagation

within a few linear and non-cyclic steps [1].

These case studies also suggest that, by analyzing the

eigenvector-based component inﬂuence and susceptibility indi-

cators, system designers may determine different types of com-

ponents, e.g., carriers, constants, multipliers and absorbers, for

different component design and interface design considerations

to manage change propagation throughout the system. For

instance, multipliers and carriers should be designed to embed

ﬂexibility to ease redesigns, and absorbers and carriers should

be designed to be robust or resilient to deal with external

changes. Interface decoupling should be considered for the

multipliers, absorbers, or carriers, in order to mitigate change

propagation from, to, or through them. The carriers with both

high inﬂuence and susceptibility are the riskiest components

in change propagation and deserve careful considerations in

system design for change propagation management. In con-

trast, constants can be prioritized for isolated redesigns and

autonomous innovations to enhance system improvements.

In addition to classifying components, one can also derive

an understanding of the change propagation proﬁles of dif-

ferent systems as a whole. For instance, Fig. 4 for the jet

engine shows clear concentrations of components as either

carriers or constants, whereas Fig.6 for Linux kernel presents a

rather spread-out distribution of components in terms of their

inﬂuence and susceptibility. With such analysis and results,

system designers may consider the architecture and interface

design choices that give rise to a higher concentration of

components toward the bottom left quadrant (i.e., constants) of

the two-dimensional distribution plot, in order to derive higher

system design stability and limit change propagation costs.

For example, given the result in Fig.2, reducing the design

dependency of component #19 on component #1 (which is

the most inﬂuential and susceptible component) or decouple

the interface between them may reduce the inﬂuence and

susceptibility of component #19 and thus move it toward the

bottom left.

In summary, complex system designers and managers may

ﬁnd a few reasons to use IRM for change propagation analysis

to support system design and management. First, the IRM

assesses the inﬂuence and susceptibility of components by

accounting for inﬁnite steps of change propagation via cyclic

or self-dependencies. The consideration of the effect of cycles

on change propagation is more necessary for complex systems

relying on cyclic dependencies. Second, the key operation of

the IRM is simply to ﬁnd only matrix eigenvectors corre-

sponding to the dominant eigenvalues and avoids the need to

trace and count speciﬁc component-to-component inﬂuences.

Third, it can be ﬂexibly applied to various types of design

dependency matrix data used to describe diverse real-world

systems. A system with a rather large number of components

and dependencies can be analyzed quickly to obtain an overall

assessment of the inﬂuence and susceptibility of components

as well as the risk plot. The ﬂexibility for data acquisition, the

simple linear algebra operation, and computational efﬁciency

of the IRM does not require specialty software tools for its

implementation.

In turn, the IRM may serve as the basis of a potential

computer tool to assess system architectures, the criticality

of the components, and systems change propagation proﬁles,

and suggest proactive design and redesign actions accordingly.

Moreover, although our framing and case examples in the

present paper are focused on a system and its components,

the IRM is naturally suitable for the analysis of system-of-

systems, such as the infrastructure systems [9] and the supply

chain or transaction networks of ﬁrms [10], [11], in which the

components themselves are semi-independent systems.

IEEE SYSTEMS JOURNAL 9

VI. CONCLUDING RE MA RK S

This paper aims to contribute to the complex system design

and management literature by modeling the inﬁnite regress ef-

fects of component change propagation, deriving eigenvector-

based component inﬂuence and susceptibility indicators, and

generating results that are useful for engineering system design

and management practices. The IRM presents advantages in

terms of its consideration of inﬁnite regresses change propaga-

tion, computational simplicity and data requirement ﬂexibility.

The IRM is an addition to the prior methods that assessed only

a limited number of steps of propagation.

However, in contrast to some of the prior methods that

can analyze the absolute level of design changes from one

component to another, the IRM only reveals the relative

inﬂuence and susceptibility of the components in a system.

Furthermore, this paper presents only three cases examples to

demonstrate the applications and utility of the IRM. For future

research, we plan to apply and test the IRM in additional

empirical contexts, for systems of diverse types and scales,

and compare it with other methods in the change propagation

literature to develop a more systematic understanding of its

strengths, weaknesses, and use conditions. In general, we hope

that the readers view this paper as an invitation for further

research, methodological improvement, and applications of the

IRM considering inﬁnite regress change propagation.

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Serhad Sarica received his BSc and MSc in Elec-

trical&Electronics Engineering from Middle East

Technical University in 2007 and 2011 respectively.

During the period 2007-2016, he worked as a senior

system designer in Aselsan Co., Turkey, where he

involved and led several naval communication sys-

tem design projects. He is currently pursuing the

PhD degree with the Engineering Product Devel-

opment Pillar, Singapore University of Technology

and Design (SUTD), Singapore. His current research

interests include change propagation in complex

systems, semantic relations in technology and innovation space, and utilization

of NLP methods for engineering design ideation.

Jianxi Luo is director of the Data-Driven Innovation

Lab at SUTD (http://ddi.sutd.edu.sg). He holds B.S.

and M.S. degrees in Mechanical Engineering from

Tsinghua University and a S.M. degree in Tech-

nology Policy and a Ph.D. degree in Engineer-

ing Systems from Massachusetts Institute of Tech-

nology (MIT). He is currently assistant professor

of engineering product development at Singapore

University of Technology and Design (SUTD) and

the associate director of the SUTD Technology

Entrepreneurship Programme (STEP). His research

is focused on developing artiﬁcial intelligences and data science methods

and tools to enhance innovation in engineering. His teaching is focused on

entrepreneurship and innovation. He was a faculty member at New York

University, a visiting scholar at Columbia University and the University of

Cambridge, and a chair emeritus of the Technology, Innovation Management

and Entrepreneurship Section of INFORMS. In practice, he is a co-founder

and advisor of several startups, an innovation consultant and public speaker.