Network embedding aims to learn the low-dimensional representations of vertexes in a network, while structure and inherent properties of the network is preserved. Existing network embedding works primarily focus on preserving the microscopic structure, such as the first- and second-order proximity of vertexes, while the macroscopic scale-free property is largely ignored. Scale-free property depicts the fact that vertex degrees follow a heavy-tailed distribution (i.e., only a few vertexes have high degrees) and is a critical property of real-world networks, such as social networks. In this paper, we study the problem of learning representations for scale-free networks. We first theoretically analyze the difficulty of embedding and reconstructing a scale-free network in the Euclidean space, by converting our problem to the sphere packing problem. Then, we propose the "degree penalty" principle for designing scale-free property preserving network embedding algorithm: punishing the proximity between high-degree vertexes. We introduce two implementations of our principle by utilizing the spectral techniques and a skip-gram model respectively. Extensive experiments on six datasets show that our algorithms are able to not only reconstruct heavy-tailed distributed degree distribution, but also outperform state-of-the-art embedding models in various network mining tasks, such as vertex classification and link prediction.
Network embedding aims to learn representations for vertexes in a given network. Some researchers regard network embedding as part of dimensionality reduction techniques. For example, Laplacian Eigenmaps (LE) @cite_17 aims to learn the vertex representation to expand the manifold where the data lie. As a variant of LE, Locality Preserving Projections (LPP) @cite_4 learns a linear projection from feature space to embedding space. Besides, there are other linear @cite_21 and non-linear @cite_13 network embedding algorithms for dimensionality reduction. Recent network embedding works take advancements in natural language processing, most notably models known as word2vec @cite_34 , which learns the distributed representations of words. Building on word2vec, define a vertex's context'' by their co-occurrence in a random walk path @cite_1 . More recently, propose a mixture of width-first and breadth-first search based procedure to generate paths of vertexes @cite_14 . further develop the model to handle heterogeneous networks @cite_16 . LINE @cite_40 decomposes a vertex's context into first-order (neighbors) and second-order (two-degree neighbors) proximity. preserve community information in their vertex representations @cite_5 . However, all of above methods focus on preserving microscopic network structure and ignore macroscopic scale-free property of networks.
@cite_14 @cite_4 @cite_21 @cite_1 @cite_16 @cite_40 @cite_5 @cite_34 @cite_13 @cite_17
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Network embedding is an important method to learn low-dimensional representations of vertexes in networks, aiming to capture and preserve the network structure. Almost all the existing network embedding methods adopt shallow models. However, since the underlying network structure is complex, shallow models cannot capture the highly non-linear network structure, resulting in sub-optimal network representations. Therefore, how to find a method that is able to effectively capture the highly non-linear network structure and preserve the global and local structure is an open yet important problem. To solve this problem, in this paper we propose a Structural Deep Network Embedding method, namely SDNE. More specifically, we first propose a semi-supervised deep model, which has multiple layers of non-linear functions, thereby being able to capture the highly non-linear network structure. Then we propose to exploit the first-order and second-order proximity jointly to preserve the network structure. The second-order proximity is used by the unsupervised component to capture the global network structure. While the first-order proximity is used as the supervised information in the supervised component to preserve the local network structure. By jointly optimizing them in the semi-supervised deep model, our method can preserve both the local and global network structure and is robust to sparse networks. Empirically, we conduct the experiments on five real-world networks, including a language network, a citation network and three social networks. The results show that compared to the baselines, our method can reconstruct the original network significantly better and achieves substantial gains in three applications, i.e. multi-label classification, link prediction and visualization. Heterogenous information network embedding aims to embed heterogenous information networks (HINs) into low dimensional spaces, in which each vertex is represented as a low-dimensional vector, and both global and local network structures in the original space are preserved. However, most of existing heterogenous information network embedding models adopt the dot product to measure the proximity in the low dimensional space, and thus they can only preserve the first-order proximity and are insufficient to capture the global structure. Compared with homogenous information networks, there are multiple types of links (i.e., multiple relations) in HINs, and the link distribution w.r.t relations is highly skewed. To address the above challenging issues, we propose a novel heterogenous information network embedding model PME based on the metric learning to capture both first-order and second-order proximities in a unified way. To alleviate the potential geometrical inflexibility of existing metric learning approaches, we propose to build object and relation embeddings in separate object space and relation spaces rather than in a common space. Afterwards, we learn embeddings by firstly projecting vertices from object space to corresponding relation space and then calculate the proximity between projected vertices. To overcome the heavy skewness of the link distribution w.r.t relations and avoid "over-sampling'' or "under-sampling'' for each relation, we propose a novel loss-aware adaptive sampling approach for the model optimization. Extensive experiments have been conducted on a large-scale HIN dataset, and the experimental results show superiority of our proposed PME model in terms of prediction accuracy and scalability. We propose a new method for embedding graphs while preserving directed edge information. Learning such continuous-space vector representations (or embeddings) of nodes in a graph is an important first step for using network information (from social networks, user-item graphs, knowledge bases, etc.) in many machine learning tasks. Unlike previous work, we (1) explicitly model an edge as a function of node embeddings, and we (2) propose a novel objective, the graph likelihood, which contrasts information from sampled random walks with non-existent edges. Individually, both of these contributions improve the learned representations, especially when there are memory constraints on the total size of the embeddings. When combined, our contributions enable us to significantly improve the state-of-the-art by learning more concise representations that better preserve the graph structure. We evaluate our method on a variety of link-prediction task including social networks, collaboration networks, and protein interactions, showing that our proposed method learn representations with error reductions of up to 76 and 55 , on directed and undirected graphs. In addition, we show that the representations learned by our method are quite space efficient, producing embeddings which have higher structure-preserving accuracy but are 10 times smaller. Network embedding methodologies, which learn a distributed vector representation for each vertex in a network, have attracted considerable interest in recent years. Existing works have demonstrated that vertex representation learned through an embedding method provides superior performance in many real-world applications, such as node classification, link prediction, and community detection. However, most of the existing methods for network embedding only utilize topological information of a vertex, ignoring a rich set of nodal attributes (such as, user profiles of an online social network, or textual contents of a citation network), which is abundant in all real-life networks. A joint network embedding that takes into account both attributional and relational information entails a complete network information and could further enrich the learned vector representations. In this work, we present Neural-Brane, a novel Neural Bayesian Personalized Ranking based Attributed Network Embedding. For a given network, Neural-Brane extracts latent feature representation of its vertices using a designed neural network model that unifies network topological information and nodal attributes; Besides, it utilizes Bayesian personalized ranking objective, which exploits the proximity ordering between a similar node-pair and a dissimilar node-pair. We evaluate the quality of vertex embedding produced by Neural-Brane by solving the node classification and clustering tasks on four real-world datasets. Experimental results demonstrate the superiority of our proposed method over the state-of-the-art existing methods.