## Abstract

We study the problem of graph summarization. Given a large graph we aim at producing a concise lossy representation (a *summary*) that can be stored in main memory and used to approximately answer queries about the original graph much faster than by using the exact representation. In this work we study a very natural type of summary: the original set of vertices is partitioned into a small number of supernodes connected by superedges to form a complete weighted graph. The superedge weights are the edge densities between vertices in the corresponding supernodes. To quantify the dissimilarity between the original graph and a summary, we adopt the *reconstruction error* and the *cut-norm error*. By exposing a connection between graph summarization and geometric clustering problems (i.e., *k*-means and *k*-median), we develop the *first polynomial-time approximation algorithms* to compute the best possible summary of a certain size under both measures. We discuss how to use our summaries to store a (lossy or lossless) compressed graph representation and to approximately answer a large class of queries about the original graph, including adjacency, degree, eigenvector centrality, and triangle and subgraph counting. Using the summary to answer queries is very efficient as the running time to compute the answer depends on the number of supernodes in the summary, rather than the number of nodes in the original graph.

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## Notes

We discuss the case of directed graphs in Sect. 3.5.

A skew-symmetric matrix (also known as antisymmetric or antimetric matrix) is a square matrix

*A*whose transpose is also its negative: \(-A = A^\intercal \).If \(v_1, \ldots , v_n \in {\mathbb R}^d\), then \(\left\| {v_i - v_j} \right\| _2^2 = \left\| {v_i} \right\| _2^2 + \left\| {v_j} \right\| _2^2 - 2 \langle v_i, v_j \rangle \). Since the quantities \(\left\| {v_i} \right\| _2^2\) can be easily precomputed, the problem reduces to computing all inner products \(\langle v_i, v_j \rangle \). These form the entries of \(A A^\intercal \), where

*A*is the \(n\times d\) matrix with rows \(v_1, \ldots , v_n\).For \(\ell _2\), we can also use the Johnson-Lindenstrauss transform (Johnson and Lindenstrauss 1984).

We denote as \(\left( {\begin{array}{c}X\\ k\end{array}}\right) \) the set of

*k*-subsets of*X*, i.e., the subsets of*X*of size*k*.Further space-saving can be achieved by storing only densities above a certain threshold using adjacency lists; the superedges removed increase the reconstruction error.

Minor modifications are needed if self-loops are allowed.

For speed reasons, we modified the algorithm by Arya et al. (2004) to try only a limited number of local improvements and did not run it to completion. It could otherwise achieve even better approximations.

The implementation is available from https://github.com/rionda/graphsumm.

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## Acknowledgments

The authors are thankful to the anonymous reviewers of the journal and of IEEE ICDM’14 for their insightful comments that contributed to improving the quality of this article. Matteo Riondato performed part of the work while affiliated to Brown University. He was supported in part by a summer internship at Yahoo Labs Barcelona and by NSF Grant IIS-1247581 and NIH Grant R01-CA180776.

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Responsible editor: G. Karypis.

A preliminary version of this work appeared in the proceedings of IEEE ICDM’14 (Riondato et al. 2014).

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Riondato, M., García-Soriano, D. & Bonchi, F. Graph summarization with quality guarantees.
*Data Min Knowl Disc* **31, **314–349 (2017). https://doi.org/10.1007/s10618-016-0468-8

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DOI: https://doi.org/10.1007/s10618-016-0468-8

### Keywords

- Graph analysis
- Cut-norm
- Approximation algorithms
- Summarization
- Regularity lemma