# A Hierarchical Generative Embedding Model for Influence Maximization in Attributed Social Networks
## 1 Introduction  
Fueled by the spectacular growth of the internet and Internet of Things, plenty of social networks such as Facebook, Twitter and WeChat have sprung up, changed the mode of interaction between people, and accelerated the development of viral marketing. Originally from the idea of word-of-mouth advertising, viral marketing takes advantage of trust among close social circles of friends, colleagues or families to promote a new product, i.e., when a friend relationship affects a user making decisions on item selection [1, 2]. Motivated by applications to early viral marketing, a new study area of influence diffusion has thrived. Therein, the problem of influence maximization is to select a fixed size set of seed nodes in a network to maximize the influence spread according to a specially designed influence diffusion model. Figure 1 gives a toy example of social influence. The nodes \(v_{1}\),\(v_{2}\),\(v_{3}\) in black are the seed nodes which are initially active, and the nodes in gray color are newly activated by the seed nodes. In terms of viral marketing, for example, if user \(v_{1}\),\(v_{2}\),\(v_{3}\) bought a product, their friends in a given social network will likely buy this product because of the friend-to-friend influence. The applications of influence maximization are prevalent in the real world and include such purposes as viral marketing, anticipating natural hazards, reducing gang violence, public opinion supervision etc., whereas the time complexity of solving the problem of influence maximization is NP hard [3]. This inspires a lot of studies on influence diffusion and influence maximization algorithms. In early time, most studies have focused on the influence maximization algorithms themselves under a general independent cascade (IC) diffusion model or a general linear threshold (LT) diffusion model [4, 5], including greedy-based algorithms [3, 6, 7, 8, 9, 10] and heuristic-based algorithms [11, 12, 13]. In terms of running time, heuristic-based algorithms are generally more efficient than various greedy-based algorithms, but they do not have any theoretical guarantee. As in the real world, influence propagation often involves latent variables that are directly unobservable, including influence strength. Estimating influence strength is a fundamental problem for influence diffusion. Some models to estimate influence strength have been proposed, including the topic model [14, 15], probabilistic methods [16, 17] and models based on deep learning [18, 19]. Although existing models and methods have achieved a lot, they demonstrate a number of major drawbacks: (1) conventional approaches propose various humanly devised rules to extract features of user interactions, the effectiveness of which heavily depends on domain expert knowledge. (2) They only consider general network structure and node attributes as factors underlying social network influence, but in the real world, the structure of a large-scale network is more complex and often involves communities, community tensility and node depth in a clustering tree. (3) They are often applicable for special scenarios or specific diffusion models, not for generalized conditions. Like most scenarios in real-world propagation [16], influence strength, in this paper, is assumed to be directly unobservable. To predict influence strength, we need to compute the similarity of node pairs, with the motivation originating from previous studies that showed that the more equivalent the network structure and node attributes of two nodes are, the more likely they make similar judgments, even if they have no direct connection reciprocally [20]. Inspired by the powerful ability of neural networks in the field of representation learning, we design a deep hierarchical network embedding model, HGE, to map nodes into latent space automatically, preserving network structures and node attributes as much as possible.