Measuring Fragility and its Implications in Network System

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. Specific to the context of networked systems, this paper proposes a new concept of fragility. Specifically, it develops and empirically tests a mathematical 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 degree and fragility, to confirm the validity of the new concept as well as enabling a contrast of Mazda and Toyota’s network structures. [RM1]Based on the findings, the managerial implications are discussed, the study limitations are identified and directions for further research are suggested.


Introduction
As in sociology and psychology, many mathematical models have been developed to increase our understanding of the organizational sciences.As an additional organizational form, within the context of networked structures, dimensions, such as degree, eigenvalue, density, block, and cluster, reveal the static nature of relationships among individual nodes and other members within the whole network.However, dynamic processes are considered to be even more salient in the organizational sciences and allied fields.Accordingly, as newer models and paradigms that illustrate dynamic processes are more desirable, this paper proposes the new P -549 concept of fragility that is grounded in the review of the relevant literature as well as rooted in a systematic repeated trial and error assessment.Drawing data from two well-known network organizations, Yokokai in Mazda and Kyohokai in Toyota, from 2004 to 2007, the purpose of this research endeavor is to calculate the relationship between degree and fragility to confirm the validity of new model.Further, the relationship between fragility and corporate performance, is also assessed that forms the basis of the managerial implications discussed in the study.Specifically, this paper makes a unique contribution to extant thought by: 1) Defining the concept of fragility, 2) Discussing the nature of the relationship between fragility and corporate performance, and 3) Empirically testing the dimensional differences between fragility and corporate performance, thus enabling Mazda's networked organization to be contrasted with Toyota.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 and fragility.Section 4 discusses our findings based on which the study limitations are identified and directions for further research are proffered in the final section.

Background and Literature Review
Indexes, such as degree, eigenvalue, density, block, and cluster have been used widely in empirical research to measure different dimensions of organizational networks.Comprising a most basic facet of centrality, degree was firstly proposed by Nieminen [1] in his research on organizational constellations.Simply put, it 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 with weight have been commonly employed to analyze different phenomena within social networks.In this context, Freeman proposed a new index of entire degree, which identifies the centrality of the whole network [2].More recently, Ito and Sakamoto proposed a new approach to identify the importance of each individual node based on Freeman's model [3], but noted that much more research is necessary to fully understand keiretsu constellations.Although the relationships between network members can be manifest in the form of equity ties, personal ties, transactional ties and work-flow ties, this study examines transactions within a network to reveal the nature of the relationships between business ties and corporate performance.More specifically, owed to the sparsity of knowledge, this investigation contributes to the literature by advancing a new procedure for measuring the interrelationships between members of a keiretsu by suggesting the new concept of fragility to be a determinant of corporate performance.

Research Method
Two concepts are applied in this paper: degree and fragility.

Degree and Fragility
Degree, as one of the basic indices of centrality, is considered as the basic index in network analysis.It can be calculated as follows [1].
where i≠j; a (pi, pk) = 1 if and only if pi and pk are connected by a line a (pi, pk) = 0 otherwise In an asymmetric network, two indexes of out-degree and in-degree should be calculated Basically fragility is a physical term, which characterizes how rapidly the dynamics of a material slow down as it is cooled toward the material transition.Accordingly, fragility is defined as the ratio of the entire degree of and the entire degree after moving a specific node.It will be illustrated as follows.
(2) where CD: Entire degree of a given network CD( p ): Entire degree after removing node p

Data Collection
Widely considered as successful examples of prominent Japanese networked organizations, data were drawn from Toyota's Kyohokai and Mazda's Yokokai from 2004 to 2007 to establish the status quo of keiretsu as well as longitudinally ascertain changes in their keiretsu structure.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 from 2004 to 2007is reported in Table 1.Table 1.Firms in Yokokai and Kyohokai.

Results and Discussion
Using regression analysis, we first tested the relationship between out-degree and in-degree as determinants of sales.The results of out-degree and in-degree-sales regression model is shown in Table 2.

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 statistically significant impact on sales.In contrasting the results of the Yokokai, the correlation coefficients as well as the coefficients of determination are higher.Thus, it can be inferred that the regression model of Kyohokai has stronger power to explain the relationship between indegree and corporate performance-as measured by sales.Table 2 indicates that the difference between Yokokai and Kyohokai looks similar, but different in degree.

Sales and Fragility
The results of out-degree and in-degree-sales regression model is shown in Table 3.
Compared with Yokokai, all the probabilities of fragility from 2004 to 2007 are significant, and coefficients of determination are higher.Thus, the model assumptions of Kyohokai hold.All correlation coefficients are negative.Thus, the assumption of higher fragility having an inverse association with sales is confirmed.Because the meaning of fragility denotes the value of entire degree of a given network after removing a specific node, it is evident that firms in Kyohokai have a higher possibility to improve sales.Conversely, all correlation coefficients are positive means that greater fragility is associated with higher sales-as predicted.However, as the probabilities of each year are larger than 0.05, additional longitudinal data should be collected.

Conclusions
Grounded in the review of the relevant literature, this paper proposed a new approach called fragility to shed light on interfirm behaviors within network organizations known as keiretsu.Data were drawn from Yokokai in Mazda and Kyohokai in Toyota to ascertain the relationship between degree and fragility.The results look similar between both the keiretsu when we investigate the relationships between degree and sales.However, different behaviors are manifest when fragility is viewed as a determinant of sales.

Directions for Future Research
Based on the results obtained, four fiscal years of data is not sufficient to support contentions of internal validity.
In addition to degree, other centrality indexes, such as closeness and betweenness, should be tested as determinants of corporate performance.Furthermore, the original definition of fragility in Physics is a derivative of a mathematical function.Thus, in the future the models tested in this study should be investigated using data drawn from other settings, such as, information technology, the ship-building industry for comparative research as well as replicating the current findings.

Table 2 .
Results of out-degree and in-degree-sales regression model.

Table 3 .
Results of fragility-sales regression model.