## 2 Materials and Methods
### 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:  
1. For two equal length time series \(\{x(t)\}\) and \(\{y(t)\}\), \(t=[1,2,\ldots,N]\), their state vectors \(X_{t}=[x_{t},x_{t+\tau},x_{t+2\tau},\ldots,x_{t+(d-1)\tau}]\) and \(Y_{t}=[y_{t},y_{t+\tau},y_{t+2\tau},\ldots,y_{t+(d-1)\tau}]\), \(t\in[1,2,\ldots,N-(d-1)\tau]\), are obtained through the phase space reconstruction procedure using the delay parameter \(\tau\) and the embedding dimension \(d\). Here, following the parameter choice of bubble entropy, \(\tau=1\);
2. Performing ascending sort on the state vector \(X_{t}\), and obtaining its position index \(\pi_{X_{t}}\). The state vector \(Y_{t}\) was rearranged using the position index \(\pi_{X_{t}}\) as a criterion and the result was recorded as \(G_{t}=Y_{t}(\pi_{X_{t}})\), \(t\in[1,2,\ldots,N-d+1]\);
3. Sorting the elements in each state vector \(G_{t}=Y_{t}(\pi_{X_{t}})\), \(t\in[1,2,\ldots,N-d+1]\) in ascending order, and calculating the necessary number of swaps \(S_{i}\), \(S_{i}\in[0,1,\ldots,d(d-1)/2]\); this is because the number of possible swaps in bubble sort for a \(d\) dimensional state vector is from \(0\) to \(d(d-1)/2\);
4. Using \(S_{i}\), \(S_{i}\in[0,1,\ldots,d(d-1)/2]\) as network nodes, a directional weighted complex network \(W\) was constructed according to the temporal adjacency relationship of \(S_{i}\) and the weight of the network \(W\) was the numbers of transition between nodes;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 \(S_{W_{i}}\), which was used to measure the probability distribution of the output strengths of each node. \[S_{W_{i}}=-\sum\limits_{j=0;j\
eq i}^{D}w_{ij}\log 2(w_{ij}) \tag{5}\]  
where \(D=d(d-1)/2\), \(w_{ij}\) was the ratio of the output strength from node \(i\) to node \(j\) to all the output strengths of node \(i\), \(\sum\limits_{j=0}^{D}w_{ij}=1\), and the normalized \(S_{W_{i}}\) was  
\[H_{W_{i}}=S_{W_{i}}/S_{i,\max} \tag{6}\]  
where \(S_{i,\max}=\log 2(D+1)\) was the normalization factor and kept the same for all nodes. The node-wise out-link transition entropy of the adjacency matrix \(W\) was  
\[H_{NOTE}=\sum\limits_{i=0}^{D}p_{i}H_{W_{i}} \tag{7}\]  
where \(p_{i}\) was the probability distribution of each node. The pseudo-code of the proposed algorithm is illustrated as follows. ```
CBTN (\(x(t)\), \(y(t)\), \(d\), \(\tau\)) // \(x(t)\), \(y(t)\) are time series. \(d\) is embedding dimensions. \(\tau\) is delay time.