## 3 Methodology  
### Algorithm of Original GNG  
The base model of the proposed method is the GNG algorithm, and the theory and process of the GNG are described briefly. Then, the vital improvements for the original GNG are detailed and demonstrated in this section. The original GNG, which combines the neural gas [21] and competitive Hebbian learning [22], is a topological graph. It aims to represent the characteristics of multivariate data by allowing the number of neurons to increase and taking into account the neighborhood relations of neurons. The learning process of the original GNG includes the adjustment of neuron parameters, adjustment of relationships between neurons, insertion of neurons, and deletion of neurons, of which the process is as follows in Algorithm 1, and the relevant parameters are shown in Table 1.  
As shown in the original GNG algorithm, a minimum number of neurons is initially created (lines 1-3), and then, new neurons and new neighborhood connections (edges) are added between them during learning, according to the input instances. For each new instance x from the data stream (lines 4-5), the two closest neurons \(s\) and \(z\) are found via Euclidean distance (line 6). A local representation error \(e_{s}\) is increased for the winning neuron (line 7), and the age of the edges connected to this neuron is updated (line 8). The winning neuron \(s\) and the neighboring neurons (linked to \(s\) by an edge) are adapted according to the learning rate \(\epsilon_{1}\) and \(\epsilon_{2}\) (line 9). Moreover, the two neurons \(s\) and \(z\) are linked by a new edge (age is 0) (lines 10-11). When the edges reach a maximum age \(a_{max}\) without being reset, they will be deleted. If any neuron belonging to the graph becomes isolated, it is also deleted (line 12). With the input of data streams, the graph periodically creates a new neuron between the two neighboring neurons that have accumulated the largest representation error (lines 13-14). Finally, the representation error of all neurons is subject to an exponential decay (line 15) in order to emphasize the importance of the recently measured errors. As stated above, the GNG model can dynamically represent the distribution characteristics of streaming data, as shown in Figure 1. However, it also suffers from several drawbacks. First, the original GNG organizes neurons to represent the streaming data using fixed learning rates, affecting the model convergence and stability. Second, inserting new neurons into the model every k step cannot guarantee the necessity of the new neurons, and it may also result in model redundancy. Third, the hyperparameter optimization of \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Parameter Meaning** \\ \hline \(A\) & The set of all neurons \\ C & The set of edge-connecting neurons \\ \(\omega\) & The weight of neuron \\ \(\epsilon_{1}\) & The learning rate of winning neuron \\ \(e_{2}\) & The learning rate of neuron in the neighborhood of winning neuron \\ \(e\) & The cumulative error of neuron \\ \(n(s)\) & The set of neurons that connect to the neuron \(s\) \\ \(a_{\text{max}}\) & The maximum of edge’s age \\ \(t\) & The current time \\ \(\lambda\) & The step size of inserting neurons \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the Growing Neural Gas (GNG) algorithm. GNG via grid search is time-consuming and imprecise, which makes it difficult to obtain the optimal solution. Several variants of GNG have been proposed to solve the previous problems, such as GNG-U [23], GWR [24], and Online GNG [25]. GNG-U defines a utility measure that removes neurons located in low-density regions and inserts them in regions of high density. However, we may need to create new neurons without the necessity of removing others. GWR judges whether new neurons need to be added according to the activity and firing thresholds of the winning neurons, but it does not consider optimizing the learning rate.