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In complex systems, design changes of one component 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.
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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.
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Complexity, cohesion and coupling have been recognized as prominent indicators of software quality. One characterization of software complexity is the existence of dependency relationships. Moreover, the degree of dependency reflects the cohesion and coupling between software elements. Dependencies in the design and implementation phase have been proven to be important predictors of software bugs. We empirically investigated how requirements dependencies correlate with and predict software integration bugs, which can provide early estimates regarding software quality and thus facilitate decision making early in the software lifecycle. We conducted network analysis on the requirements dependency networks of three commercial software projects. Significant correlation is observed between most of our network measures and the number of bugs. Furthermore, many network measures demonstrate significantly greater values for higher severity (or a higher fixing workload). Afterward, we built bug prediction models using these network measures and found that bugs can be predicted with high accuracy and sensitivity, even in cross-project and cross-company contexts. We further identified the dependency type that contributes most to bug correlation, as well as the network measures that contribute more to bug prediction. These observations show that the requirements dependency network can be used as an early indicator and predictor of software integration bugs.
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