1. Graph representation learning
(network embedding / graph embedding / network representation learning) tries to embed each node of a graph into a low-dimensional vector space, which preserves the structural similarities or distances among the nodes in the original graph.
2. Graph分类：
按照Input：
- Homogeneous graph (e.g., citation network)
- Heterogeneous graph
Multimedia network
Knowledge graph (entity,relation)
- Graph with side information（辅助信息）
Node/edge label (categorical)
Node/edge attribute (discrete or continuous)
Node feature (e.g., texts)
- Graph transformed from non-relational data (从非关系型数据中转换成的图)
Manifold learning
按照Output:
- Node embedding (the most common case)
- Edge embedding
Relations in knowledge graph
Link prediction
- Sub-graph embedding
Substructure embedding
Community embedding
- Whole-graph embedding
Multiple small graphs, e.g., molecule, protein
按照Method:
- Matrix factorization
Singular value decomposition
Spectral decomposition (eigen-decomposition)
- Random walk
- Deep learning
Auto-encoder(SDNE)
Convolutional neural network
- Self-defined loss (LINE)
Maximizing edge reconstruction probability
Minimizing distance-based loss
Minimizing margin-based ranking loss
3. Motivation
网络表示学习方法可以分成两个类别。
一种是Generative model（生成式模型），假定对于每一个顶点，在图中存在一个潜在的、真实的连续性分布 Ptrue(v|vc)， 图中的每条边都可以看作是从Ptrue里采样的一些样本。生成式方法都试图将边的似然概率最大化，来学习vertex embedding。例如DeepWalk (KDD 2014) and node2vec (KDD 2016)。
Discriminative Model（判别式模型）将两顶点联合作为feature，预测两点之间存在边的概率。例如SDNE (KDD 2016) and PPNE (DASFAA, 2017)。
LINE (WWW 2015) 尝试将两者结合起来。而最近非常popular的GAN设计了一个 game-theoretical minimax game 将两者结合。
4. GraphGAN Framework
Generator G(v|vc) tries to fit the underlying true connectivity distribution ptrue(v|vc)，generates the most likely vertices to be connected with vc; Discriminator D(v; vc) tries to distinguish well-connected vertex pairs from ill-connected ones, outputs a single scalar representing the probability of an edge existing between v and vc.
对于上式，第一项的点是和vc真实相连的点sample出来的，第二项是从G生成的sample出来。给定，想minimize这个式子，学习G的参数，使G生成的点尽量像真实分布；给定
，maximize这个式子，学习D的参数，使得D给真实连接的pair值大，G生成的值小。
5. Discriminator
Given positive samples from true connectivity distribution and negative samples from the generator, the objective for the discriminator is to maximize the log-probability of assigning the correct labels, which could be solved by stochastic gradient ascent. D 定义为输入的两个顶点的内积的sigmoid函数， update only dv and dvc by ascending the gradient.
6. Generator
Because the sampling of v is discrete, we propose computing the gradient of V (G; D) with respect to θG by policy gradient:
一种最直观的想法是用softmax来实现G，也就是将G(v|VC)定义成一个softmax函数。这种定义有如下两个问题：首先是计算复杂度过高，计算会涉及到图中所有的节点，而且求导也需要更新图中所有节点。这样一来，大规模图将难以适用。 另一个问题是没有考虑图的结构特征，即这些点和Vc的距离未被纳入考虑范围内。
在GraphGAN 中，目标是设计出一种softmax方法，让其满足如下三个要求。第一个要求是正则化，即概率和为 1，它必须是一个合法的概率分布。第二个要求是能感知图结构，并且能充分利用图的结构特征信息。最后一个要求是计算效率高，也就是G概率只能涉及到图中的少部分节点。
7. Algorithm
备注：内容部分参考原作者PPT