## Appendix A
### DeepWalk [23]:  
it tries to learn a model by the following optimization function to maximize the probability of any node appearing in the \(v_{i}\)'s neighborhood with no knowledge of its offset from \(v_{i}\):  
\[\min_{f}\quad-\log\ \Pr\left(v_{i-n},...,v_{i+n}\setminus v_{i}\mid f(v_{i}) \right), \tag{11}\]  
where the Skip-gram model equipped with a hierarchical softmax and the stochastic gradient descent (SGD) [47] are utilized to optimize the objective function. **Line**[25]: first-order and second-order proximities are preserved by the two following objective functions, respectively:  
\[O_{1}=-\sum_{(i,j)\in E}\frac{w_{ij}}{1+\exp(-\overline{u}_{i}^{T}\cdot \overline{u}_{j})}, \tag{12}\]  
\[O_{2}=-\sum_{(i,j)\in E}\frac{w_{ij}\cdot\exp(\overline{\underline{u}}_{j}^{ T}\cdot\overline{u}_{i})}{\Sigma_{k=1}^{|V|}\exp(\overline{\underline{u}}_{k}^{ T}\cdot\overline{u}_{i})}, \tag{13}\]  
where \(w_{ij}\) is the weight of the link between \(v_{i}\) and \(v_{j}\) (in unweighted graphs, it is set as one), \(\overline{u}_{i}\) is the vector representation of \(v_{i}\), and \(\overline{\underline{u}}_{i}\) is the vector representation of \(v_{i}\) when it is regarded as a neighbor in the second-order proximity of other nodes. Line employs a negative sampling technique based on the links' weights and the asynchronous stochastic gradient descent (ASGD) [53]. Two models for the above objective functions are trained _separately_, and their results are concatenated as the final result. **node2 vec**[26]: it employs the following objective function:  
\[\max_{f}\quad\sum_{v\in V}[-\log Z_{v}+\sum_{u\in N(v)}f(u)\cdot f(v)], \tag{14}\]  
\[Z_{v}=\sum_{u\in V}\exp(f(v).f(u)),\]  
where \(N(v)\) is a set of nodes as \(v\)'s neighborhood. The above objective function is optimized by utilizing the negative sampling and the stochastic gradient ascent. **graphGAN**[27]: the objective is to train the two models as \(\mathbb{G}(u|v;\theta_{\mathbb{G}})\) (tries to approximate the underlying true connectivity distribution of \(v\), \(p_{t}(u|v)\)) and \(\mathbb{D}(u,v;\theta_{\mathbb{D}})\) (tries to discriminate the connectivity for the node-pair \((u,v)\)) by the following two-play minimax game with a value function \(\mathbb{V}(\mathbb{G},\mathbb{D})\):\[\min_{\theta_{G}}\max_{\theta_{D}}\mathbb{V}(G,D)=\sum_{v\in V}\big{(}\mathbb{E}_{u \sim p_{i}(.|v)}[\log\mathbb{D}(u,v;\theta_{D})]+\mathbb{E}_{u\sim G(.|v;\theta_ {D})}[\log(1-\mathbb{D}(u,v;\theta_{D}))]\big{)}, \tag{101}\]  
where \(\theta_{D}\) and \(\theta_{G}\) are the union of _all_ vector representations of nodes \(u\) constructed by \(\mathbb{D}\) and \(\mathbb{G}\), respectively. It employs a negative sampling technique where nodes _truly_ connected to \(v\) are used as positive samples and some _fabricated_ nodes are used as negative ones; \(\mathbb{D}\) is implemented by a softmax function, while \(\mathbb{G}\) is implemented by a graph softmax function [27]. **NetMF**[29]: it proposes the following low-rank connectivity matrix for DeepWalk:  
\[M=\frac{S}{s\cdot W}\cdot\big{(}\sum_{r=1}^{W}P^{r}\big{)}\cdot D^{-1} \tag{102}\]  
where \(S\) denotes the volume of \(G\) (i.e., summation of all entries in \(A\) as the adjacency matrix of \(G\)), \(s\) is the number for negative samples, \(W\) is the window size, \(D\) is a diagonal matrix containing values of \(d_{1},...,d_{|V|}\) as its diagonal entries where \(d_{i}\) is the degree of node \(v_{i}\), and \(P\) is defined as \(D^{-1}\cdot A\); \(M\) is explicitly factorized by the singular value decomposition (SVD) technique [43] (i.e., \(M=\mathbf{U}\Sigma\mathbf{V}\)) where row \(i\) in matrix \(\mathbf{U}\sqrt{\Sigma}\) contains the representation vector of node \(v_{i}\).