The first implication is that the number of trade ties is very uneven across firms. Turning to the number of supplier connections, the sample mean is about 60 each year, which is much smaller than the number of customer connections. The number of customers and suppliers listed is 6.7 and 6.4 for a typical firm, which is much less than the average number of total customer and supplier ties shown in Table 1.

The number of links is uneven between firms with respect to both customer and supplier links, as we have seen in Table 1. The slope associated with supplier links is approximately -1.5, so that the CDFs for the number of supplier links can be characterized by . It is important to remember that the number of links, in the case of both customer and supplier links, follows a fat-tailed distribution.

## 5 The Evolution of Customer-Supplier Networks

Next, we examine changes in the total number of links, i.e. NC and NS, over time. We saw in Figure 2 that the distribution of the total number of links, for both customer and supplier connections, does not change much over the five years. However, this does not necessarily mean that the number of links for each company does not change much.

We see that the dots are concentrated around the 45 degree line for both customer and supplier links, indicating that the number of links for most companies remained unchanged. To examine in more detail how the companies' number of links changes over time, we show in figure 6 the distribution of the annual growth rates for the number of customer links, logNiC(t+ 1)/NiC(t), and for the number of customers. supplier connections, logNiS(t+ 1)/NiS(t), with the growth rates on the horizontal axis and the corresponding densities on the vertical axis. Note that there are a total of eleven distributions in the two panels, each corresponding to a group of companies with a certain number of links in year t.

For example, the distribution labeled 103.5 ≤NC(t)<104.0 represents the distribution of the growth rates of the number of customer links from year to year+1 for companies with a number of customer links within the specified range. To show this more clearly, figure 7 plots the number of joints in year t against the standard deviation of the growth rates of joints from t tot+ 1. The figure shows that although the standard deviation is relatively high, when the number of joints in year is either very small (i.e. . below 10) or very large (above 104), it is relatively small and almost uniform for intermediate values.

We assume that the number of attempts a firm makes to acquire new customers int+ 1 is proportional to the number of customers the firm has in t. In this simple setting, the growth rate frattot+1 of the number of customers for a firm is on average unity, which is consistent with the empirical result shown in Figure 6.

## 6 Implications for Firm Sales and Growth

### The relationship between customer links and firm sales

In economics terminology, the number of links to customers is the extensive margin, while the average size of links to customers is the intensive margin. In the context of international trade, many studies have addressed this issue, including [31, 32], some of which show the relative importance of a large margin [32]. In the context of firm dynamics, some studies argue that the number of customer ties plays a dominant role in explaining differences in firm sales [15], while some anecdotal evidence suggests that ties have larger sizes that may reflect a closer and longer lasting bond with a particular partner enables companies to sell more.

However, to the best of our knowledge, researchers have access neither to information that makes it possible to decompose sales into final and intermediate demand nor to information on the size of customer ties. However, we are still able to investigate how the number of customers for a firm is related to the firm's sales. To this end, Figure 8 shows the relationship between the two, plotting a firm's number of customers on the horizontal axis and the firm's sales on the vertical axis.

More specifically, we define 14 bins of the same size in logarithm for the number of customer links and show different percentiles of the sales distribution for companies belonging to each bin, namely the 25th (×), 50th (◦), 90th (N), 99th ( ) and 99.9th () percentiles. As can be clearly seen in the figure, sales are positively correlated with the number of customer links. Furthermore, a simple regression indicates that the median of the sales distribution in logarithm, denoted by m, depends on the number of customer links.

At the same time, Eq. 10) also shows that a 10 percent increase in the number of customer connections increases firm sales by only 5 percent, suggesting that other determinants of firm sales that are not controlled for in the regression may be inversely correlated with the number of customer connections. . For example, the size of customer connections may be negatively related to the number of users.

### Can customer-supplier links predict firm growth correlations?

Alternatively, companies with a larger customer base may be located further upstream in customer-supplier chains, so that they may have less opportunity to sell their products to consumers etc. as end product. wherelij is the shortest path length between ﬁrmsiandj and⟨ρij |lij. =l⟩ is the average of ρij conditional on the shortest path length between them being l. The positive constant term in Eq indicates that the growth rates of ﬁrmsi and jare were positively correlated even when they are very large, meaning that part of the growth wheel correlations may be due to factors that have nothing to do with customer-supplier chains.

In fact, the growth rate correlation for pairs of firms not connected to the network at all (i.e., SP L=∞) averages 0.056, which is close to the constant term in Eq. To investigate the relationship between the growth correlation and the shortest path length in more detail, we follow the recent literature on supply chains [33, 34] and assume that. We denote the uncorrelated new perturbation vector by ˆϵt and define the new growth rate vector ˆgt as ˆgt = (I −A)˜ −1ˆϵt. The growth rates for i and j cannot be correlated through common shocks, but they can be correlated through customer-supplier relationships.

The result of this exercise is shown in Figure 10, where the horizontal axis shows the shortest path length, while the vertical axis depicts the correlation of conditional growth with the shortest path length. The result using actual growth rate data, gt, is represented by. and it shows that⟨ρij |lij =l⟩decreases mel, as we saw in Eq. Then, the result for the correlations of the growth rate through links alone, which are calculated using ˆgt, is shown by. The result shows that ⟨ρij |lij =l⟩ again decreases with l, but this time it decreases very close to zero when l≥7.

Finally, we add the estimate for the correlations of the growth rates due to common shocks, 0.045 in Eq. 11), to the correlations of the growth rates via links. Doing this reveals that the sum of the two, represented by , successfully generates the growth rate correlations observed in the data.

## 7 Conclusion

For example, it is known that the correlations between stock prices during stock price bubbles do not necessarily correspond one-to-one with customer-supplier relationships. Much remains to be done regarding how these two inter-firm networks are connected.

Acknowledgments

Hubs and Authorities on Japanese Interfirm Network: Characterization of Nodes in Very Large Directed Networks. Biased distribution on the Japanese interfirm trade network: estimating sales from the network structure. Too Interconnected to Fail Financial Network of the US CDS Market: Topological Fragility and Systemic Risk.

The red arrows in the figure show the flow of money, while the black arrows show the flow of products that each firm produces. The horizontal axis represents the total number of connections, i.e., NCandNS, while the vertical axis represents the corresponding cumulative density. Firms T,R,K and D are randomly selected from the sample, which consists of all firms on the extended customer/supplier lists.

The number of links in yeart on the horizontal axis versus the number of links in year t+1 on the vertical axis. Distributions of link growth rates from year to year + 1 for customer links (top panel) and for supplier links (bottom panel). The dotted curve in the top panel represents a normal distribution with a standard deviation of 0.12, which is the estimated standard deviation for customer link growth rate, while the dotted curve in the bottom panel represents a normal distribution with a standard deviation of 0.10, which is the estimated standard deviation for the growth rate of supplier relationships.

The ratio of the number of links in year t, shown on the horizontal axis, to the standard deviation of link growth rates from year to year t+ 1, shown on the vertical axis. The thin dotted line labeled "random shuffling" represents the distribution for a random shuffling case in which (1) we randomly select two years for a given firm, swap the growth rates for those two years, and repeat for other pairs of years; (2) we perform the same random mixing for other firms until we completely remove any correlation between the growth rates of any pair of firms. The average of the growth rate correlation between pairs of firms, depending on the shortest path length between the pairs.

The figure shows correlations obtained from data (), correlations through common shocks (△), correlations through customer-supplier ties (⋄), and correlations through the sum of both.