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. Over the past few years, some embedding methods have been proposed for feature extraction and dimensionality reduction in various machine learning and pattern classification tasks. Among the methods proposed are Neighborhood Preserving Embedding (NPE), Locality Preserving Projection (LPP) and Local Discriminant Embedding (LDE) which have been used in such applications as face recognition and image video retrieval. However, although the data in these applications are more naturally represented as higher-order tensors, the embedding methods can only work with vectorized data representations which may not capture well some useful information in the original data. Moreover, high-dimensional vectorized representations also suffer from the curse of dimensionality and the high computational demand. In this paper, we propose some novel tensor embedding methods which, unlike previous methods, take data directly in the form of tensors of arbitrary order as input. These methods allow the relationships between dimensions of a tensor representation to be efficiently characterized. Moreover, they also allow the intrinsic local geometric and topological properties of the manifold embedded in a tensor space to be naturally estimated. Furthermore, they do not suffer from the curse of dimensionality and the high computational demand. We demonstrate the effectiveness of the proposed tensor embedding methods on a face recognition application and compare them with some previous methods. Extensive experiments show that our methods are not only more effective but also more efficient. Since the invention of word2vec, the skip-gram model has significantly advanced the research of network embedding, such as the recent emergence of the DeepWalk, LINE, PTE, and node2vec approaches. In this work, we show that all of the aforementioned models with negative sampling can be unified into the matrix factorization framework with closed forms. Our analysis and proofs reveal that: (1) DeepWalk empirically produces a low-rank transformation of a network's normalized Laplacian matrix; (2) LINE, in theory, is a special case of DeepWalk when the size of vertices' context is set to one; (3) As an extension of LINE, PTE can be viewed as the joint factorization of multiple networks» Laplacians; (4) node2vec is factorizing a matrix related to the stationary distribution and transition probability tensor of a 2nd-order random walk. We further provide the theoretical connections between skip-gram based network embedding algorithms and the theory of graph Laplacian. Finally, we present the NetMF method as well as its approximation algorithm for computing network embedding. Our method offers significant improvements over DeepWalk and LINE for conventional network mining tasks. This work lays the theoretical foundation for skip-gram based network embedding methods, leading to a better understanding of latent network representation learning. Network embedding task aims at learning low-dimension latent representations of vertices while preserving the structure of a network simultaneously. Most existing network embedding methods mainly focus on static networks, which extract and condense the network information without temporal information. However, in the real world, networks keep evolving, where the linkage states between the same vertex pairs at consequential timestamps have very close correlations. In this paper, we propose to study the network embedding problem and focus on modeling the linkage evolution in the dynamic network setting. To address this problem, we propose a deep dynamic network embedding method. More specifically, the method utilizes the historical information obtained from the network snapshots at past timestamps to learn latent representations of the future network. In the proposed embedding method, the objective function is carefully designed to incorporate both the network internal and network dynamic transition structures. Extensive empirical experiments prove the effectiveness of the proposed model on various categories of real-world networks, including a human contact network, a bibliographic network, and e-mail networks. Furthermore, the experimental results also demonstrate the significant advantages of the method compared with both the state-of-the-art embedding techniques and several existing baseline methods. Network embedding, aiming to learn the low-dimensional representations of nodes in networks, is of paramount importance in many real applications. One basic requirement of network embedding is to preserve the structure and inherent properties of the networks. While previous network embedding methods primarily preserve the microscopic structure, such as the first- and second-order proximities of nodes, the mesoscopic community structure, which is one of the most prominent feature of networks, is largely ignored. In this paper, we propose a novel Modularized Nonnegative Matrix Factorization (M-NMF) model to incorporate the community structure into network embedding. We exploit the consensus relationship between the representations of nodes and community structure, and then jointly optimize NMF based representation learning model and modularity based community detection model in a unified framework, which enables the learned representations of nodes to preserve both of the microscopic and community structures. We also provide efficient updating rules to infer the parameters of our model, together with the correctness and convergence guarantees. Extensive experimental results on a variety of real-world networks show the superior performance of the proposed method over the state-of-the-arts. Network embedding leverages the node proximity manifested to learn a low-dimensional node vector representation for each node in the network. The learned embeddings could advance various learning tasks such as node classification, network clustering, and link prediction. Most, if not all, of the existing works, are overwhelmingly performed in the context of plain and static networks. Nonetheless, in reality, network structure often evolves over time with addition deletion of links and nodes. Also, a vast majority of real-world networks are associated with a rich set of node attributes, and their attribute values are also naturally changing, with the emerging of new content patterns and the fading of old content patterns. These changing characteristics motivate us to seek an effective embedding representation to capture network and attribute evolving patterns, which is of fundamental importance for learning in a dynamic environment. To our best knowledge, we are the first to tackle this problem with the following two challenges: (1) the inherently correlated network and node attributes could be noisy and incomplete, it necessitates a robust consensus representation to capture their individual properties and correlations; (2) the embedding learning needs to be performed in an online fashion to adapt to the changes accordingly. In this paper, we tackle this problem by proposing a novel dynamic attributed network embedding framework - DANE. In particular, DANE first provides an offline method for a consensus embedding and then leverages matrix perturbation theory to maintain the freshness of the end embedding results in an online manner. We perform extensive experiments on both synthetic and real attributed networks to corroborate the effectiveness and efficiency of the proposed framework. The graph embedding paradigm projects nodes of a graph into a vector space, which can facilitate various downstream graph analysis tasks such as node classification and clustering. To efficiently learn node embeddings from a graph, graph embedding techniques usually preserve the proximity between node pairs sampled from the graph using random walks. In the context of a heterogeneous graph, which contains nodes from different domains, classical random walks are biased towards highly visible domains where nodes are associated with a dominant number of paths. To overcome this bias, existing heterogeneous graph embedding techniques typically rely on meta-paths (i.e., fixed sequences of node types) to guide random walks. However, using these meta-paths either requires prior knowledge from domain experts for optimal meta-path selection, or requires extended computations to combine all meta-paths shorter than a predefined length. In this paper, we propose an alternative solution that does not involve any meta-path. Specifically, we propose JUST, a heterogeneous graph embedding technique using random walks with JUmp and STay strategies to overcome the aforementioned bias in an more efficient manner. JUST can not only gracefully balance between homogeneous and heterogeneous edges, it can also balance the node distribution over different domains (i.e., node types). By conducting a thorough empirical evaluation of our method on three heterogeneous graph datasets, we show the superiority of our proposed technique. In particular, compared to a state-of-the-art heterogeneous graph embedding technique Hin2vec, which tries to optimally combine all meta-paths shorter than a predefined length, our technique yields better results in most experiments, with a dramatically reduced embedding learning time (about 3x speedup). This work develops a representation learning method for bipartite networks. While existing works have developed various embedding methods for network data, they have primarily focused on homogeneous networks in general and overlooked the special properties of bipartite networks. As such, these methods can be suboptimal for embedding bipartite networks. In this paper, we propose a new method named BiNE, short for Bipartite Network Embedding, to learn the vertex representations for bipartite networks. By performing biased random walks purposefully, we generate vertex sequences that can well preserve the long-tail distribution of vertices in the original bipartite network. We then propose a novel optimization framework by accounting for both the explicit relations (i.e., observed links) and implicit relations (i.e., unobserved but transitive links) in learning the vertex representations. We conduct extensive experiments on several real datasets covering the tasks of link prediction (classification), recommendation (personalized ranking), and visualization. Both quantitative results and qualitative analysis verify the effectiveness and rationality of our BiNE method. Celebrated and its fruitful variants are powerful models to achieve excellent performance on the tasks that map sequences to sequences. However, these are many machine learning tasks with inputs naturally represented in a form of graphs, which imposes significant challenges to existing Seq2Seq models for lossless conversion from its graph form to the sequence. In this work, we present a general end-to-end approach to map the input graph to a sequence of vectors, and then another attention-based LSTM to decode the target sequence from these vectors. Specifically, to address inevitable information loss for data conversion, we introduce a novel graph-to-sequence neural network model that follows the encoder-decoder architecture. Our method first uses an improved graph-based neural network to generate the node and graph embeddings by a novel aggregation strategy to incorporate the edge direction information into the node embeddings. We also propose an attention based mechanism that aligns node embeddings and decoding sequence to better cope with large graphs. Experimental results on bAbI task, Shortest Path Task, and Natural Language Generation Task demonstrate that our model achieves the state-of-the-art performance and significantly outperforms other baselines. We also show that with the proposed aggregation strategy, our proposed model is able to quickly converge to good performance. Graph Embedding methods are aimed at mapping each vertex into a low dimensional vector space, which preserves certain structural relationships among the vertices in the original graph. Recently, several works have been proposed to learn embeddings based on sampled paths from the graph, e.g., DeepWalk, Line, Node2Vec. However, their methods only preserve symmetric proximities, which could be insufficient in many applications, even the underlying graph is undirected. Besides, they lack of theoretical analysis of what exactly the relationships they preserve in their embedding space. In this paper, we propose an asymmetric proximity preserving (APP) graph embedding method via random walk with restart, which captures both asymmetric and high-order similarities between node pairs. We give theoretical analysis that our method implicitly preserves the Rooted PageRank score for any two vertices. We conduct extensive experiments on tasks of link prediction and node recommendation on open source datasets, as well as online recommendation services in Alibaba Group, in which the training graph has over 290 million vertices and 18 billion edges, showing our method to be highly scalable and effective. Attributed network embedding aims to seek low-dimensional vector representations for nodes in a network, such that original network topological structure and node attribute proximity can be preserved in the vectors. These learned representations have been demonstrated to be helpful in many learning tasks such as network clustering and link prediction. While existing algorithms follow an unsupervised manner, nodes in many real-world attributed networks are often associated with abundant label information, which is potentially valuable in seeking more effective joint vector representations. In this paper, we investigate how labels can be modeled and incorporated to improve attributed network embedding. This is a challenging task since label information could be noisy and incomplete. In addition, labels are completely distinct with the geometrical structure and node attributes. The bewildering combination of heterogeneous information makes the joint vector representation learning more difficult. To address these issues, we propose a novel Label informed Attributed Network Embedding (LANE) framework. It can smoothly incorporate label information into the attributed network embedding while preserving their correlations. Experiments on real-world datasets demonstrate that the proposed framework achieves significantly better performance compared with the state-of-the-art embedding algorithms. Learning low-dimensional representations of networks has proved effective in a variety of tasks such as node classification, link prediction and network visualization. Existing methods can effectively encode different structural properties into the representations, such as neighborhood connectivity patterns, global structural role similarities and other high-order proximities. However, except for objectives to capture network structural properties, most of them suffer from lack of additional constraints for enhancing the robustness of representations. In this paper, we aim to exploit the strengths of generative adversarial networks in capturing latent features, and investigate its contribution in learning stable and robust graph representations. Specifically, we propose an Adversarial Network Embedding (ANE) framework, which leverages the adversarial learning principle to regularize the representation learning. It consists of two components, i.e., a structure preserving component and an adversarial learning component. The former component aims to capture network structural properties, while the latter contributes to learning robust representations by matching the posterior distribution of the latent representations to given priors. As shown by the empirical results, our method is competitive with or superior to state-of-the-art approaches on benchmark network embedding tasks. In this paper, we study the problem of author identification under double-blind review setting, which is to identify potential authors given information of an anonymized paper. Different from existing approaches that rely heavily on feature engineering, we propose to use network embedding approach to address the problem, which can automatically represent nodes into lower dimensional feature vectors. However, there are two major limitations in recent studies on network embedding: (1) they are usually general-purpose embedding methods, which are independent of the specific tasks; and (2) most of these approaches can only deal with homogeneous networks, where the heterogeneity of the network is ignored. Hence, challenges faced here are two folds: (1) how to embed the network under the guidance of the author identification task, and (2) how to select the best type of information due to the heterogeneity of the network. To address the challenges, we propose a task-guided and path-augmented heterogeneous network embedding model. In our model, nodes are first embedded as vectors in latent feature space. Embeddings are then shared and jointly trained according to task-specific and network-general objectives. We extend the existing unsupervised network embedding to incorporate meta paths in heterogeneous networks, and select paths according to the specific task. The guidance from author identification task for network embedding is provided both explicitly in joint training and implicitly during meta path selection. Our experiments demonstrate that by using path-augmented network embedding with task guidance, our model can obtain significantly better accuracy at identifying the true authors comparing to existing methods. Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones. We consider learning representations of entities and relations in KBs using the neural-embedding approach. We show that most existing models, including NTN (, 2013) and TransE (, 2013b), can be generalized under a unified learning framework, where entities are low-dimensional vectors learned from a neural network and relations are bilinear and or linear mapping functions. Under this framework, we compare a variety of embedding models on the link prediction task. We show that a simple bilinear formulation achieves new state-of-the-art results for the task (achieving a top-10 accuracy of 73.2 vs. 54.7 by TransE on Freebase). Furthermore, we introduce a novel approach that utilizes the learned relation embeddings to mine logical rules such as "BornInCity(a,b) and CityInCountry(b,c) => Nationality(a,c)". We find that embeddings learned from the bilinear objective are particularly good at capturing relational semantics and that the composition of relations is characterized by matrix multiplication. More interestingly, we demonstrate that our embedding-based rule extraction approach successfully outperforms a state-of-the-art confidence-based rule mining approach in mining Horn rules that involve compositional reasoning. This paper proposes a novel translation-based knowledge graph embedding that preserves the logical properties of relations such as transitivity and symmetricity. The embedding space generated by existing translation-based embeddings do not represent transitive and symmetric relations precisely, because they ignore the role of entities in triples. Thus, we introduce a role-specific projection which maps an entity to distinct vectors according to its role in a triple. That is, a head entity is projected onto an embedding space by a head projection operator, and a tail entity is projected by a tail projection operator. This idea is applied to TransE, TransR, and TransD to produce lppTransE, lppTransR, and lppTransD, respectively. According to the experimental results on link prediction and triple classification, the proposed logical property preserving embeddings show the state-of-the-art performance at both tasks. These results prove that it is critical to preserve logical properties of relations while embedding knowledge graphs, and the proposed method does it effectively. In this paper, we introduce a feature-preserving denoising algorithm. It is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions. A vertex close to a sharp feature is likely to have a neighborhood that includes distinct smooth segments. By defining the consistent subneighborhood as the segment whose geometry and normal orientation most consistent with those of the vertex, we can completely remove the influence from neighbors lying on other segments during denoising. Our method identifies piecewise smooth subneighborhoods using a robust density-based clustering algorithm based on shared nearest neighbors. In our method, we obtain an initial estimate of vertex normals and curvature tensors by robustly fitting a local quadric model. An anisotropic filter based on optimal estimation theory is further applied to smooth the normal field and the curvature tensor field. This is followed by second-order bilateral filtering, which better preserves curvature details and alleviates volume shrinkage during denoising. The support of these filters is defined by the consistent subneighborhood of a vertex. We have applied this algorithm to both generic and CAD models, and sharp features, such as edges and corners, are very well preserved.