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Understanding graph embedding methods and their applications

Mengjia Xu

arXiv:2012.08019v1cs.LGcs.ITcs.SI

TL;DR

Industrial-scale graph analytics is limited by high computational and memory demands in high-dimensional, heterogeneous networks. This review surveys graph embedding methods that learn compact structure-preserving representations, including uncertainty-aware Gaussian and dynamic approaches. It highlights applications across four domains and reports graph embedding at scale on a 68-million-node, 2.5-billion-edge Friendster graph.

  • Problem

    Traditional graph analytics can suffer high computational cost and excessive memory requirements on industrial-size networks with high dimensionality and heterogeneous characteristics.

  • Method

    The review surveys graph embedding methods, emphasizing scalable random walk-based and neural network-based approaches alongside Gaussian uncertainty-aware and dynamic graph embeddings.

  • Results

    Graph embedding at scale was demonstrated on the Friendster graph with 68 million nodes and 2.5 billion edges.

  • Takeaways & Limitations

    Graph embeddings provide representations used across four application domains and can support analysis of very large graphs at reasonable computational cost on a single GPU.

  • Takeaways & Limitations

    Adjacency-matrix-based similarity has O(|V|^2) computational complexity, considers only local connections, and may perform poorly for inconspicuous connection detection.

Abstract

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Graph analytics can lead to better quantitative understanding and control of complex networks, but traditional methods suffer from high computational cost and excessive memory requirements associated with the high-dimensionality and heterogeneous characteristics of industrial size networks. Graph embedding techniques can be effective in converting high-dimensional sparse graphs into low-dimensional, dense and continuous vector spaces, preserving maximally the graph structure properties. Another type of emerging graph embedding employs Gaussian distribution-based graph embedding with important uncertainty estimation. The main goal of graph embedding methods is to pack every node's properties into a vector with a smaller dimension, hence, node similarity in the original complex irregular spaces can be easily quantified in the embedded vector spaces using standard metrics. The generated nonlinear and highly informative graph embeddings in the latent space can be conveniently used to address different downstream graph analytics tasks (e.g., node classification, link prediction, community detection, visualization, etc.). In this Review, we present some fundamental concepts in graph analytics and graph embedding methods, focusing in particular on random walk-based and neural network-based methods. We also discuss the emerging deep learning-based dynamic graph embedding methods. We highlight the distinct advantages of graph embedding methods in four diverse applications, and present implementation details and references to open-source software as well as available databases in the Appendix for the interested readers to start their exploration into graph analytics.

1. Introduction.

Traditional graph analytics can become costly and memory-intensive on large, heterogeneous networks. Graph embeddings address this by learning compact representations that preserve graph structure and support multiple downstream tasks; this review focuses on scalable static and dynamic approaches.

  • Traditional graph analytics may incur high computational and memory costs on industrial-scale networks because of high dimensionality and heterogeneous characteristics.
  • Hand-engineered graph features are often task-specific and may not perform equivalently across different graph analytics tasks.
  • Graph embedding converts high-dimensional sparse graphs into low-dimensional, dense, continuous vector spaces while maximally preserving graph structure properties.
  • Latent graph embeddings can support node classification, link prediction, community detection, visualization, and other downstream analytics tasks.
  • Graph embedding methods include matrix factorization-based, random walk-based, and neural network-based categories.
  • The review emphasizes random walk-based and neural network-based methods because matrix factorization-based methods do not easily scale to large network embeddings.
  • The review also discusses emerging deep learning-based dynamic graph embedding methods.
  • The paper presents application highlights across four domains and provides open-source software, available data, and implementation details in supplementary materials.

2. Mathematical Formulation of the Graph Embedding Problem.

Graph embedding formulates graph structure and node similarity in compact latent representations, using graph types, representations, proximity measures, and neighborhood sampling to define the learning problem.

  • 2.1. Preliminaries.: Graphs may be directed or undirected, homogeneous or heterogeneous, and weighted or binary, depending on edge direction, node and edge types, and edge costs.
  • 2.1. Preliminaries.: Adjacency matrices encode node connectivity in a |V| × |V| matrix, with weights for weighted edges and symmetry determined by graph directionality.
  • 2.1. Preliminaries.: Adjacency lists and incidence matrices provide alternative static-graph representations; adjacency lists use O(|E|) space and suit large sparse graphs.
  • 2. Mathematical Formulation of the Graph Embedding Problem.: The embedding problem learns a projection from graph nodes to low-dimensional point vectors or stochastic Gaussian distributions while preserving graph structure.
  • 2.2. Graph embedding problem setting.: Embedded node similarity approximates original graph similarity through vector dot products or Gaussian mean-vector dot products.
  • 2.2. Graph embedding problem setting.: Graph embedding generally optimizes low-dimensional representations under an unsupervised schema that is independent of downstream prediction tasks.
  • 2.2.1. Graph structure property preservation.: First-order proximity captures local pairwise similarity through edge weights, whereas second-order proximity captures similarity between neighborhood structures.
  • 2.2.2. Node similarity measures in the original graphs.: Multi-hop neighborhoods sample vertices reachable within k hops, while random-walk similarity uses the probability of reaching one node from another.

3. Overview of graph embedding methods.

Graph embedding methods convert complex graphs into lower-dimensional representations for scalable analysis, with random-walk, neural, Gaussian, and dynamic approaches preserving structure or uncertainty for downstream tasks.

  • Vector point-based graph embedding methods: Vector point-based methods project high-dimensional graph nodes into low-dimensional latent vectors while preserving original graph structure properties.The reviewed types include matrix factorization-, random walk-, and deep learning-based methods.
  • Random walk-based methods: Random walk-based methods generate fixed-length node contexts, encode them with language models such as SkipGram, and produce embeddings for downstream graph tasks.Embedding distances approximate original graph similarity, and vectors can support link prediction, node classification, community detection, and visualization.
  • Random walk-based methods: Biased random walks combine BFS and DFS through return and exploration parameters, balancing local neighborhood sampling with deeper exploration of structurally similar nodes.The parameters p and q control backtracking, exploration, and the preservation of 2nd-order proximity.
  • Random walk-based methods: Longer random walks cover more neighbors but introduce more noisy node co-occurrences, making walk length and window size important design choices.
  • Gaussian embedding methods: Gaussian graph embeddings represent nodes as distributions whose variances quantify uncertainty while preserving multi-scale structure and incorporating attributes.Graph2Gauss uses multi-hop neighborhood sampling and energy-based personalized ranking for attributed directed or undirected graphs.
  • Dynamic graph embedding methods: Dynamic graph embedding must address temporal representations and learn stable mappings when adjacent snapshots change only subtly.Continuous-time formulations preserve temporal order and flow more finely than discrete snapshots.

4. Applications.

Graph embeddings support applications across social, citation, brain, and genomic networks by providing compact node or network representations for classification, intervention assessment, disease characterization, clustering, and gene-set analysis.

  • Social network applications: DeepWalk embeds the 34-node karate club network into 128-dimensional vectors whose distances capture relationships for membership prediction.The embeddings are projected into two-dimensional Euclidean space for visualization.
  • Citation network applications: Graph2Gauss represents attributed citation-network nodes as multivariate Gaussian distributions, capturing uncertainty through mean and variance vectors.Its uncertainty estimates support neighborhood-diversity analysis and intrinsic-dimensionality inference; effective dimensionality L = 6 was close to the seven ground-truth communities.
  • Brain network applications: MG2G uses Gaussian embeddings and Wasserstein distance to quantify fMRI brain-network changes at region, system, and subject levels after three months of cognitive training.Embedding variances provide uncertainty quantification and support patient-specific evaluation; the inferred brain-network dimensionality was approximately the 14-community count.
  • Brain network applications: Brain-network embeddings can serve as latent features for classifying Alzheimer’s disease progression and identifying regions with network alterations related to mild cognitive impairment.The described classifiers include random forest, SVM, and SVD applied to MEG-derived embeddings.
  • Genomic network applications: LINE embedding with k-means predicts nine genomic sub-compartments versus five from HMM and surpasses HMM, HOPE, and DeepWalk on reported clustering metrics.The Silhouette index ranges from −1 to 1, with higher values indicating better clustering performance.
  • Genomic network applications: Set2Gauss encodes gene sets as multivariate Gaussian distributions, modeling functional diversity through uncertainty rather than standard average embeddings.It uses a biological network and gene sets as input, representing genes as points and gene sets as distributions in latent space.

5. Summary and Discussion.

Graph embedding provides a more compatible representation for efficient downstream graph analytics, while this Review surveys scalable methods, dynamic extensions, and applications across diverse domains. The surveyed applications show that embeddings can transform high-dimensional heterogeneous networks into compact vectors or probability density functions.

  • Graph embedding transforms high-dimensional graphs into representations suited to efficient downstream tasks such as node classification, link prediction, and community detection.
  • Embedding methods can quantify uncertainty, providing useful interpretations in applications such as genetic network analysis.
  • The Review focuses on random walk-based and neural network-based methods, including LINE, DeepWalk, node2vec, and G2G, while noting matrix factorization methods do not easily scale.
  • The Review supplies implementation details, open-source software information, and available data to support further exploration of graph analytics.
  • Dynamic graph embedding methods are also discussed, but their computational expense requires careful method selection.
  • Results from four application domains demonstrate embeddings' effectiveness for processing high-dimensional heterogeneous networks as compact vectors or probability density functions.The applications include social, citation, brain, and genomic networks.

Appendix A. Similarity measures used in the latent space.

Appendix A identifies common similarity measures for graph embeddings in latent space and summarizes complexity information for selected Gaussian embedding methods.

  • Table A1 summarizes common similarity measures for vector point-based and Gaussian distribution-based graph embeddings in latent space.
  • G2G has O(N) complexity, whereas DVNE has O(T × N × (d × S + S × L + L)) complexity.T is the iteration count, N the number of nodes, d the average degree, S the hidden-layer size, and L the embedding size.

Appendix B. Summary of Graph Gaussian embedding methods.

Appendix B organizes surveyed Gaussian graph embedding methods by graph type and reports their method-specific applicability and iteration notation.

  • Table B1 covers KG2E, DVNE, Graph2Gauss, and MG2G across directed, undirected, attributed, non-attributed, and multiple-graph settings.KG2E targets directed knowledge graphs; DVNE targets undirected non-attributed graphs; Graph2Gauss supports attributed or non-attributed directed or undirected graphs; MG2G supports multiple undirected attributed graphs.

Appendix C. Comparisons of random walk-based graph embedding methods.

Appendix C compares DeepWalk, LINE, and node2vec across five properties relevant to random walk-based graph node embedding.

  • DeepWalk, LINE, and node2vec are compared by neighborhood sampling, proximity, learning mode, optimization, and labeled-data requirements.

Appendix D. Overview of recent dynamic graph embedding methods.

The appendix surveys five main classes of dynamic graph embedding models and introduces implementations based on node2vec and G2G.

  • Table D1 organizes the surveyed dynamic graph embedding models into five main classes.
  • The appendix introduces implementations based on the node2vec and G2G graph embedding methods.

Appendix E. Implementation Details.

The implementation appendix compares random walk-based methods and dynamic graph embedding methods, while documenting the software used for the experiments. It begins the node2vec procedure on an undirected, non-attributed karate club network.

  • Random walk-based methods: Table C1 compares three random walk-based graph embedding approaches, including their structural preservation and network architecture characteristics.DeepWalk supports online embedding learning, and all three methods use shallow networks.
  • Dynamic graph embedding methods: Table D1 presents the dynamic graph embedding methods surveyed in the study.
  • Software and datasets: The two experiments use Python with NetworkX, Gensim, scikit-learn, and TensorFlow.
  • Software and datasets: The node2vec implementation is performed on the undirected and non-attributed karate club network.

E.1. Karate club network embedding based on node2vec.

The karate club implementation embeds an undirected graph by preprocessing transition probabilities, generating node2vec walks, training Word2Vec embeddings, and visualizing the result in two dimensions.

  • The procedure reads the undirected karate graph, preprocesses transition probabilities, simulates walks, learns embeddings, and visualizes them.
  • Walk generation: Node2vec walks are generated from every node across repeated iterations and appended to the training collection.
  • Embedding learning: The Word2Vec model converts the generated walks into karate graph embeddings using embedding size L, context size c, and a specified iteration count.
  • Embedding learning: The learned embeddings are saved in word2vec format before being returned by the procedure.
  • Visualization: The implementation plots the resulting embedded node positions as points in two-dimensional space.

E.2. CORA-ML network embedding based on stochastic graph embedding.

The CORA-ML implementation applies G2G to a directed, attributed citation graph, representing nodes as Gaussian distributions and evaluating link prediction, node classification, and visualization.

  • Dataset and representation: CORA-ML contains 2,995 nodes, 8,416 directed edges, and 2,879-dimensional attributes per node.The graph is represented by adjacency matrix A and attributed matrix X.
  • Dataset and representation: Each node is represented by a multivariate Gaussian with a mean vector and covariance matrix.The mean and covariance dimensions are determined by the embedding size L.
  • Stochastic embedding: The G2G procedure loads graph data, samples anchor-positive-negative triplets from k-hop neighborhoods, and trains the stochastic graph embedding model.
  • Evaluation: The trained model outputs Gaussian embeddings and evaluates them through link prediction and node classification.Link prediction returns AUC and average precision, while node classification returns micro-F1 and macro-F1.
  • Visualization: Gaussian embedding visualization projects means and covariance information into two dimensions when L exceeds two, then plots positions and uncertainty.
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