# NodeVector: A Novel Network Node Vectorization with Graph Analysis and Deep Learning
## 3 Materials and Methods
### Node Similarities
This added edge also makes it different in its representations of nodes \(n_{x}\) and \(n_{y}\), whose neighbors are all the same. Figure 2 summarizes the \(L_{1}\), \(L_{2}\), and \(L_{3}\) level sample node representation. For given nodes\(n_{x}\) and \(n_{y}\) on target network \(G\), we use the Matching Index [34] to measure how similar (in terms of neighbors) the representations of nodes \(n_{x}\) and \(n_{y}\) are. We formulate the matching index between the representations of nodes \(n_{x}\) and \(n_{y}\) as the ratio of the number of common neighbors to that of all their neighbors. For a given level, \(k\), let us denote the set of neighbors of nodes \(n_{x}\) and \(n_{y}\) in network \(G\) with \(n_{x}^{n}\), \(n_{y}^{n}\), respectively. We compute the representation similarity (RS) of these given nodes with the following Equation (1). Due to the symmetric structure of the data structure, the computational complexity is \(O(n*(\dfrac{n+1}{2}))\) (n is node size). RS takes the value of 1 only when the target node is compared to itself, \(RS_{k}(n_{x},n_{x})\), and is less than 1 in all other cases. \[RS_{k}(n_{x},n_{y})=\dfrac{n_{x}^{n}\cap n_{y}^{n}}{n_{x}^{n}\cup n_{y}^{n}}. \tag{1}\]  
Nodes in networks tend to form clusters to perform various functions. With the RS similarity measurement we defined, we measure the similarity of the clusters formed by the neighbors of the nodes located at a predefined distance such as \(L_{1}\), \(L_{2}\), and \(L_{3}\) where \(k\) is 1, 2 and 3, respectively. With the help of the representation and similarity measurement, we define the clusters formed by nodes; we move from a narrow node definition (there is interaction between nodes, there is no interaction between nodes, and nodes are located on the same path, etc.) to a broader node definition. We perform network enrichment by finding possible edges that are not yet detected or that are likely to occur with the calculated similarities. Networks tend to feature smallworld. That is, interaction between two nodes can be established with a minimal number of edges. Due to this feature of the networks, a maximum of 3 is used for the number of edges parameter \(k\). As a first step in identifying candidate interactions, we calculate \(L_{1}\), \(L_{2}\), and \(L_{3}\) similarities between all paired nodes. We then sort the similarities at each level, with the most similar pair of nodes first. For a given network, \(G\), let us denote the candidate edge ratio with \(\epsilon\), the number of nodes with \(nn\), and the candidate edge count by adjacent similarity with \(cec_{adjsim}\). We calculate the number of new edges to be added from each level to the given network, \(G\), with the following Equation (2):  
\[cec_{adjsim}=nn*\epsilon/100. \tag{2}\]  
Finally, we select the first \(cec_{adjsim}\) pairs of nodes ordered by similarity at \(L_{1}\), \(L_{2}\), and \(L_{3}\) levels and add them as edges to target network \(G\). Algorithm 1 presents the pseudocode of our network enrichment method by our adjacent similarity. Figure 2: \(L_{1}\), \(L_{2}\), and \(L_{3}\) level sample network node representation. Each row in the table is related to \(L_{1}\), \(L_{2}\), and \(L_{3}\) level representation for Node 2. Each different level neighborhood is colored separately.