## 4 Simulation of Interaction Telecommunication in Virtual Concerts  
Simulations were conducted to validate the model for clustering audiences in the Zone. Better, a framework for transmitting audience interaction data. To find the parameters that perform optimal clustering, we generated audiences located within the virtual performance world and performed fuzzy c-means (FCM) clustering based on their location values. The hyperparameters to be considered in the FCM algorithm to cluster the zones are as follows:  
* **Audience placement in the virtual concert world**: FCM clustering is an algorithm that updates the probability of belonging to each cluster by generating a vector with a preset number of columns for each piece of data. The performance and execution time of clustering according to the number of audiences has a linear relationship, and the realistic placement of audiences can determine the reliability of the clustering results. * **Number of zones (number of clusters)**: As this study uses partitioned clustering, the number of clusters must be set in advance to perform clustering. In the actual meta-verse environment, the number of zones may vary depending on the specifications of the network equipment and the environment, so the number of server environments that can be allocated varies depending on the performance. * **Number of clustering iterations (iter)**: As iterations increases, the FCM algorithm is performed once. Initially, a centroid is assigned at the same coordinates or random coordinates, and the centroid is updated by the value of iter. Because clustering in the Zone Setter needs to be fast to provide the audience with a lag-free immersive experience, it is important to quickly capture the iteration that approaches the optimal clustering result. * **Fuzzifier**: Fuzzifier is a hyperparameter that has a great impact on the performance of FCM, and researchers usually use experimental or heuristic methods to set the fuzzifier [34]. Through simulation, various experiments were conducted in this study to determine the appropriate fuzzifier to port to the Zone Setter. To recreate the environment where the audience is concentrated around the performer's coordinates in a real performance, we used a hypergeometric distribution to generate random audience placement. The number of success states and the number of failure states were adjusted to be equal to each other to create a distribution of coordinates clustered around a central coordinate, which was set to be the performer's coordinate. For this simulation, we set up a total of four random audience placements for a certain number of audience members, as shown in Figure 9.  
Figure 8: CDN-based network architecture for transmitting interaction to large audiences. We performed FCM clustering on audiences of 500 people to see how the centroid changed with fuzzifier. We used the scenario of case 01 in Figure 9, and the number of clusters was set to 5. The total number of iterations was 50, and the FCM algorithm was performed with 5 different fuzzifier values. The results showed that FCM clustering with a fuzzifier between 1.1 and 1.5 showed a high initial change in each iteration of the algorithm, and then the change converged to 0 after a certain number of iterations, while, with a fuzzifier above 1.7, there was almost no change in the initial centroid coordinates overall. Therefore, it is recommended to set the fuzzifier to at least 1.5 or less to achieve fast clustering results in fewer iterations. We also conducted an experiment to check the variation in the optimal number of clusters depending on the fuzzifier when exploring the optimal number of clusters using the elbow method. As in the previous experiment, we set five fuzzifier values to check the change in inertia (sum of Euclidean distance between data and clusters) from the minimum number of clusters of 2 to the maximum number of clusters of 27. As a result, we found that the change in inertia was quite similar for fuzzifiers from 1.1 to 1.5, and the change in inertia decreased from 1.7.