Figure 5.6 A graph autoencoder (GAE) for network embedding. The encoder uses graph convolutional layers to get a network embedding for each node. The decoder computes the pairwise distance given network embeddings. After applying a nonlinear activation function, the decoder reconstructs the graph adjacency matrix. The network is trained by minimizing the discrepancy between the real adjacency matrix and the reconstructed adjacency matrix.

      Source: Wu et al. [38].

      (5.60)

      where x_v = A_{v, :} and enc(·) is an encoder that consists of a multi‐layer perceptron. The second loss function enables the learned network embeddings to preserve the node’s second‐order proximity by minimizing the distance between a node’s inputs and its reconstructed inputs and is defined as

      (5.61)

      where b_{v, u} = 1 if A_{v, u} = 0, b_{v, u} = β > 1 if A_{v, u} = 1, and dec(·) is a decoder that consists of a multi‐layer perceptron.

      DNGR [54] and SDNE [55] only consider node structural information about the connectivity between pairs of nodes. They ignore the fact that the nodes may contain feature information that depicts the attributes of nodes themselves. Graph Autoencoder (GAE*) [56] leverages GCN [14] to encode node structural information and node feature information at the same time. The encoder of GAE* consists of two graph convolutional layers, which takes the form

(5.62)

      where Z denotes the network embedding matrix of a graph, f(·) is a ReLU activation function, and the Gconv (·) function is a graph convolutional layer defined by Eq. (5.44). The decoder of GAE* aims to decode node relational information from their embeddings by reconstructing the graph adjacency matrix, which is defined as

      (5.63)

where z_v is the embedding of node v. GAE* is trained by minimizing the negative cross‐entropy given the real adjacency matrix A and the reconstructed adjacency matrix

      Simply reconstructing the graph adjacency matrix may lead to overfitting due to the capacity of the autoencoders. The variational graph autoencoder (VGAE) [56] is a variational version of GAE that was developed to learn the distribution of data. The VGAE optimizes the variational lower bound L:

(5.64)

      where KL(·) is the Kullback–Leibler divergence function, which measures the distance between two distributions; p(Z) is a Gaussian prior , with q(z_i ∣ X, A) = N(z_i ∣ μ_i , diag .

The mean vector μ_i is the i‐th row of an encoder’s outputs defined by Eq. (5.62), and log σ_i is derived similarly as μ_i with another encoder. According to Eq. (5.64), VGAE assumes that the empirical distribution q(Z ∣ X, A) should be as close as possible to the prior distribution p(Z). To further enforce this, the empirical distribution q(Z ∣ X, A) is chosen to approximate the prior distribution p(Z).

      Like GAE*, GraphSAGE [23] encodes node features with two graph convolutional layers. Instead of optimizing the reconstruction error, GraphSAGE shows that the relational information between two nodes can be preserved by negative sampling with the loss:

      (5.65)

      where node u is a neighbor of node v, node v_n is a distant node to node v and is sampled from a negative sampling distribution P_n(v), and Q is the number of negative samples. This loss function essentially imposes similar representations on close nodes and dissimilar representations on distant nodes.

      Deep Recursive Network Embedding (DRNE) [57] assumes that a node’s network embedding should approximate the aggregation of its neighborhood network embeddings. It adopts an LSTM network [26] to aggregate a node’s neighbors. The reconstruction error of DRNE is defined as

(5.66)

where z_v is the network embedding of node v obtained by a dictionary look‐up,