## 2 Materials and Methods
### Proposed Method  
Each graph consists of nodes and edges that form a social network. Each node can be a member of the network that communicates with other members. This study used the HEA interaction network that was created in our previous work. To construct the HEA interaction network, the properties of each compound were first investigated, and the content cosine similarity of each pair of descriptors was calculated via Equation (1). Then, the structural Jaccard similarity of the descriptor of each compound was obtained via Equation (2). The hybrid similarity criterion was created from the cosine content similarity parameter and the Jacquard structural similarity parameter via Equation (3). The alpha coefficient determined the effect of each similarity parameter on the hybrid similarity. The similarity showed the weight between two compounds obtained from the hybrid similarity. A threshold (Th) was then applied to the HEA interaction network which was obtained by trial and error, and it eliminated weaker connections with less weight to obtain a network with strong connections. Relationships between HEA compounds were interactions based on hybrid similarity. Greater weight implied more similarities called strong connections. Thus, in the built-in HEA interaction network, each node represented an HEA compound, and their relationship was the hybrid similarity of the HEA compounds with each other [19]. \[content\ cosine\ similarity(x,y)=\frac{\sum_{i}(x_{i}y_{i})}{\sqrt{\sum_{i}x_{i}^{2} \sum_{i}y_{i}^{2}}} \tag{1}\]\[structural\text{ }]\text{ } Jaccard\text{ }similarity\left(y_{i},y_{j}\right)=\frac{\left|N_{i} \cap\text{ }N_{j}\right|}{\left|N_{i}\cup\text{ }N_{j}\right|} \tag{2}\]  
\[Hybrid\text{ }similarity=\alpha\left(\frac{\sum_{i}(x_{i}y_{i})}{\sqrt{\sum_{i}x_{i }^{2}\sum_{i}y_{i}^{2}}}\right)+(1-\alpha)\left(\frac{\left|N_{i}\cap\text{ }N_{j}\right|}{\left|N_{i}\cup\text{ }N_{j}\right|}\right) \tag{3}\]  
where \(x_{i}\) represents the \(i\)th descriptor of the first compound, and \(y_{i}\) is the \(i\)th descriptor of the second compound. Additionally, \(\left|N_{i}\cap\text{ }N_{j}\right|\) denotes the common features of the two compounds, and \(\left|N_{i}\cup\text{ }N_{j}\right|\) are all the descriptors. The \(N_{i}\) is defined as \(N_{i}=\left\{j\text{ }|\text{ }e(i,j)\geq Th\right\}\) in the HEA interaction network. We set \(Th=0.98\) for content cosine similarity to determine more similar descriptors. The traditional KNN algorithm is a lazy algorithm that does not make any models until the unseen data are imported. It uses the dataset to calculate the similarity between new data and records in the dataset. Then, it predicts an unseen data label by voting the most similar samples. The novelty of this study is the HEA compound phase prediction via the graph-based \(k\)-nearest neighbor (KNN) algorithm using the HEA interaction network. The proposed approach first selects all the neighbors of each compound in the network. The compound neighbors are described as \(\left[v_{1}\text{ }v_{2}\text{ }\ldots\text{ }v_{n}\right]^{\top}\), where the compounds \(v_{i}\)s are sorted in a descending order based on their weight with the target compound. Then, we used the \(k\) first compounds of \(v_{i}\)s and predicted the target compound phase by plurality vote. Figure 1 illustrates the proposed approach. The proposed method uses the set of related descriptors and creates an HEA interaction network based on the similarity between HEA compounds. The same colors in Figure 1b indicate the extracted communities from the HEA network. Figure 1c shows an extracted community as an example. Figure 1d illustrates the selection of the most similar neighbors of the target compound for phase prediction. Finally, it predicts the composition phase. The challenge considered is, if there are a compound's neighbors in the testing set, it cannot used for voting in phase prediction because its phase is unknown due to their presence on the testing set.