## 2 Related Research
### Fuzzy C-Means Algorithm  
In this paper, we propose a metaverse network architecture for virtual concerts utilizing SDN and XDN, as previously introduced. To efficiently operate a network targeted at a massive multiplayer base, such as a virtual concert or game, clustering users (nodes) according to an appropriate standard is required. Clustering techniques are broadly divided into hierarchical clustering and partitional clustering. Hierarchical clustering uses a hierarchical tree model to perform clustering by integrating individual objects into sequentially and hierarchically similar objects or clusters. Partitional clustering initially sets the number of clusters and then searches for the optimal center of each cluster to perform clustering. In this study, we chose partitional clustering because we need to distribute network resources according to the clustering results.  
Figure 6: Metaverse virtual concert network design [16].  
Partitional clustering is divided into hard clustering when each data must belong to a single cluster and soft clustering when it can belong to multiple clusters. In soft clustering, each data can potentially belong to multiple clusters. The fuzzy c-means algorithm is a representative soft clustering technique. [PERSON] has devised a method using fuzzy segmentation by expanding the k-means algorithm and presenting the objective function as shown in Equation (1) below [25; 26].  
\[\sum_{k=1}^{K}\mu_{ik}=1,\ \ \ 0<\sum_{i=1}^{n}\mu_{ik}<n \tag{1}\]  
\[\min\!J_{m}=\sum_{k=1}^{K}\sum_{i=1}^{n}(\mu_{ik})^{m}d(x_{i},\ c_{k})^{2} \tag{2}\]  
As fuzzy c-means clustering is an algorithm that calculates the probability of each data belonging to a certain cluster as a probability, the sum of \(\mu_{ik}\), which is the weight of the probability that data \(x_{i}\) belongs to each cluster \(c_{1},\ c_{2},\ \ldots c_{k}\), is 1. In addition, to ensure the effectiveness of the clustering analysis using fuzzy logic, each cluster \(c_{k}\) has at least one data with a non-zero weight, and all of the weights of the cluster cannot be 1. Therefore, the partitioning condition is satisfied as shown in Equation (1). The clustering algorithm, like any other hard clustering, proceeds by minimizing \(d(x_{i},c_{k})\), which is the distance between the observation and the fuzzy cluster centroid, i.e., the optimal cluster is found by minimizing the \(J_{m}\) objective function, as shown in Equation (2). Here, \(m\) is a constant greater than 1 and is the fuzzifier that controls the value of the weights.  
\[C_{k}=\frac{\sum_{i=1}^{n}\mu_{ik}^{m}x_{i}}{\sum_{i=1}^{n}\mu_{ik}^{m}},\ \ k\in[1,K] \tag{3}\]  
\[\mu_{ik}=\frac{\left\{\frac{1}{d(x_{i}c_{k})^{2}}\right\}^{\frac{1}{m-1}}}{ \sum_{j=1}^{K}\left\{\frac{1}{d(x_{i}c_{j})^{2}}\right\}^{\frac{1}{m-1}}} \tag{4}\]  
We need to calculate a partial derivative to estimate each variable that minimizes \(J_{m}\). Equations (3) and (4) are the solutions of the system of equations after calculating the partial derivative with respect to \(c_{k}\) and \(w_{ik}\), respectively, and setting their values to zero. As the value of the fuzzifier \(m\) increases, the mean of each cluster moves closer to the overall mean, so it becomes increasingly fuzzier to classify the data into one cluster. Generally, \(m\) is set to 2 for computational convenience, but in this study, fuzzy c-means clustering is performed through various m values. Because the fuzzy c-means algorithm specifies the number of clusters and weighting coefficients and has a different optimization procedure from the k-means algorithm, it offers superior results in various experiments compared with the k-means algorithm [27; 28; 29]. This algorithm has been employed for clustering in a wide range of industries; in this study, we utilized it to cluster audiences in the virtual concert environment [30; 31; 32].