11.4.1 Basic Properties of Survival Rate
In this section, we introduce the survival rate for each node and provide its basic properties. At the transition point ( f
c
= 0. 994), the survival rate is defined as the ratio of the number of trials, in which the node belongs to the largest cluster, to the total number of trials. In this study, 100,000 trials were performed to estimate the value of the survival rate for each node. This parameter is widely distributed, and its large-scale behaviour approximates a power law [11]. It should be noted that this index is able to characterise the global connectivity of each node in the network as we explain in Sect. 11.4.3.
Next, we discuss the correlation between the survival rate and important parameters characterising firms, such as degree, shell number, sales and the number of employees. Spearman’s rank correlation coefficient was chosen for this purpose, because Pearson’s correlation coefficient is susceptible to outliers. As shown in Table 11.1, there is a positive correlation between the survival rate and all the parameters, and this is especially strong for values characterising the network, such as the degree and shell number.
Table 11.1 Spearman’s rank correlation coefficient between the survival rate and principal parameters (degree, shell number, sales, and number of employees)
The correlation between degree k and the survival rate P
s
was investigated in more detail. The variation of the survival rate P
s
was clarified by plotting its distribution for degree k as shown in Fig. 11.2. We found that the survival rate P
s
varies even in the same range of degree k. Therefore, the survival rate P
s
is not completely determined by information relating to the local connectivity, such as degree k; the degree k can explain this value roughly. This fact suggests that the survival rate P
s
is determined by the critical cluster, which includes the information of the whole network topology. In this sense, this robust index includes information about the global connectivity, such as the shell number, as opposed to local connectivity such as the link number.
11.4.2 Practical Meaning of P
s
It is important to note that the nodes with the same survival rate P
s
were confirmed to have widely distributed link numbers as shown in Fig. 11.2. We subsequently investigated the features of the nodes with a high survival rate for small link numbers and those with a low survival rate for large link numbers. First, we specified a range of degree k from 1 to 10 as a set of small link numbers within a certain range of the survival rate (\(5.0 \times 10^{-4} \leq P_{s} \leq 5.0 \times 10^{-3}\)), which includes approximately 20,000 nodes. When we investigated the industry these nodes represent, it was revealed that the nodes categorised as belonging to the construction industry captured 27 % of the share, whereas the share was 21 % of the network in its initial state. This result means that nodes belonging to the construction category have a higher survival rate than nodes in other categories.
Next, we focused on nodes with a large number of links within the same range of survival rate (\(5.0 \times 10^{-4} \leq P_{s} \leq 5.0 \times 10^{-3}\)). These nodes are characterised by a low survival rate and are fragile in spite of their many links. As an example, we paid attention to the node with large k and relatively small P
s
, \((k,P_{s}) = (448,3.3 \times 10^{-3})\). As shown in Fig. 11.3, most of its linking nodes are located on the same island, Hokkaido, and there are only 13 links (about 3 %) connecting to firms outside this island. There are not many links connecting to nodes located outside of this island. This result suggests that this type of node bundles firms in a local region.
11.4.3 Theoretical Estimation
A theoretical estimation of the survival rate was derived by using the degree and the rates of linking nodes as follows. In the case of a node that only has one link we have the following exact relation.
$$\displaystyle{ P_{s,i} = (1 - f_{c})P_{s,j} }$$
(11.2)
where, the subscript i represents the focusing node, and the subscript j represents the its linking node.
We next extended this formulation to the general case for nodes with multiple links. The probability of the focusing node being connected to the giant component, P
s, i
is approximated as follows.
$$\displaystyle{ P_{s,i} = 1 -\prod _{j=1}^{k}\{1 - (1 - f_{ c})P_{s,j}\} }$$
(11.3)
On condition that the survival rate P
s
is sufficiently small, we can approximate Eq. (11.3) by the following equation.
$$\displaystyle{ P_{s,i} \simeq (1 - f_{c})Q_{s,j} }$$
(11.4)
Here, Q
s, j
is defined as \(\sum _{j=1}^{k}P_{s,j}\). This equation shows that the survival rate P
s
is explained by the summation of the survival rates of linking nodes. In Fig. 11.4, we confirm that this relation is in good agreement. This examination revealed that the survival rate P
s
depends on the link number k and the survival rates of the linking nodes, P
s, j
. This result shows that the value P
s, i
is determined from the global network topology, and that the mean field approach used in Eq. (11.3) works well in deriving Eq. (11.4).