Compressing Graphs and Indexes with Recursive Graph Bisection

Dhulipala, Laxman; Kabiljo, Igor; Karrer, Brian; Ottaviano, Giuseppe; Pupyrev, Sergey; Shalita, Alon

doi:10.48550/arxiv.1602.08820

Cited by 3 publications

(4 citation statements)

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“…Since compression and summarization are distinct fields, we only give a few fundamental methods in the former, including: the so-called "Eulerian data structure" to handle neighbor queries in social networks [Maserrat and Pei 2010] and extensions of this work to community-preserving compression [Maserrat and Pei 2012]; node reordering techniques, such as zip block encoding in GBASE [Kang et al 2011], bipartite minimum logarithmic arrangement [Dhulipala et al 2016] for inverted indices, and techniques for real graphs with power-law degree distributions [Lim et al 2014]; edge reordering techniques [Goonetilleke et al 2017]; compression of web graphs using lexicographic localities [Boldi and Vigna 2004]; extensions to social networks [Grabowski and Bieniecki 2014;Chierichetti et al 2009]; breadth first search-based approaches [Apostolico and Drovandi 2009]; lossy edge encoding per triangle [Feng et al 2013]; weighted graph compression to maintain edge weights up to a certain number of hops [Toivonen et al 2011]; provably optimal compression of Erdös-Rényi random graphs using structural entropy (SZIP [Choi and Szpankowski 2012]); and minimal probabilistic tile cover mining [Liu and Chen 2016] that has applications to binary matrices and bipartite graphs.…”

Section: Bit Compression-based Methodsmentioning

confidence: 99%

Graph Summarization Methods and Applications

et al. 2018

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While advances in computing resources have made processing enormous amounts of data possible, human ability to identify patterns in such data has not scaled accordingly. Efficient computational methods for condensing and simplifying data are thus becoming vital for extracting actionable insights. In particular, while data summarization techniques have been studied extensively, only recently has summarizing interconnected data, or graphs, become popular. This survey is a structured, comprehensive overview of the state-of-the-art methods for summarizing graph data. We first broach the motivation behind, and the challenges of, graph summarization. We then categorize summarization approaches by the type of graphs taken as input and further organize each category by core methodology. Finally, we discuss applications of summarization on real-world graphs and conclude by describing some open problems in the field.

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Section: Bit Compression-based Methodsmentioning

confidence: 99%

Graph Summarization Methods and Applications

et al. 2018

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“…It is NP-hard and one of the most important problems in the field of approximation algorithms; through the years, it has led to the design of new, powerful algorithmic techniques (e.g. the O( √ log n)-approximation of Arora, Rao, and Vazirani [12]), and is also increasingly becoming a keystone of divide-and-conquer strategies for a variety of problems arising in graph compression [33], clustering [32,29,26], and beyond. The goal is to cut the graph into two roughly balanced parts while cutting as few edges as possible.…”

Section: Introductionmentioning

confidence: 99%

New hardness results for planar graph problems in p and an algorithm for sparsest cut

Abboud

Cohen-Addad

Klein

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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The Sparsest Cut is a fundamental optimization problem that has been extensively studied. For planar inputs the problem is in P and can be solved inÕ(n 3) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90's and can only achieve O(log n)-approximation inÕ(n) time or a constant factor approximation inÕ(n 2) time [Rao, STOC92]. Our main result is an Ω(n 2−ε) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)-Convolution conjecture, showing that approximations are inevitable in the near-linear time regime. To complement the lower bound, we provide a constant factor approximation in near-linear time, improving upon the 25-year old result of Rao in both time and accuracy. Our lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P. Moreover, we prove near-quadratic lower bounds under SETH for variants of the closest pair problem in planar graphs, and use them to show that the popular Average-Linkage procedure for Hierarchical Clustering cannot be simulated in truly subquadratic time. At the core of our constructions is a diamond-like gadget that also settles the complexity of Diameter in distributed planar networks. We prove an Ω(n/ log n) lower bound on the number of communication rounds required to compute the weighted diameter of a network in the CONGEST model, even when the underlying graph is planar and all nodes are D = 4 hops away from each other. This is the first poly(n) + ω(D) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter [STOC 2019] for (exact) unweighted diameter and for (1 + ε) approximate weighted diameter.

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“…However, the recovered communities do not preserve much of the edge information since the communities themselves are sparsely connected. Hence, effective reconstruction of the original graph from the summarized graph is a meaningful task that enjoys applications in graph compression [Dhabu et al, 2013, Dhulipala et al, 2016, graph sampling [Orbanz, 2017, Leskovec andFaloutsos, 2006] and so on.…”

Section: Introductionmentioning

confidence: 99%

“…Previous works have considered related problems in undirected graph settings [Shahaf et al, 2013, Navlakha et al, 2008, which aim to define compressed nodes by preserving particular structures. Graph compression literature [Maneth and Peternek, 2015, Fan et al, 2012, Dhulipala et al, 2016 is also related, while the goal is to minimize the storage space, irrespective of preserving feature patterns of the graph. In addition, another line of related work, under the theme of influence maximization [Li et al, 2018b], studies directed influence of a set of vertices to the rest of the network.…”

Section: Introductionmentioning

confidence: 99%

Direction Matters: On Influence-Preserving Graph Summarization and Max-cut Principle for Directed Graphs

Niu

Hyvärinen

et al. 2019

Preprint

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Summarizing large-scaled directed graphs into small-scale representations is a useful but less studied problem setting. Conventional clustering approaches, which based on "Min-Cut"-style criteria, compress both the vertices and edges of the graph into the communities, that lead to a loss of directed edge information. On the other hand, compressing the vertices while preserving the directed edge information provides a way to learn the small-scale representation of a directed graph. The reconstruction error, which measures the edge information preserved by the summarized graph, can be used to learn such representation. Compared to the original graphs, the summarized graphs are easier to analyze and are capable of extracting group-level features which is useful for efficient interventions of population behavior. In this paper, we present a model, based on minimizing reconstruction error with non-negative constraints, which relates to a "Max-Cut" criterion that simultaneously identifies the compressed nodes and the directed compressed relations between these nodes. A multiplicative update algorithm with column-wise normalization is proposed. We further provide theoretical results on the identifiability of the model and on the convergence of the proposed algorithms. Experiments are conducted to demonstrate the accuracy and robustness of the proposed method.

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Compressing Graphs and Indexes with Recursive Graph Bisection

Cited by 3 publications

References 0 publications

Graph Summarization Methods and Applications

Graph Summarization Methods and Applications

New hardness results for planar graph problems in p and an algorithm for sparsest cut

Direction Matters: On Influence-Preserving Graph Summarization and Max-cut Principle for Directed Graphs

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