*2.2. Cross-Bubble Transition Network (CBTN)*

In the original CPE, the process of counting the intersection number of each state vector is essentially a symbolization of it. In the calculation of entropy, only the probability distribution of symbols is considered and the transition behavior between adjacent symbols is ignored. Therefore, the transition network is introduced, in which each symbol is taken as a node and a directional weighted complex network is constructed based on the temporal adjacency of the symbols, with the network weights being the number of transitions between nodes. In addition, to limit the impact of parameter selection on the analysis results, the symbolization process of the bubble entropy was referenced by replacing the intersection number corresponding to each state vector with the number of swaps necessary to sort the state vector in ascending order. The specific implementation process of the cross-bubble transition entropy (Algorithm 1) is as follows:


5. In order to reflect the connection relationship between nodes as much as possible, the node-wise out-link transition entropy (NOTE) of the adjacency matrix *W* was proposed to be used as an indicator parameter. The NOTE was obtained as follows.

The Shannon entropy of each row of the adjacency matrix *W* was calculated to obtain the local node out-link entropy *SWi* , which was used to measure the probability distribution of the output strengths of each node.

$$S\_{W\_i} = -\sum\_{j=0; j\neq i}^{D} w\_{ij} \log \mathbf{2}(w\_{ij}) \tag{5}$$

where *D* = *d*(*d* − 1)/2, *wij* was the ratio of the output strength from node *i* to node *j* to all the output strengths of node *i*, *D* ∑ *j*=0 *wij* = 1, and the normalized *SWi* was

$$H\_{\mathbb{W}\_i} = \mathbb{S}\_{\mathbb{W}\_i} / \mathbb{S}\_{i, \text{max}} \tag{6}$$

where *Si*,max = log 2(*D* + 1) was the normalization factor and kept the same for all nodes. The node-wise out-link transition entropy of the adjacency matrix *W* was

$$H\_{\rm NOTE} = \sum\_{i=0}^{D} p\_i H\_{W\_i} \tag{7}$$

where *pi* was the probability distribution of each node.

The pseudo-code of the proposed algorithm is illustrated as follows.


To demonstrate the performance of the NOTE to track the deterministic dynamical variation in time series, the values with the NOTE and the original CPE were obtained separately for the symbolized sets A = [2 2 4 3 5 1 2] and B = [1 2 2 5 3 2 4 ]. The probability distributions of the elements in the sets A and B were the same. The probability of individual elements sorted in an ascending order were [0.143, 0.428, 0.143, 0.143, 0.143]. The original CPE method would yield an entropy value of 2.128 for both sets. The directional weighted adjacency matrices constructed for the elements in sets A and B according to their temporal adjacency relationship are shown in Figure 1a,b, respectively. The two adjacency matrices exhibited distinct differences. The NOTE value of these two adjacency matrices was 0.1846 and 0.2925, respectively, which shows the different dynamical variations contained in the sets A and B.


**Figure 1.** The directional weighted adjacency matrix constructed from the temporal adjacency relationship for symbol sets A and B. (**a**) The adjacency matrix for the symbolized set A; (**b**) The adjacency matrix for the symbolized set B.
