A General Framework for Graph Sparsification

Fung, Wai Shing; Hariharan, Ramesh; Harvey, Nicholas J. A.; Panigrahi, Debmalya

doi:10.1137/16m1091666

Cited by 44 publications

(88 citation statements)

References 12 publications

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“…We note that our method above does have a potential drawback for very small graphs. This is because we need to sample Θ(n log(n)) edges to avoid the graph being disconnected [12]. As graphs become large this should be a nonissue because we have lim n→∞ (pn−qn) 2 −2(pn+qn) nlogn = ∞, which indicates that our sampling criteria will require more edges than needed to ensure connectivity.…”

Section: Methods and Technical Solutionsmentioning

confidence: 99%

Fast Community Detection with Graph Sparsification

Laeuchli

2020

Advances in Knowledge Discovery and Data Mining

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A popular model for detecting community structure in large graphs is the Stochastic Block Model (SBM). The exact parameters to recover the community structure of a SBM has been well studied, and many methods have been proposed to recover a nodes' community membership. A popular approach is to use spectral methods where the Graph Laplacian L of the given graph is created, and the Fiedler vector of the graph is found. This vector is then used to cluster nodes in the same community. While a robust method, it can be expensive to compute the Fiedler vector exactly. In this paper we examine the types of errors that can be tolerated using spectral methods while still recovering the communities. The two sources of error considered are: (i) dropping edges using different sparsification strategies; and (ii) inaccurately computing the eigenvectors. In this way, spectral clustering algorithms can be tuned to be far more efficient at detecting community structure for these community models.

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Section: Methods and Technical Solutionsmentioning

confidence: 99%

Fast Community Detection with Graph Sparsification

Laeuchli

2020

Advances in Knowledge Discovery and Data Mining

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“…Generally speaking, most graph properties of a dense graph can be approximated from its [sparsified] sparse graph. Cut sparsifiers [1,8,13] ensure the total weight of cuts in the sparsified graph approximates that of cuts in the original graph within some bounded distance. Spectral sparsifiers [23,24] ensure sparsified graphs preserve spectral properties of the graph Laplacian.…”

Section: Related Workmentioning

confidence: 99%

SGCN: A Graph Sparsifier Based on Graph Convolutional Networks

Zhang

Tian

et al. 2020

Advances in Knowledge Discovery and Data Mining

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Graphs are ubiquitous across the globe and within science and engineering. With graphs growing in size, node classification on large graphs can be space and time consuming, even with powerful classifiers such as Graph Convolutional Networks (GCNs). Hence, some questions are raised, particularly, whether one can keep only some of the edges of a graph while maintaining prediction performance for node classification, or train classifiers on specific subgraphs instead of a whole graph with limited performance loss in node classification. To address these questions, we propose Sparsified Graph Convolutional Network (SGCN), a neural network graph sparsifier that sparsifies a graph by pruning some edges. We formulate sparsification as an optimization problem, which we solve by an Alternating Direction Method of Multipliers (ADMM)based solution. We show that sparsified graphs provided by SGCN can be used as inputs to GCN, leading to better or comparable node classification performance with that of original graphs in GCN, DeepWalk, and GraphSAGE.

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“…The existing methods assume integer weights and differ mainly in the first step, i.e., that of choosing pe for each edge e = (u, v). Benczúr and Karger [7] assign probabilities inversely proportional to the k-strong connectivity 3 of u, v. Fung et al [14] simplify analysis by utilizing sampling probabilities inversely proportional to the size of the minimum cut separating u and v. Nagamochi and Ibaraki [27] estimate the edge connectivities using the NI index λe of an edge e. The NI index is generated by iteratively constructing spanning forests of the initial graph, while reducing the weights of the selected edges. In essence, the NI index is the last spanning forest that contains e, given that an edge with weight we needs to participate in we contiguous forests.…”

Section: Cut-based Sparsifiersmentioning

confidence: 99%

“…According to [8], e 1/λe = O(n log n); thus, E(|E ′ |) = e pe = ρ e 1/λe = O(n log 2 n/ǫ 2 ). A more refined analysis [14] reduces this to O(n log n/ǫ 2 ). Section 3.3 modifies [27] as a benchmark for our experiments.…”

Section: Cut-based Sparsifiersmentioning

confidence: 99%

Uncertain Graph Sparsification

Parchas

Papailiou

Papadias

et al. 2018

IEEE Trans. Knowl. Data Eng.

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Uncertain graphs are prevalent in several applications including communications systems, biological databases and social networks. The ever increasing size of the underlying data renders both graph storage and query processing extremely expensive. Sparsification has often been used to reduce the size of deterministic graphs by maintaining only the important edges. However, adaptation of deterministic sparsification methods fails in the uncertain setting. To overcome this problem, we introduce the first sparsification techniques aimed explicitly at uncertain graphs. The proposed methods reduce the number of edges and redistribute their probabilities in order to decrease the graph size, while preserving its underlying structure. The resulting graph can be used to efficiently and accurately approximate any query and mining tasks on the original graph. An extensive experimental evaluation with real and synthetic datasets illustrates the effectiveness of our techniques on several common graph tasks, including clustering coefficient, page rank, reliability and shortest path distance.

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A General Framework for Graph Sparsification

Cited by 44 publications

References 12 publications

Fast Community Detection with Graph Sparsification

Fast Community Detection with Graph Sparsification

SGCN: A Graph Sparsifier Based on Graph Convolutional Networks

Uncertain Graph Sparsification

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