Table 1 The process of achieving the criterion \(\hat{d}\)

Input: the total number of stocks N, the daily lead–lag network G_t achieved from real data (t = 1, 2, …, T), and the statistical significance level δ

Output: the criterion \(\hat{d}\)

Steps:

(1) Random directed network RG_t on day t is generated by retaining the node degree distribution of G_t via the configuration model. The code can be directly achieved from https://networkx.org/documentation/stable/modules/networkx/generators/degree_seq.html#directed_configuration_modmo. Note that the case of following oneself is also considered

(2) By repeating the steps (1) from t = 1 to T, a series of random networks RG₁, RG₂, …, RG_T can be obtained. Then, the accumulated following days of each pair (denoted as rd_ij, i, j = 1, 2, …, N) are achieved based on the above series of random networks via Eq. (3). As a result, one group of simulation is completed with the obtained set {rd_ij}_i,j=1

(3) Hundred groups of simulations can be conducted (e.g., 500 groups) as above and then all the accumulated following days of each pair obtained from each group are put together to get their distribution. Given the statistical significance level δ, the corresponding criterion \(\hat{d}\) can be immediately achieved