In this study, we investigate interfirm networks by employing a unique dataset containing information on more than 800,000 Japanese firms, about half of all corporate firms currently operating in Japan. First, we find that the number of relationships, measured by the indegree, has a fat-tail distribution, implying that there exist "hub" firms with a large number of relationships. Moreover, the indegree distribution for those hub firms also exhibits a fat tail, suggesting the existence of "super-hub" firms. Second, we find that larger firms tend to have more counterparts, but that the relationship between firms' size and the number of their counterparts is not necessarily proportional; firms that already have a large number of counterparts tend to grow without proportionately expanding it.

IWAMURA Mitsuru

SAITO (UMENO) Yukiko

WATANABE Tsutomu

English

Research Papers in Economics

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RIETI Discussion Paper Series 07-E-028
Do Larger Firms Have More Interfirm Relationships?
SAITO (UMENO) Yukiko
Fujitsu Research Institute
WATANABE Tsutomu
RIETI
IWAMURA Mitsuru
Waseda University
The Research Institute of Economy, Trade and Industry
http://www.rieti.go.jp/en/Do Larger Firms Have More Interﬁrm Relationships?
Yukiko Umeno Saito∗
Fujitsu Research Institute
Tsutomu Watanabe
Hitotsubashi University
RIETI
Mitsuru Iwamura
Waseda University
April 12, 2007
Abstract
In this study, we investigate interﬁrm networks by employing a unique dataset containing
information on more than 800,000 Japanese ﬁrms, about half of all corporate ﬁrms currently op-
erating in Japan. First, we ﬁnd that the number of relationships, measured by the indegree, has
a fat-tail distribution, implying that there exist “hub” ﬁrms with a large number of relationships.
Moreover, the indegree distribution for those hub ﬁrms also exhibits a fat tail, suggesting the
existence of “super-hub” ﬁrms. Second, we ﬁnd that larger ﬁr m st e n dt oh a v em o r ec o u n t e r p a r t s ,
b u tt h a tt h er e l a t i o n s h i pb e t w e e nﬁrms’ size and the number of their counterparts is not neces-
sarily proportional; ﬁrms that already have a large number of counterparts tend to grow without
proportionately expanding it.
Keywords: Network structure; interﬁrm relationships; hub ﬁrms; ﬁrm size; fat-tail distributions
1 Introduction
When examining interﬁrm networks, it comes as little surprise to ﬁnd that larger ﬁrms tend to have
more interﬁrm relationships than smaller ﬁrms. For example, Toyota purchases intermediate products
and raw materials from a large number of ﬁrms, located inside and outside the country, and sells ﬁnal
products to a large number of customers; it has close relationships with numerous commercial and
investment banks; it also has a large number of aﬃliated ﬁrms. Somewhat surprisingly, however, we
do not know much about the statistical relationship between the size of a ﬁrm and the number of its
relationships. The main purpose of this paper is to take a closer look at the linkage between the two
variables.
Such an exercise is useful to learn more about the evolution of ﬁrms’ networks. Speciﬁcally, one
may be interested in whether a large number of interﬁrm relationships is a necessary condition for a
ﬁrm to grow rapidly. In a previous study on ﬁrms’ growth [1], we found that ﬁrms successfully achieve
continuous growth once their scale of production exceeds a certain threshold; otherwise, they are not
able to achieve continuous growth. Based on this empirical ﬁnding, we suggested the hypothesis that
∗Corresponding author. Economic Research Center, Fujitsu Research Institute, 1-16-1 Kaigan, Minato-ku, Tokyo
105-0022, Japan. E-mail address: saito.yukiko@jp.fujitsu.com (Yukiko Umeno Saito)
RIETI Discussion Paper Series 07-E -028 
 whether a ﬁrm successfully builds a good and stable relationship with other ﬁr m sa n db a n k si st h e
key to the ﬁrm’s continuous growth.
Interﬁrm networks have been investigated by several papers [1,2,4,5]. In particular, shareholding
networks have been examined for those ﬁrms listed in the United States, Japan, and Italy [2,5],
to ﬁnd that these networks have scale-free characteristics, and that the indegree exhibits power-law
distributions. But these existing studies have only looked at a small subset of ﬁrms, typically listed
ﬁrms or ﬁrms in a speciﬁc industry or region. In this paper, therefore, we look at 800,000 ﬁrms,
covering about half of all corporate ﬁrms currently operating in Japan.
2D a t a
The dataset we use is compiled by TOKYO SHOKO RESEARCH, LTD. (TSR). In this dataset, each
ﬁrm reports three types of linkages with other ﬁrms: suppliers (i.e., ﬁrms from which it purchases raw
materials and intermediate products, etc.); customers (i.e., ﬁrms to which it sells its products); and
owners (i.e., ﬁrms by which it is owned). The maximum number of ﬁrms each ﬁrm reports in terms
of each of the above three categories is 24 and the dataset consequently contains about four million
relationships in total. The dataset also provides background information on each ﬁrm, such as its
industry classiﬁcation, region, sales, proﬁts, and the year of establishment.
Table 1: Example of relationships
#1 #2 #3 #4 #5 #6 #7
Reporting ﬁrm Firm A Firm A Firm A Firm B Firm B Firm C Firm D
Reported ﬁrm Firm B Firm C Firm D Firm A Firm D Firm D Firm B
Type of relationship supplier supplier customer customer supplier supplier customer
There are two diﬀerent ways to count the number of relationships. One may count the number of
ﬁrms which, say, ﬁrm A reports as its counterpart. We refer to this as the “outdegree.” In the example
presented in Table 1, the outdegree for ﬁrm A is 2 in terms of “supplier” relationships and 1 in terms
of “customer” relationships, while the outdegree for ﬁrm B is 2 in total, and the corresponding number
for ﬁrms C and D is 1 respectively. Note that it is highly possible that the outdegree is truncated at
24, simply reﬂecting the reporting rule set by the TSR.
Alternatively, one may count the number of ﬁrms which report ﬁrm A as their counterpart. We
refer to this as the “indegree.” In the example presented in Table 1, the indegree for ﬁrm A is 1
(i.e., ﬁrm B reports ﬁrm A as its customer), while the indegree for ﬁrm B is 2, and the corresponding
numbers for ﬁrms C and D are 1 and 3 respectively. Note that the indegree could be truncated as
well, given that the number of ﬁrms each ﬁrm reports as its counterpart is less than 24. But the bias
caused by that would be smaller as compared with the outdegree, so that the indegree could be seen
as a more accurate and reliable measure of the number of relationships.
3 Network structure
We investigate the network structure in terms of the indegree distribution. Speciﬁcally, we ﬁrst look
at its unconditional distribution, then proceed to the distribution for a subgroup of ﬁrms that are
2identiﬁed as “hub” ﬁrms, and ﬁnally look at the distribution conditional on ﬁrms’ scale of produc-
tion/sales.
[Insert Figure 1]
Figure 1 shows three kinds of indegree distribution, namely, the cumulative density function (CDF)
of the indegree in terms of “supplier” relationships, the CDF in terms of “customers” relationships,
and the CDF in terms of “owner” relationships. As can be clearly seen in the ﬁg u r e ,e a c ho ft h e
three distributions exhibits a fat tail and, more interestingly, each line has an almost identical slope.
The reference line shown in the ﬁgure represents the slope associated with the power-law distribution
whose cumulative density function is given by R(x)=x−(α−1) with α=2.3. A simple comparison
with the reference line indicates that each of the three obeys a power-law distribution with α=2.3,
implying that the indegree is concentrated on a small number of ﬁrms.
[Insert Figure 2]
Firms with a large number of relationships may be regarded as “hub” ﬁrms. Given this interpre-
tation, we are interested in the network among these hub ﬁrms. We deﬁne ﬁr m sw i t ha ni n d e g r e e
greater than N (N=50, 100, 200, 500, and 1000) as hub ﬁrms and then count the indegree among
these hub ﬁrms. Note that we sum up all of the three relationships (namely, “suppliers,” “customers,”
and “owners”) in this exercise. Figure 2 shows the result of this exercise. Again, we see that each
distribution exhibits a fat tail and that each distribution has an almost identical slope. This result
suggests the existence of a layered structure in ﬁrms’ networks; that is, among the hub ﬁrms, there
are “super-hub” ﬁrms.
4 Firms’ size and the number of their relationships
Figure 3 shows the CDF for ﬁrms’ size. Here, we measure ﬁrms’ size by their sales. As can be
seen in the ﬁgure, the CDF is again fairly close to a power-law distribution, implying that a small
number of ﬁrms in the top group occupy a disproportionately large share of total sales. However,
if one compares Figures 1 and 3, one can see that the ﬁrm-size distribution has a fatter tail than
the indegree distributions, suggesting that the degree of concentration is weaker for ﬁrm size than
for network. In fact, the parameter α is now about 2.0, substantially smaller than in the case of the
indegree distributions.
[Insert Figure 3]
In Figures 4 and 5, we take a closer look at the relationship between ﬁrms’ size and the number of
their relationships. Figure 4 shows the indegree distributions for four subgroups identiﬁed by ﬁrm size.
This conditional distribution shows that ﬁrms with larger sales tend to display a greater indegree. At
the same time, Figure 4 does not necessarily indicate a linear relationship between the two variables.
To ﬁnd out more about the relationship between the two variables, we calculate the points at which
the CDF of the indegree distribution equals 1/2, 1/4, and 1/8 for each of 10 subgroups identiﬁed
by ﬁrm size. The result of this exercise is presented in Figure 5. We see that the points associated
with 1/2 (namely, the median of the distributions) are almost parallel to the linear reference line,
implying that the relationship between the two is indeed a proportional one as far as the median is
concerned. However, we also see that the points associated with 1/4 and 1/8 are not parallel to the
3linear reference line, but are signiﬁcantly ﬂatter than the reference line. This indicates that ﬁrms that
have already acquired a large number of relationships do not necessarily increase this ﬁgure even when
they grow in terms of sales.
[Insert Figures 4 and 5]
One may wonder why ﬁrms with a large number of relationships do not proportionately increase this
ﬁgure as they grow. One possibility is that those ﬁrms may want to save on the “maintenance costs”
of their networks. For example, several studies on the costs and beneﬁts of bank-ﬁrm networks [4]
have shown that multiple relationships with banks (namely, a ﬁrm has deposit/borrowing relationships
with more than one bank) are beneﬁcial to ﬁrms since they reduce the risk of a liquidity shortage
due to bank failure. However, such relationships are not free; ﬁrms have to pay maintenance costs in
the form of, say, preparing reporting documents to multiple banks. Given such costs, ﬁrms typically
choose not to monotonically increase the number of banks with which they have transactions even as
they become larger in terms of sales/production. This is easy to see if one considers, for example, that
the number of banks Toyota has transactions with is approximately the same as the corresponding
number for much smaller ﬁrms. Generally speaking, it would be costly for ﬁrms to maintain their
interﬁrm relationships, and if this is the case, ﬁrms that have already a large number of relationships
may choose not to expand their relationships even as they grow.1
5 Conclusion
We have found in this paper that larger ﬁrms tend to have more interﬁrm relationships, but that the
relationship between ﬁrms’ size and the number of their counterparts is not necessarily proportional;
ﬁrms that already have a large number of counterparts tend to grow without proportionately expanding
it. The next step in this line of investigation would be to examine more carefully the causality running
between ﬁrms’ size and the size of their networks, i.e., whether a large network is a cause or just a
result of a larger scale of production. Also, it would be an interesting task to investigate other aspects
of interﬁrm networks, including clustering indices and mean distances. Furthermore, it would provide
important policy implications if we were to look at interﬁrm networks by industry, by region, and by
the main product which each ﬁrm produces.2 We leave these topics to our future work.
Acknowledgement
We would like to thank the Research Institute of Economy, Trade and Industry (RIETI) for allowing
us to use the TSR data. We also would like to thank Hideki Takayasu, Iichiro Uesugi, and the members
of the research group on corporate ﬁnance at the RIETI, for useful comments.
References
[1] Caldarelli G., Garlaschelli D. and Battiston S. (2006), “The Skelton of the Shareholders Net-
works,” H. Takayasu (ed.), Practical Fruits of Econophysics, Tokyo: Springer-Verlag, 297-300.
1Another possibility is that ﬁrms with a large number of relationships may become hub ﬁrms or even super-hub
ﬁrms, thereby saving on the number of spokes (namely, the indegree) they directly have with other ﬁrms.
2The dataset we use in this paper describes interﬁrm network as of 2003. We would be able to study its dynamic
evolution if we have an additional data describing it as of, say 2006. For example, we would be able to look at how the
network was reconstructed when one of hub ﬁrms was forced to exit from the market.
4[2] Garlaschelli D., Battiston S., Castri M, Servedio V.D.P. and Caldarelli G. (2005), “The Scale
Free Nature of Market Investment Network,” Physica A 350, 491-499.
[3] Saito, Y., T. Watanabe, M. Iwamura (2006), “Path-Dependency of Firm Growth,” Bussei
Kenkyu, Vol. 86, No. 4, in Japanese.
[4] Souma, W., Y. Fujiwara, H. Aoyama (2005), “Shareholding Networks in Japan,”
http://arxiv.org/abs/physics/0503177.
[5] Souma, W., Y. Fujiwara, H. Aoyama (2006), “Change of Ownership Networks in Japan,” in H.
Takayasu (ed.), Practical Fruits of Econophysics, Tokyo: Springer-Verlag, 307-11.
[6] Detragiache, E., P. G. Garellia, L. Guiso (2000), “Multiple versus Single Banking Relationships:
Theory and Evidence,” Journal of Finance 55(3), 1133-1161.
51.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04
Indegree
C
D
F
"Supplier''
"Customer''
"Owner''
Reference line
Figure 1: Indegree distributions in log-log plot. Open circles, cross symbols, and open triangles
represent “suppliers”, “customers”, and “owners”relationships, respectively. The solid line represents
a reference line for the power-law distribution whose cumulative density function is given by R(x)=
x−(α−1) with α =2 .3.
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04
Indegree
C
D
F
50-
100-
200-
500-
1000-
all
Figure 2: Indegree distributions for hub ﬁrms. Each represents a distribution for hub ﬁrms with an
indegree greater than 50, 100, 200, 500, and 1000.
61.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06
Sales
C
D
F
Figure 3: Distribution of ﬁrms’ sales in log-log plot. The solid and dotted lines represents reference
lines for the power-law distributions whose cumulative density function is given by R(x)=x−(α−1)
with α =2 .0a n dα =2 .3, respectively.
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04
Indegree
C
D
F
10-50
50-250
250-1250
1250-6250
all
Figure 4: Indegree distributions conditional on ﬁrms’ sales. The entire sample is divided into four
sub-groups identiﬁed by sales.
71.E+00
1.E+01
1.E+02
1.E+03
1.E+02 1.E+03 1.E+04 1.E+05
Sales
I
n
d
e
g
r
e
e
 1/2
 1/4
 1/8
Reference line
Figure 5: Sales versus indegree. The ﬁgure shows the points at which the CDF of the indegree
distribution equals 1/2, 1/4, and 1/8 for each of ten subgroups identiﬁed by ﬁrm size.
8

Do Larger Firms Have More Interfirm Relationships?

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