Network (or graph) embedding is the task to map the nodes of a graph to a lower dimensional vector space, such that it preserves the graph properties and facilitates the downstream network mining tasks. Real world networks often come with (community) outlier nodes, which behave differently from the regular nodes of the community. These outlier nodes can affect the embedding of the regular nodes, if not handled carefully. In this paper, we propose a novel unsupervised graph embedding approach (called DMGD) which integrates outlier and community detection with node embedding. We extend the idea of deep support vector data description to the framework of graph embedding when there are multiple communities present in the given network, and an outlier is characterized relative to its community. We also show the theoretical bounds on the number of outliers detected by DMGD. Our formulation boils down to an interesting minimax game between the outliers, community assignments and the node embedding function. We also propose an efficient algorithm to solve this optimization framework. Experimental results on both synthetic and real world networks show the merit of our approach compared to state-of-the-arts.
Various non-trivial spaces are becoming popular for embedding structured data such as graphs, texts, or images. Following spherical and hyperbolic spaces, more general product spaces have been proposed. However, searching for the best configuration of product space is a resource-intensive procedure, which reduces the practical applicability of the idea. We generalize the concept of product space and introduce an overlapping space that does not have the configuration search problem. The main idea is to allow subsets of coordinates to be shared between spaces of different types (Euclidean, hyperbolic, spherical). As a result, parameter optimization automatically learns the optimal configuration. Additionally, overlapping spaces allow for more compact representations since their geometry is more complex. Our experiments confirm that overlapping spaces outperform the competitors in graph embedding tasks. Here, we consider both distortion setup, where the aim is to preserve distances, and ranking setup, where the relative order should be preserved. The proposed method effectively solves the problem and outperforms the competitors in both settings. We also perform an empirical analysis in a realistic information retrieval task, where we compare all spaces by incorporating them into DSSM. In this case, the proposed overlapping space consistently achieves nearly optimal results without any configuration tuning. This allows for reducing training time, which can be significant in large-scale applications.
Attributed graph embedding, which learns vector representations from graph topology and node features, is a challenging task for graph analysis. Recently, methods based on graph convolutional networks (GCNs) have made great progress on this task. However,existing GCN-based methods have three major drawbacks. Firstly,our experiments indicate that the entanglement of graph convolutional filters and weight matrices will harm both the performance and robustness. Secondly, we show that graph convolutional filters in these methods reveal to be special cases of generalized Laplacian smoothing filters, but they do not preserve optimal low-pass characteristics. Finally, the training objectives of existing algorithms are usually recovering the adjacency matrix or feature matrix, which are not always consistent with real-world applications. To address these issues, we propose Adaptive Graph Encoder (AGE), a novel attributed graph embedding framework. AGE consists of two modules: (1) To better alleviate the high-frequency noises in the node features, AGE first applies a carefully-designed Laplacian smoothing filter. (2) AGE employs an adaptive encoder that iteratively strengthens the filtered features for better node embeddings. We conduct experiments using four public benchmark datasets to validate AGE on node clustering and link prediction tasks. Experimental results show that AGE consistently outperforms state-of-the-art graph embedding methods considerably on these tasks.
Extracting information from real-world large networks is a key challenge nowadays. For instance, computing a node centrality may become unfeasible depending on the intended centrality due to its computational cost. One solution is to develop fast methods capable of approximating network centralities. Here, we propose an approach for efficiently approximating node centralities for large networks using Neural Networks and Graph Embedding techniques. Our proposed model, entitled Network Centrality Approximation using Graph Embedding (NCA-GE), uses the adjacency matrix of a graph and a set of features for each node (here, we use only the degree) as input and computes the approximate desired centrality rank for every node. NCA-GE has a time complexity of $O(|E|)$, $E$ being the set of edges of a graph, making it suitable for large networks. NCA-GE also trains pretty fast, requiring only a set of a thousand small synthetic scale-free graphs (ranging from 100 to 1000 nodes each), and it works well for different node centralities, network sizes, and topologies. Finally, we compare our approach to the state-of-the-art method that approximates centrality ranks using the degree and eigenvector centralities as input, where we show that the NCA-GE outperforms the former in a variety of scenarios.
Knowledge graph completion is an important task that aims to predict the missing relational link between entities. Knowledge graph embedding methods perform this task by representing entities and relations as embedding vectors and modeling their interactions to compute the matching score of each triple. Previous work has usually treated each embedding as a whole and has modeled the interactions between these whole embeddings, potentially making the model excessively expensive or requiring specially designed interaction mechanisms. In this work, we propose the multi-partition embedding interaction (MEI) model with block term format to systematically address this problem. MEI divides each embedding into a multi-partition vector to efficiently restrict the interactions. Each local interaction is modeled with the Tucker tensor format and the full interaction is modeled with the block term tensor format, enabling MEI to control the trade-off between expressiveness and computational cost, learn the interaction mechanisms from data automatically, and achieve state-of-the-art performance on the link prediction task. In addition, we theoretically study the parameter efficiency problem and derive a simple empirically verified criterion for optimal parameter trade-off. We also apply the framework of MEI to provide a new generalized explanation for several specially designed interaction mechanisms in previous models. The source code is released at https://github.com/tranhungnghiep/MEI-KGE.
Various classes of Graph Neural Networks (GNN) have been proposed and shown to be successful in a wide range of applications with graph structured data. In this paper, we propose a theoretical framework able to compare the expressive power of these GNN architectures. The current universality theorems only apply to intractable classes of GNNs. Here, we prove the first approximation guarantees for practical GNNs, paving the way for a better understanding of their generalization. Our theoretical results are proved for invariant GNNs computing a graph embedding (permutation of the nodes of the input graph does not affect the output) and equivariant GNNs computing an embedding of the nodes (permutation of the input permutes the output). We show that Folklore Graph Neural Networks (FGNN), which are tensor based GNNs augmented with matrix multiplication are the most expressive architectures proposed so far for a given tensor order. We illustrate our results on the Quadratic Assignment Problem (a NP-Hard combinatorial problem) by showing that FGNNs are able to learn how to solve the problem, leading to much better average performances than existing algorithms (based on spectral, SDP or other GNNs architectures). On a practical side, we also implement masked tensors to handle batches of graphs of varying sizes.
Much of biomedical and healthcare data is encoded in discrete, symbolic form such as text and medical codes. There is a wealth of expert-curated biomedical domain knowledge stored in knowledge bases and ontologies, but the lack of reliable methods for learning knowledge representation has limited their usefulness in machine learning applications. While text-based representation learning has significantly improved in recent years through advances in natural language processing, attempts to learn biomedical concept embeddings so far have been lacking. A recent family of models called knowledge graph embeddings have shown promising results on general domain knowledge graphs, and we explore their capabilities in the biomedical domain. We train several state-of-the-art knowledge graph embedding models on the SNOMED-CT knowledge graph, provide a benchmark with comparison to existing methods and in-depth discussion on best practices, and make a case for the importance of leveraging the multi-relational nature of knowledge graphs for learning biomedical knowledge representation. The embeddings, code, and materials will be made available to the communitY.
The heterogeneity in recently published knowledge graph embedding models' implementations, training, and evaluation has made fair and thorough comparisons difficult. In order to assess the reproducibility of previously published results, we re-implemented and evaluated 21 interaction models in the PyKEEN software package. Here, we outline which results could be reproduced with their reported hyper-parameters, which could only be reproduced with alternate hyper-parameters, and which could not be reproduced at all as well as provide insight as to why this might be the case. We then performed a large-scale benchmarking on four datasets with several thousands of experiments and 24,804 GPU hours of computation time. We present insights gained as to best practices, best configurations for each model, and where improvements could be made over previously published best configurations. Our results highlight that the combination of model architecture, training approach, loss function, and the explicit modeling of inverse relations is crucial for a model's performances, and not only determined by the model architecture. We provide evidence that several architectures can obtain results competitive to the state-of-the-art when configured carefully. We have made all code, experimental configurations, results, and analyses that lead to our interpretations available at https://github.com/pykeen/pykeen and https://github.com/pykeen/benchmarking
In this paper, we propose an end-to-end graph learning framework, namely Iterative Deep Graph Learning (IDGL), for jointly and iteratively learning graph structure and graph embedding. The key rationale of IDGL is to learn a better graph structure based on better node embeddings, and vice versa (i.e., better node embeddings based on a better graph structure). Our iterative method dynamically stops when the learned graph structure approaches close enough to the graph optimized for the downstream prediction task. In addition, we cast the graph learning problem as a similarity metric learning problem and leverage adaptive graph regularization for controlling the quality of the learned graph. Finally, combining the anchor-based approximation technique, we further propose a scalable version of IDGL, namely IDGL-Anch, which significantly reduces the time and space complexity of IDGL without compromising the performance. Our extensive experiments on nine benchmarks show that our proposed IDGL models can consistently outperform or match the state-of-the-art baselines. Furthermore, IDGL can be more robust to adversarial graphs and cope with both transductive and inductive learning.
We present Wasserstein Embedding for Graph Learning (WEGL), a novel and fast framework for embedding entire graphs in a vector space, in which various machine learning models are applicable for graph-level prediction tasks. We leverage new insights on defining similarity between graphs as a function of the similarity between their node embedding distributions. Specifically, we use the Wasserstein distance to measure the dissimilarity between node embeddings of different graphs. Unlike prior work, we avoid pairwise calculation of distances between graphs and reduce the computational complexity from quadratic to linear in the number of graphs. WEGL calculates Monge maps from a reference distribution to each node embedding and, based on these maps, creates a fixed-sized vector representation of the graph. We evaluate our new graph embedding approach on various benchmark graph-property prediction tasks, showing state-of-the-art classification performance while having superior computational efficiency. The code is available at https://github.com/navid-naderi/WEGL.
Recently, a considerable literature has grown up around the theme of Graph Convolutional Network (GCN). How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly propagating and updating the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the Knowledge Embedding based Graph Convolutional Network (KE-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge embedding (a.k.a. knowledge graph embedding) methods, and goes beyond. Our theoretical analysis shows that KE-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of KE-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.
Graph embedding techniques, which learn low-dimensional representations of a graph, are achieving state-of-the-art performance in many graph mining tasks. Most existing embedding algorithms assign a single vector to each node, implicitly assuming that a single representation is enough to capture all characteristics of the node. However, across many domains, it is common to observe pervasively overlapping community structure, where most nodes belong to multiple communities, playing different roles depending on the contexts. Here, we propose persona2vec, a graph embedding framework that efficiently learns multiple representations of nodes based on their structural contexts. Using link prediction-based evaluation, we show that our framework is significantly faster than the existing state-of-the-art model while achieving better performance.
Learning distributions of graphs can be used for automatic drug discovery, molecular design, complex network analysis, and much more. We present an improved framework for learning generative models of graphs based on the idea of deep state machines. To learn state transition decisions we use a set of graph and node embedding techniques as memory of the state machine. Our analysis is based on learning the distribution of random graph generators for which we provide statistical tests to determine which properties can be learned and how well the original distribution of graphs is represented. We show that the design of the state machine favors specific distributions. Models of graphs of size up to 150 vertices are learned. Code and parameters are publicly available to reproduce our results.
Sampling graphs is an important task in data mining. In this paper, we describe Little Ball of Fur a Python library that includes more than twenty graph sampling algorithms. Our goal is to make node, edge, and exploration-based network sampling techniques accessible to a large number of professionals, researchers, and students in a single streamlined framework. We created this framework with a focus on a coherent application public interface which has a convenient design, generic input data requirements, and reasonable baseline settings of algorithms. Here we overview these design foundations of the framework in detail with illustrative code snippets. We show the practical usability of the library by estimating various global statistics of social networks and web graphs. Experiments demonstrate that Little Ball of Fur can speed up node and whole graph embedding techniques considerably with mildly deteriorating the predictive value of distilled features.
Several state-of-the-art neural graph embedding methods are based on short random walks (stochastic processes) because of their ease of computation, simplicity in capturing complex local graph properties, scalability, and interpretibility. In this work, we are interested in studying how much a probabilistic bias in this stochastic process affects the quality of the nodes picked by the process. In particular, our biased walk, with a certain probability, favors movement towards nodes whose neighborhoods bear a structural resemblance to the current node's neighborhood. We succinctly capture this neighborhood as a probability measure based on the spectrum of the node's neighborhood subgraph represented as a normalized laplacian matrix. We propose the use of a paragraph vector model with a novel Wasserstein regularization term. We empirically evaluate our approach against several state-of-the-art node embedding techniques on a wide variety of real-world datasets and demonstrate that our proposed method significantly improves upon existing methods on both link prediction and node classification tasks.
As communities represent similar opinions, similar functions, similar purposes, etc., community detection is an important and extremely useful tool in both scientific inquiry and data analytics. However, the classic methods of community detection, such as spectral clustering and statistical inference, are falling by the wayside as deep learning techniques demonstrate an increasing capacity to handle high-dimensional graph data with impressive performance. Thus, a survey of current progress in community detection through deep learning is timely. Structured into three broad research streams in this domain - deep neural networks, deep graph embedding, and graph neural networks, this article summarizes the contributions of the various frameworks, models, and algorithms in each stream along with the current challenges that remain unsolved and the future research opportunities yet to be explored.
There has been a surge of recent interest in learning representations for graph-structured data. Graph representation learning methods have generally fallen into three main categories, based on the availability of labeled data. The first, network embedding (such as shallow graph embedding or graph auto-encoders), focuses on learning unsupervised representations of relational structure. The second, graph regularized neural networks, leverages graphs to augment neural network losses with a regularization objective for semi-supervised learning. The third, graph neural networks, aims to learn differentiable functions over discrete topologies with arbitrary structure. However, despite the popularity of these areas there has been surprisingly little work on unifying the three paradigms. Here, we aim to bridge the gap between graph neural networks, network embedding and graph regularization models. We propose a comprehensive taxonomy of representation learning methods for graph-structured data, aiming to unify several disparate bodies of work. Specifically, we propose a Graph Encoder Decoder Model (GRAPHEDM), which generalizes popular algorithms for semi-supervised learning on graphs (e.g. GraphSage, Graph Convolutional Networks, Graph Attention Networks), and unsupervised learning of graph representations (e.g. DeepWalk, node2vec, etc) into a single consistent approach. To illustrate the generality of this approach, we fit over thirty existing methods into this framework. We believe that this unifying view both provides a solid foundation for understanding the intuition behind these methods, and enables future research in the area.
Influence maximization is a widely studied topic in network science, where the aim is to reach the maximum possible number of nodes, while only targeting a small initial set of individuals. It has critical applications in many fields, including viral marketing, information propagation, news dissemination, and vaccinations. However, the objective does not usually take into account whether the final set of influenced nodes is fair with respect to sensitive attributes, such as race or gender. Here we address fair influence maximization, aiming to reach minorities more equitably. We introduce Adversarial Graph Embeddings: we co-train an auto-encoder for graph embedding and a discriminator to discern sensitive attributes. This leads to embeddings which are similarly distributed across sensitive attributes. We then find a good initial set by clustering the embeddings. We believe we are the first to use embeddings for the task of fair influence maximization. While there are typically trade-offs between fairness and influence maximization objectives, our experiments on synthetic and real-world datasets show that our approach dramatically reduces disparity while remaining competitive with state-of-the-art influence maximization methods.
In this research, we improve upon the current state of the art in entity retrieval by re-ranking the result list using graph embeddings. The paper shows that graph embeddings are useful for entity-oriented search tasks. We demonstrate empirically that encoding information from the knowledge graph into (graph) embeddings contributes to a higher increase in effectiveness of entity retrieval results than using plain word embeddings. We analyze the impact of the accuracy of the entity linker on the overall retrieval effectiveness. Our analysis further deploys the cluster hypothesis to explain the observed advantages of graph embeddings over the more widely used word embeddings, for user tasks involving ranking entities.
In recent years, knowledge graph embedding becomes a pretty hot research topic of artificial intelligence and plays increasingly vital roles in various downstream applications, such as recommendation and question answering. However, existing methods for knowledge graph embedding can not make a proper trade-off between the model complexity and the model expressiveness, which makes them still far from satisfactory. To mitigate this problem, we propose a lightweight modeling framework that can achieve highly competitive relational expressiveness without increasing the model complexity. Our framework focuses on the design of scoring functions and highlights two critical characteristics: 1) facilitating sufficient feature interactions; 2) preserving both symmetry and antisymmetry properties of relations. It is noteworthy that owing to the general and elegant design of scoring functions, our framework can incorporate many famous existing methods as special cases. Moreover, extensive experiments on public benchmarks demonstrate the efficiency and effectiveness of our framework. Source codes and data can be found at \url{https://github.com/Wentao-Xu/SEEK}.