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1. Embedding Directed Graphs in Potential Fields Using FastMap-D

   Gopalakrishnan, S.; Cohen, L.; Koenig, S.; Kumar, S. (January 2020, Symposium on Combinatorial Search (SoCS))

   Embedding undirected graphs in a Euclidean space has many computational benefits. FastMap is an efficient embedding algorithm that facilitates a geometric interpretation of problems posed on undirected graphs. However, Euclidean distances are inherently symmetric and, thus, Euclidean embeddings cannot be used for directed graphs. In this paper, we present FastMap-D, an efficient generalization of FastMap to directed graphs. FastMap-D embeds vertices using a potential field to capture the asymmetry between the pairwise distances in directed graphs. FastMap-D learns a potential function to define the potential field using a machine learning module. In experiments on various kinds of directed graphs, we demonstrate the advantage of FastMap-D over other approaches. Errata: This version of the paper corrects a programming mistake, resulting in even better experimental results than those reported in the original paper. 

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2. Towards Scalable Spectral Embedding and Data Visualization via Spectral Coarsening

   https://doi.org/10.1145/3437963.3441767  

   Zhao, Zhiqiang; Zhang, Ying; Feng, Zhuo (March 2021, ACM International Conference on Web Search and Data Mining)

   null (Ed.)

   This paper proposes a scalable multilevel framework for the spectral embedding of large undirected graphs. The proposed method first computes much smaller yet sparse graphs while preserving the key spectral (structural) properties of the original graph, by exploiting a nearly-linear time spectral graph coarsening approach. Then, the resultant spectrally-coarsened graphs are leveraged for the development of much faster algorithms for multilevel spectral graph embedding (clustering) as well as visualization of large data sets. We conducted extensive experiments using a variety of large graphs and datasets and obtained very promising results. For instance, we are able to coarsen the "coPapersCiteseer" graph with 0.43 million nodes and 16 million edges into a much smaller graph with only 13K (32X fewer) nodes and 17K (950X fewer) edges in about 16 seconds; the spectrally-coarsened graphs allow us to achieve up to 1,100X speedup for multilevel spectral graph embedding (clustering) and up to 60X speedup for t-SNE visualization of large data sets. 

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3. A Simple Deterministic Near-Linear Time Approximation Scheme for Transshipment with Arbitrary Positive Edge Costs

   https://doi.org/10.4230/LIPIcs.ESA.2024.56  

   Fox, Emily (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Chan, Timothy; Fischer, Johannes; Iacono, John; Herman, Grzegorz (Ed.)

   We describe a simple deterministic near-linear time approximation scheme for uncapacitated minimum cost flow in undirected graphs with positive real edge weights, a problem also known as transshipment. Specifically, our algorithm takes as input a (connected) undirected graph G = (V, E), vertex demands b ∈ R^V such that ∑_{v ∈ V} b(v) = 0, positive edge costs c ∈ R_{≥ 0}^E, and a parameter ε > 0. In O(ε^{-2} m log^{O(1)} n) time, it returns a flow f such that the net flow out of each vertex is equal to the vertex’s demand and the cost of the flow is within a (1 ± ε) factor of optimal. Our algorithm is combinatorial and has no running time dependency on the demands or edge costs. With the exception of a recent result presented at STOC 2022 for polynomially bounded edge weights, all almost- and near-linear time approximation schemes for transshipment relied on randomization to embed the problem instance into low-dimensional space. Our algorithm instead deterministically approximates the cost of routing decisions that would be made if the input were subject to a random tree embedding. To avoid computing the Ω(n²) vertex-vertex distances that an approximation of this kind suggests, we also take advantage of the clustering method used in the well-known Thorup-Zwick distance oracle. 

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4. Efficient Exact Subgraph Matching via GNN-based Path Dominance Embedding

   Ye, Y; Lian, X; Chen, M (August 2024, Proceedings of the International Conference on Very Large Data Bases)

   The classic problem of exact subgraph matching returns those subgraphs in a large-scale data graph that are isomorphic to a given query graph, which has gained increasing importance in many real-world applications such as social network analysis, knowledge graph discovery in the Semantic Web, bibliographical network mining, and so on. In this paper, we propose a novel and effective graph neural network (GNN)-based path embedding framework (GNN-PE), which allows efficient exact subgraph matching without introducing false dismissals. Unlike traditional GNN-based graph embeddings that only produce approximate subgraph matching results, in this paper, we carefully devise GNN-based embeddings for paths, such that: if two paths (and 1-hop neighbors of vertices on them) have the subgraph relationship, their corresponding GNN-based embedding vectors will strictly follow the dominance relationship. With such a newly designed property of path dominance embeddings, we are able to propose effective pruning strategies based on path label/dominance embeddings and guarantee no false dismissals for subgraph matching. We build multidimensional indexes over path embedding vectors, and develop an efficient subgraph matching algorithm by traversing indexes over graph partitions in parallel and applying our pruning methods. We also propose a cost-model-based query plan that obtains query paths from the query graph with low query cost. Through extensive experiments, we confirm the efficiency and effectiveness of our proposed GNN-PE approach for exact subgraph matching on both real and synthetic graph data. 

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5. MagNet: A Neural Network for Directed Graphs

   Zhang, Xitong; He, Yixuan; Brugnone, Nathan; Perlmutter, Michael; Hirn, Matthew (January 2021, Advances in neural information processing systems)

   Ranzato, M.; Beygelzimer, A.; Dauphin, Y.; Liang, P.S.; Vaughan, J. Wortman (Ed.)

   The prevalence of graph-based data has spurred the rapid development of graph neural networks (GNNs) and related machine learning algorithms. Yet, despite the many datasets naturally modeled as directed graphs, including citation, website, and traffic networks, the vast majority of this research focuses on undirected graphs. In this paper, we propose MagNet, a GNN for directed graphs based on a complex Hermitian matrix known as the magnetic Laplacian. This matrix encodes undirected geometric structure in the magnitude of its entries and directional information in their phase. A charge parameter attunes spectral information to variation among directed cycles. We apply our network to a variety of directed graph node classification and link prediction tasks showing that MagNet performs well on all tasks and that its performance exceeds all other methods on a majority of such tasks. The underlying principles of MagNet are such that it can be adapted to other GNN architectures. 

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