Efficient Graph Encoder Embedding for Large Sparse Graphs in Python

Graph is a ubiquitous representation of data in various research fields, and graph embedding is a prevalent machine learning technique for capturing key features and generating fixed-sized attributes. However, most state-of-the-art graph embedding methods are computationally and spatially expensive. Recently, the Graph Encoder Embedding (GEE) has been shown as the fastest graph embedding technique and is suitable for a variety of network data applications. As real-world data often involves large and sparse graphs, the huge sparsity usually results in redundant computations and storage. To address this issue, we propose an improved version of GEE, sparse GEE, which optimizes the calculation and storage of zero entries in sparse matrices to enhance the running time further. Our experiments demonstrate that the sparse version achieves significant speedup compared to the original GEE with Python implementation for large sparse graphs, and sparse GEE is capable of processing millions of edges within minutes on a standard laptop.

Fast and Scalable Multi-Kernel Encoder Classifier

This paper introduces a new kernel-based classifier by viewing kernel matrices as generalized graphs and leveraging recent progress in graph embedding techniques. The proposed method facilitates fast and scalable kernel matrix embedding, and seamlessly integrates multiple kernels to enhance the learning process. Our theoretical analysis offers a population-level characterization of this approach using random variables. Empirically, our method demonstrates superior running time compared to standard approaches such as support vector machines and two-layer neural network, while achieving comparable classification accuracy across various simulated and real datasets.

Encoder Embedding for General Graph and Node Classification

Graph encoder embedding, a recent technique for graph data, offers speed and scalability in producing vertex-level representations from binary graphs. In this paper, we extend the applicability of this method to a general graph model, which includes weighted graphs, distance matrices, and kernel matrices. We prove that the encoder embedding satisfies the law of large numbers and the central limit theorem on a per-observation basis. Under certain condition, it achieves asymptotic normality on a per-class basis, enabling optimal classification through discriminant analysis. These theoretical findings are validated through a series of experiments involving weighted graphs, as well as text and image data transformed into general graph representations using appropriate distance metrics.

Refined Graph Encoder Embedding via Self-Training and Latent Community Recovery

This paper introduces a refined graph encoder embedding method, enhancing the original graph encoder embedding using linear transformation, self-training, and hidden community recovery within observed communities. We provide the theoretical rationale for the refinement procedure, demonstrating how and why our proposed method can effectively identify useful hidden communities via stochastic block models, and how the refinement method leads to improved vertex embedding and better decision boundaries for subsequent vertex classification. The efficacy of our approach is validated through a collection of simulated and real-world graph data.

Edge-Parallel Graph Encoder Embedding

New algorithms for embedding graphs have reduced the asymptotic complexity of finding low-dimensional representations. One-Hot Graph Encoder Embedding (GEE) uses a single, linear pass over edges and produces an embedding that converges asymptotically to the spectral embedding. The scaling and performance benefits of this approach have been limited by a serial implementation in an interpreted language. We refactor GEE into a parallel program in the Ligra graph engine that maps functions over the edges of the graph and uses lock-free atomic instrutions to prevent data races. On a graph with 1.8B edges, this results in a 500 times speedup over the original implementation and a 17 times speedup over a just-in-time compiled version.

Learning sources of variability from high-dimensional observational studies

Causal inference studies whether the presence of a variable influences an observed outcome. As measured by quantities such as the "average treatment effect," this paradigm is employed across numerous biological fields, from vaccine and drug development to policy interventions. Unfortunately, the majority of these methods are often limited to univariate outcomes. Our work generalizes causal estimands to outcomes with any number of dimensions or any measurable space, and formulates traditional causal estimands for nominal variables as causal discrepancy tests. We propose a simple technique for adjusting universally consistent conditional independence tests and prove that these tests are universally consistent causal discrepancy tests. Numerical experiments illustrate that our method, Causal CDcorr, leads to improvements in both finite sample validity and power when compared to existing strategies. Our methods are all open source and available at github.com/ebridge2/cdcorr.

Discovering Communication Pattern Shifts in Large-Scale Networks using Encoder Embedding and Vertex Dynamics

The analysis of large-scale time-series network data, such as social media and email communications, remains a significant challenge for graph analysis methodology. In particular, the scalability of graph analysis is a critical issue hindering further progress in large-scale downstream inference. In this paper, we introduce a novel approach called "temporal encoder embedding" that can efficiently embed large amounts of graph data with linear complexity. We apply this method to an anonymized time-series communication network from a large organization spanning 2019-2020, consisting of over 100 thousand vertices and 80 million edges. Our method embeds the data within 10 seconds on a standard computer and enables the detection of communication pattern shifts for individual vertices, vertex communities, and the overall graph structure. Through supporting theory and synthesis studies, we demonstrate the theoretical soundness of our approach under random graph models and its numerical effectiveness through simulation studies.

Synergistic Graph Fusion via Encoder Embedding

In this paper, we introduce a novel approach to multi-graph embedding called graph fusion encoder embedding. The method is designed to work with multiple graphs that share a common vertex set. Under the supervised learning setting, we show that the resulting embedding exhibits a surprising yet highly desirable "synergistic effect": for sufficiently large vertex size, the vertex classification accuracy always benefits from additional graphs. We provide a mathematical proof of this effect under the stochastic block model, and identify the necessary and sufficient condition for asymptotically perfect classification. The simulations and real data experiments confirm the superiority of the proposed method, which consistently outperforms recent benchmark methods in classification.

Graph Encoder Ensemble for Simultaneous Vertex Embedding and Community Detection

In this paper we propose a novel and computationally efficient method to simultaneously achieve vertex embedding, community detection, and community size determination. By utilizing a normalized one-hot graph encoder and a new rank-based cluster size measure, the proposed graph encoder ensemble algorithm achieves excellent numerical performance throughout a variety of simulations and real data experiments.

Graph Encoder Embedding

In this paper we propose a lightning fast graph embedding method called graph encoder embedding. The proposed method has a linear computational complexity and the capacity to process billions of edges within minutes on standard PC -- an unattainable feat for any existing graph embedding method. The speedup is achieved without sacrificing embedding performance: the encoder embedding performs as good as, and can be viewed as a transformation of the more costly spectral embedding. The encoder embedding is applicable to either adjacency matrix or graph Laplacian, and is theoretically sound, i.e., under stochastic block model or random dot product graph, the graph encoder embedding asymptotically converges to the block probability or latent positions, and is approximately normally distributed. We showcase three important applications: vertex classification, vertex clustering, and graph bootstrap; and the embedding performance is evaluated via a comprehensive set of synthetic and real data. In every case, the graph encoder embedding exhibits unrivalled computational advantages while delivering excellent numerical performance.