NETWORK CLUSTERING METHOD FOR PREVENTING THE SPREAD OF COVID-19 IN INDONESIAN SCHOOLS

,


INTRODUCTION
Indonesia's population in August 2022 reached 275,361,267 people (www.bps.go.id). It is a formidable challenge for the Indonesian government to continue to anticipate the wave of the Coronavirus Desease (COVID-19) pandemic that may occur next time. Considering that Indonesia has an extensive area with a large population, if a pandemic occurs, it will decrease the quality of education. For this reason, unique learning strategies are needed to continue the educational process. In addition, research results show a relationship between learning strategies and student motivation [1]. A good learning strategy can increase student motivation.
Several studies report that learning with blended learning can overcome problems in education in schools. Ashraf et al. [2] reported that several studies on blended learning are very effective in overcoming educational problems, especially distance learning. Distance learning and face-to-face learning can be combined simultaneously through information and communication technology, namely, using software for online learning [3].
Blended learning is one of the most frequently used approaches to applying information and communication technology in education [4]. One of the effective blended learning models of learning for teachers and students is hybrid learning. Hybrid learning is a learning method that combines or combines online learning with face-to-face learning or offline learning [5].
Furthermore, students are arranged or set a schedule to enter the class and vice versa, learning from home online. Students continue to attend classes by using online learning methods. Then they will take turns with other students to have the same opportunity. Through this hybrid learning, it is hoped that it can overcome the limitations of distance learning or online learning. There are times when social interaction occurs when students come face to face with the teacher. For example, students meet teachers in class during the COVID-19 pandemic.
Online learning has been implemented since the pandemic entered Indonesia. It has caused many problems. The lack of social interaction between students and teachers leaves psychological problems. Stress is one of the effects caused by the learning system. So many parties want this 3 NETWORK CLUSTERING METHOD FOR PREVENTING THE SPREAD OF COVID-19 online learning to end, especially with its dependence on devices to go online and the internet network. So, problems arise more quickly during learning than in face-to-face or offline learning.
However, online and offline learning solutions can reduce stress for students.
The COVID-19 pandemic allows it to happen again. So, until now, there has yet to be a full face-to-face meeting in every region in Indonesia. In the end, the government implemented a limited offline learning policy. One of these policies is hybrid learning, combining online and offline learning.
Many papers report the benefits of the clustering method in research. The benefits of clustering include education [6] [7], item delivery [8], traffic density prediction [9], Etc. In this paper, the clustering method is used in education. The next problem is determining the group of students who do online learning and those who do offline learning. The solution offered in the study in this paper is to form a community. The community is an area of student groups in a network community. This community network is obtained from the results of network clustering. The network represents the map of the student's presence area. Through the school community network, it is easy to obtain information about the high or low conditions of people with COVID-19. The community also provides information on which students are in the area.

Network
Based on several references, a network is closely related to a graph, which is part of mathematics [10] [11], especially graph theory. Furthermore, the graph is defined in the definition below. The graph G = G(V, E) is a set pair (V, E) with V = {v 1 , v 2 , ⋯ , v } called the set of vertices/nodes that are finite and non-empty, and E = { = (v , v ): v , v ∈ V, ≠ } is the set of pairs between two points called edges [12].
The graph is classified into connected graphs and unconnected graphs. This study focuses on connected graphs. A graph is connected if it only consists of one component [13]. This definition implies that a connected graph will have a path connecting a vertex to other vertices. Newman [10] defines a network as a connected graph. Furthermore, in this study, the network is represented as a simple graph, see Figure 4.
The focus of this study is related to problems in dynamic networks. Dynamic networks experience changes in structure and weight from time to time [14]. A network ( ) = (V, E) with edge weight is called a weighted network if each edge is mapped to a positive real number and reaches the edge weight [13].
In this case, the network weight is the number of people with COVID-19 in the area that connects the nodes. The node is the location or address of the student's residence. The network's weight is dynamic because, along with the information, there are many people with COVID-19.
Network clustering in graph theory is a graph partition that divides a graph into more than one subgraph [15] [11]. This partition divides G( ) into subsets of independent parts, and it is determined by considering the minimum number of side weights between subnets. Mathematically, such that it is obtained Generally, a network can be partitioned into more than one subnet consisting of several subgroups with group members having similar characteristics. The group is a network community, in the future, referred to as a community.

Research Design
There are many algorithms for obtaining the community [10], but in this study, the algorithm is focused on the spectral bisection method. Community search using the spectral bisection method is based on the eigenvalues and eigenvectors of the Laplace matrix of a network [16] [17]. Spectral

NETWORK CLUSTERING METHOD FOR PREVENTING THE SPREAD OF COVID-19
bisection is a clustering method that divides the network into two communities. The resulting community formation is based on the Fiedler vector of the Laplace matrix, which corresponds to the nodes on the network [18]. In this study, network clustering uses spectral bisection recursively through Algorithm 1 [17]. ALGORITHM 1. Spectral bisection recursively: 2) Get the Fiedler vector 2 from the second smallest eigenvalue 2 .

Analyzing of Data
For the data, we used 65 students for simulation. The network represents a map of student locations or addresses in this case. The purpose of making a network is to facilitate the process of grouping students into several communities. We can create a community of the network by using the network clustering method. In communities with a high frequency of COVID-19 sufferers, students are advised to take online learning. Students are advised to take offline learning for communities with a low frequency of COVID-19 sufferers.
Hybrid learning is applied when a network experiences edge weight fluctuation. If the sum of network weights is too large, it is advisable to use online learning. If the sum of network weights is not large, it is advisable to apply offline learning. Furthermore, the hybrid learning strategy is illustrated in Figure 1.

Statstical Test
We use a multinomial distribution as a simulation. Because this distribution represents the number of events at a certain time. The test used is statistical which refers to Theorem 1 and Corollary 1 in Indratno et al. [9]. , and variance ⃛ 2 =̈2.
The multinomial goodness-of-fit test applies the following hypothesis test: ) Follows a normal distribution with the mean ⃛ and variance ⃛ 2 , The hypothesis test used is the statistical test with the formula The margin of error for the confidence interval (1 − ) in statistics on Equation (3) can be determined through Theorem 2 [9]. This theorem provides a formula for the error tolerance limit for the population average at the confidence interval (1 − ).
, then margin of error ( ) for the confidence For testing data based on historical data are presented in Algorithm 2 [9]. 6) Return to step 3.

RESULTS AND DISCUSSION
We have 65 students' locations mapped on the map; see Table 1 and Figure 4. Then, we take a map from google maps (https://www.google.com/maps/@-6.9159506,107.6513172,16z) based on the addresses or areas of all students; the results are shown in Figure 2 and Figure 3. The map with satellite mode is presented in Figure 2. The map with terrain mode is shown in Figure 3.  . The parameter is generated randomly from the standard uniform distribution ( (0,1)) . For estimation parameter ( ) of historical data, it is obtained in Table 2. We conduct simulations on the first day (time-1) to see the condition of the number of COVID-19 sufferers in all communities. The results are shown in Figure 5. The simulation on the second day (time-2) is shown in Figure 6.

CONCLUSION
The learning strategies produced in this study are aimed at schools or universities located in urban areas, especially in densely populated Indonesia. Combining the hybrid learning model with the community network generated through network clustering effectively anticipates the next wave of the COVID-19 pandemic.
Detecting communities with network clustering makes it easier for us to implement hybrid learning in education. The combination of a hybrid learning model with network clustering is one 13 NETWORK CLUSTERING METHOD FOR PREVENTING THE SPREAD OF COVID-19 form of machine learning product. Machine learning is a science that many people are working on around the world [19]. Through machine learning, artificial intelligence products can be produced [19] [20].
For future research, it can combine this model for categorical data by adding a correspondence analysis [21]. Furthermore, the development of the model will be made in the form of a program to become an artificial intelligence product [20]. The learning strategy in this study assumes that roads connect all student locations. If a highway connecting these locations is broken or not functioning, this research study does not apply. This strategy can also be used if there are fluctuations in people with COVID-19 with a fixed total population.