Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 81–86___________________________________________________________________________________________________________

Fragility in Network Systems:
An Empirical Investigation
Tsutomu Ito*, Katsuhiko Takahashi, Katsumi Morikawa, Takao Ito

Graduate School of Engineering, Hiroshima University,1-4-1 Kagamiyama,
Higashi-Hiroshima, 739-8527, Japan
E-mail: fw.eldorado.500cuin@gmail.com
Rajiv Mehta

School of Management, New Jersey Institute of Technology, University Heights,
Newark, New Jersey, 07102-1982, U.S.
Seigo Matsuno

Department of Business and Administration, Ube National College of Technology, Tokiwadai 2-14-1
Ube, Yamaguchi, 755-0096, Japan
Makoto Sakamoto

Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai-Nishi,
Miyazaki, 889-2192, Japan

Abstract
A survey of the relevant literature reveals that a plethora of network analytic facets, such as degree, eigenvalue,density, block, cluster, have been developed and employed to further our understanding of network structures. To
extend our understanding of network systems, additional dimensions need to be identified that shed light on the
dynamic processes among individual member nodes within network structures. Within the context of network
systems, this paper proposes a new concept of fragility. More specifically, it develops and empirically tests amathematical model of fragility from the standpoint of how ties among network members significantly influence
corporate performance, thus uniquely contributing to extant knowledge. Using data drawn from two well-known
network organizations, Mazda and Toyota, this research attempts to shed light on the relationship between degreeand fragility, to confirm the validity of the new concept as well as enabling a contrast of Mazda and Toyota’s
network structures. Based on the findings, the managerial implications are discussed, the study limitations are
identified and directions for further research are suggested.
Keywords: degree, fragility, Keiretsu, relationship, Yokokai.

organizations, which play a dominant role in theJapanese economy, researchers have not devoted muchattention on conducting empirical investigations toincrease our understanding of these networks. Whilestudies on Keiretsu have garnered greater attention andempirical studies have started to emerge in scholarly
research journals, still these investigations can be

1. Introduction
As in sociology and psychology, many mathematicalmodels have been developed to increase ourunderstanding of the organizational sciences. For
instance, relative to the prominence of Keiretsu

Copyright © 2017, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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considered fragmentary and sparse [1-2]. Being an
additional organizational form within the context of
networked structures, more recently dimensions, such as
momentum [3], centrality [4-5], transactional
relationships [6-7], capital relationships [8-9], Euclidean
distance [10], cross shareholdings [11], and capacity [5],have revealed the static nature of relationships amongindividual actors relative to other members within thewhole network. However, dynamic processes are
considered to be even more salient in the organizational
sciences and allied fields. A review of the literaturesuggests that newer models and paradigms illustrating
the dynamic processes between network participants are
more desirable. Accordingly, this paper proposes,
measures and illustrates how the new concept offragility, which is theoretically grounded as well as
rooted in a systematic repeated trial and error
assessment, has an impact on degree and corporate
performance.
Drawing data from two well-known networkorganizations, Mazda’s Yokokai and Toyota’s
Kyohokai from 2004 to 2007, the purpose of thisresearch endeavor is to calculate and ascertain therelationship between fragility, degree and sales, thus
confirming the validity of new model. Further, therelationshipbetweenfragilityandcorporate
performance is also assessed, which forms the basis of
the managerial implications discussed in the manuscript.
More specifically, this paper makes a uniquecontribution to extant thought by: 1) Defining theconcept of fragility, 2) Discussing the nature of therelationship between degree and corporate performance,and 3) Empirically testing the dimensional differences
between fragility and corporate performance. Thus, thisempirical investigation enables Mazda’s networked
organization to be compared and contrasted with
Toyota’s network constellation.
This paper is structured as follows. Section 2 reviews
the literature focusing on network analysis. In section 3,
the paper explicates the calculation of degree andfragility. Section 4 discusses our findings based on
which the study limitations are identified and directions
for further research are proffered in the final section.

Simply put, degree is defined as the number of links
incident upon a node in a graph. In real society, most of
the relationships between members are considered to be
mutual. Furthermore, nodes within a network are known
to interact with each other. Consequently, direction withweight have been commonly employed to analyze
different phenomena within social networks. In thiscontext, Freeman proposed a new index of entire
degree, which identifies the centrality of the wholenetwork [13].
More recently, Ito and Sakamoto proposed a new
approach to identify the importance of each individual
node based on Freeman’s model [14], but noted thatmuch more research is necessary to fully understandkeiretsu constellations. Although the relationships
between network members can be manifest in the formof equity ties, personal ties, transactional ties andworkflow ties, this study examines transactions within a
network to reveal the nature of the relationships
between business ties and corporate performance. Morespecifically, owed to the sparsity of knowledge, thisinvestigation contributes to the literature by advancing a
new procedure for measuring the interrelationships
between members of a keiretsu by suggesting the newconcept of fragility to be a determinant of corporate
performance.
3. Research Method and Hypotheses
Two concepts are applied in this paper: degree andfragility.
3.1. Degree and Fragility
Degree, as one of the basic indices of centrality, isconsidered as the basic index in network analysis. It canbe calculated as follows [12].
n

C D ( pk ) = ∑ a( pi , pk )

(1)
i=1
where
i≠k;
a (pi, pk) = 1 if and only if pi and pk are connected by aline; (the percentage data are adopted in this paper.)a (pi, pk) = 0 otherwise.
In an asymmetric network, two indexes of out-degree
and in-degree should be calculated.
In transactional network, the value of degree is thepercentage of transaction of parts maker i with any otherfirms. Accordingly, out-degree means parts maker isells parts to other firm j and/or car makers, and indegree refers to parts maker i purchases parts from other

2. Background and Literature Review
As noted earlier, indexes, such as momentum,
centrality,transactional
relationships,
capitalrelationships, Euclidean distance, cross shareholdings,
and capacity, have been used widely in empiricalresearch to measure different dimensions oforganizational networks. Comprising a most basic facetof centrality, degree was firstly proposed by Nieminen[12] in his research on organizational constellations.

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H3: Higher levels of fragility is associated with lower
sales.

firm k and/or car makers. For instance, NSK Ltd., anenterprise specializing producing bearing, purchases
parts of anti-vibration rubber from Kurashiki Kako Co.,
Ltd., and sells its products to car makers. Therefore,
out-degree is associated with sales while in-degree isassociated with sales indirectly. Thus, selling parts tokeiretsu members will increase sales revenues whilebuying parts from other firms in the same keiretsu will
be associated with greater sales indirectly. Accordingly,
the following hypotheses are posited.

3.2. Data Collection
Widely considered as successful examples of prominent
Japanese networked organizations, data were drawn
from Toyota’s Kyohokai and Mazda’s Yokokai from2004 to 2007 to establish the status quo of keiretsu as
well as longitudinally ascertain changes in their keiretsu
structure.
Japanese Keiretsu (e.g., Toyota, Nissan, and Mazda’s
networks) are known to comprise multifarious
members. It is widely acknowledged that firms are
reluctant to share financial information about
themselves and their members due to confidentialconsiderations that may have competitive implications.
Moreover, key informants from the focal and member
firms have to painstakingly research and provide
relevant data from their management information
systems, which is extremely time consuming andexpensive. Further complicating these matters is thatthe data have to be cross-validated from other member
firms within each respective network. Moreover, thelongitudinal data were even cross-validated from on-line
financial databases, thus enhancing the reliability of thedata-set. As a result of this albeit onerous task, acomplete set of valid, reliable and useable data were
only available for the specified 2004-2007 period fromToyota’s Yokokai and Mazda’s Kyohokai.
Both of these keiretsu organizations include singletons,
which refers to a partner firm in the keiretsu that has no
relationship with other member firms. However,singletons were removed from the data-set because they
have no impact on the calculation of network indexes.
Data on Toyota’s Kyohokai and Mazda’s Yokokai from2004 to 2007is reported in Table 1.

H1: Higher levels of out-degree is associated withincreased sales.
H2: Higher levels of in-degree is associated withincreased sales.
Fragility is a physical term that characterizes how
rapidly the dynamics of a material slow down as it iscooled toward the material transition [15]. Accordingly,
fragility, which is defined as the ratio of the entire
degree of and the entire degree after moving a specificnode, is mathematically expressed as:−F( p) =CD ( p) −CD
CD

(2)
where
CD: Entire degree of a given network;CD( p− ): Entire degree after removing node p.
The equation of the entire network is defined as below[13].
n∑ [C (P*) − C (P )]C D = i =1

where

D

D

i

(3)

n − 3n + 2
2

C D ( p*) = max C D ( pi )Fragility could be considered as a concept of structural
importance. High fragility means the entire degree after
moving a specific node i is larger than the entire degree.
Node i is a weak point if the entire degree after moving
the node i is larger than the entire degree including thenode i. Therefore, if a member of the keiretsu, which isconsidered to be vulnerable, is removed from theconstellation, the value of degree of the keiretsu isreduced, which, in turn, can negatively impact sales
revenues. Thus, the following hypothesis is proposed:

Table 1. Firms in Yokokai and Kyohokai.
Firm in Yokokai

83

Firm in Kyohokai
Singleton
Total

Total

Singleton

2004

188

97

215

93

2005

191

89

216

97

2006

190

92

213

93

2007

189

104

213

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 81–86___________________________________________________________________________________________________________

Table 2. Results of out-degree and in-degree-sales regression model.
ModelsYokokai
Sales
Out-degree
Partial regression coefficientStandard coefficientt value
Probability
Correlation coefficientPartial correlation coefficient
In-degree
Partial regression coefficientStandard coefficientt value
Probability
Correlation coefficientPartial correlation coefficient
Coefficient of determination (R2)
Multiple correlation coefficient
F value
Degree of freedom
AIC
DW ratio
Data number

2004

2005

2006

2007

2004

Kyohokai
2005
2006

2007

5153.172
0.06450.86110.3919-0.4908
0.09837124.583
0.0904
1.49570.1384-0.3858
0.1601-8079.95-0.0964-1.04250.3003-0.4298-0.1151-15695.4-0.1447-1.59710.1146-0.3185-0.8137-7537.79
0.1133-2.7502
0.007-0.4726-0.2615-4468.46-0.0624-1.5524
0.1238-0.4668-0.1557-3779.68-0.0471-1.2655
0.2087-0.4388-0.1262-3619.42-0.0389-0.9838
0.3277-0.448-0.101514162.85
0.896811.9655
00.85690.80820.73677
0.85835
106.3622, 762441.18
1.9331797124.583
0.922115.2559
00.87540.85590.772310.87881144.1595
2, 852706.92
1.6531
889322.689
0.6465
6.991700.6962
0.61350.491550.7011139.15432, 812678.46
2.56648420726.5
0.61416.775900.655
0.62140.448360.669629.66672, 73
2466.462.4682766465.408
0.872
21.159200.91870.90160.85474
0.92452
303.038
2, 1033182.65
1.5609106

8166.002
0.905722.5426
00.93360.91640.874640.93522338.3998
2, 973007.131.507
1009597.788
0.921124.734100.94110.92770.88753
0.94209390.6121
2, 99
3078.191.4393
10211958.24
0.922
23.341500.93930.92420.88346
0.93992352.4917
2, 932929.59
1.381796

Table 3. Results of fragility-sales regression model.
ModelsYokokai
Sales
Fragility
Partial regression coefficientStandard coefficientt value
Probability
Correlation coefficientPartial correlation coefficient
Coefficient of determination (R2)
Multiple correlation coefficient
F value
Degree of freedom
AIC
DW ratio
Data number

Kyohokai
2006

2004

2005

2006

2007

2004

2005

-2286318-0.2154-1.9353
0.0566-0.2154-0.21540.04638
0.215373.745291, 772540.88
1.230279-2246640-0.1958-1.85130.0676-0.1958-0.1958
0.03833
0.19577
3.42737
1, 86
2831.71
1.115588-102605-0.0177-0.1603
0.873-0.0177-0.01770.000310.017700.02571, 822733.250.9671
84-1996633-0.1262-1.09450.2773-0.1262-0.12620.015930.12621
1.197871, 74
2508.45
1.1641
76-19942684.1-0.7684
12.24380-0.7684-0.76840.590410.76838
149.9108
1, 1043290.540.9158106-22446825.7-0.7656-11.7819
0-0.7656-0.7656
0.586170.76562138.81668
1, 98
3124.560.9119100

A diagrammatic representation of the inter-firmtransactional relationships in 2007 is illustrated in Fig.
1.

-26007968.4-0.7752-12.27050-0.7752-0.77520.60090.77518150.56441
1, 1003205.68
0.8992102

2007-29733934.04-0.7798-12.0763
0-0.7798-0.7798
0.608070.77979145.83724
1, 943044.020.905296

4. Results and Discussion
Using regression analysis, we first tested therelationship between out-degree and in-degree as
determinants of sales. The results of out-degree and indegree-sales regression model is shown in Table 2.
4.1. Sales and out-degree and in-degree
All the probabilities of in-degree are significant while
out-degree are not significant except for Kyohokai in
2004. Thus, evidently in-degree has a statisticallysignificant impact on sales. In contrasting the results of
the Yokokai, the correlation coefficients as well as thecoefficients of determination are higher. Thus, it can beinferred that the regression model of Kyohokai has

Fig. 1. Transaction network of Yokokai in 2007.

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6. Directions for Future Researchstronger power to explain the relationship between indegree and corporate performance—as measured bysales. Hence, the results with regard to H1 show that
there is no association between out-degree and sales
revenues. However, H2 is verified because there is astrong association between sales and in-degree.
Specifically, Table 2 indicates that the difference
between Yokokai and Kyohokai looks similar, butdifferent in-degree.

Based on the results obtained, four fiscal years of data is
not sufficient to support contentions of internal validity.
Because there is lack of support for H1 which posits apositive relationship between out-degree and sales,
additional research should be conducted to replicate thefindings, thus shedding greater light on the realassociation between these constructs. In addition todegree, other centrality indexes, such as closeness andbetweenness, should be tested as determinants ofcorporate performance. Furthermore, the original
definition of fragility in Physics is a derivative of amathematical function. Thus, in the future the models
tested in this study should be investigated using data
drawn from other settings, such as, information
technology, and the ship-building industry forcomparative research as well as replicating the findingsof this study.

4.2. Sales and Fragility
Fragility can also help distinguish the difference
between Yokokai and Kyohokai. The results offragility-sales regression model is shown in Table 3.
Compared with Yokokai, all the probabilities of fragility
from 2004 to 2007 are statistically significant, andcoefficients of determination are higher. Thus, themodel assumptions of Kyohokai hold, which confirms
H3. However, the findings with regard to Yokokai are
mixed. Specifically, the linkage between fragility andsales revenues is verified in only two of the four years.
This is because, all correlation coefficients are negative.
Thus, the assumption of higher fragility having an
inverse association with sales is partly confirmed.
Because the meaning of fragility denotes the value ofentire degree of a given network after removing a
specific node, it is evident that firms in Kyohokai have ahigher possibility to improve sales. Conversely, allcorrelation coefficients in Mazda are negative meansthat lower fragility is associated with higher sales—aspredicted. However, as the probabilities of each year arelarger than 0.05, additional longitudinal data should becollected.

Acknowledgements
This work was supported by JSPS KAKENHI Grant
Number 16K01243.
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Grounded in the review of the relevant literature, thispaper proposed a new approach called fragility to shedlight on interfirm behaviors within networkorganizations known as keiretsu. Data were drawn from
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