Skip to main content
Springer Nature Link
Log in

Graph summarization with quality guarantees

  • Published:
Data Mining and Knowledge Discovery Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

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

  2. A skew-symmetric matrix (also known as antisymmetric or antimetric matrix) is a square matrix A whose transpose is also its negative: \(-A = A^\intercal \).

  3. 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 \(v_1, \ldots , v_n\).

  4. For \(\ell _2\), we can also use the Johnson-Lindenstrauss transform (Johnson and Lindenstrauss 1984).

  5. 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.

  6. Further space-saving can be achieved by storing only densities above a certain threshold using adjacency lists; the superedges removed increase the reconstruction error.

  7. Minor modifications are needed if self-loops are allowed.

  8. http://snap.stanford.edu/data/.

  9. http://irefindex.org.

  10. 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.

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

References

  • Aggarwal A, Deshpande A, Kannan R (2009) Adaptive sampling for k-means clustering. Approximation, randomization, and combinatorial optimization. Algorithms and techniques, APPROX-RANDOM. Springer, Berlin, pp 15–28

    Chapter  Google Scholar 

  • Aloise D, Deshpande A, Hansen P, Popat P (2009) NP-hardness of Euclidean sum-of-squares clustering. Mach Learn 75(2):245–248

    Article  Google Scholar 

  • Alon N, Duke RA, Lefmann H, Rödl V, Yuster R (1994) The algorithmic aspects of the regularity lemma. J Algorithms 16(1):80–109

    Article  MathSciNet  Google Scholar 

  • Alon N, Naor A (2006) Approximating the cut-norm via Grothendieck’s inequality. SIAM J Comput 35(4):787–803

    Article  MathSciNet  Google Scholar 

  • Arthur D, Vassilvitskii S (2007) \(k\)-means++: the advantages of careful seeding. In: Proceedings of the 18th annual ACM-SIAM symposium on discrete algorithms, SIAM, SODA ’07, pp 1027–1035

  • Arya V, Garg N, Khandekar R, Meyerson A, Munagala K, Pandit V (2004) Local search heuristics for \(k\)-median and facility location problems. SIAM J Comput 33(3):544–562

    Article  MathSciNet  Google Scholar 

  • Bahmani B, Moseley B, Vattani A, Kumar R, Vassilvitskii S (2012) Scalable \(k\)-means++. Proc VLDB Endow 5(7):622–633

    Article  Google Scholar 

  • Boldi P, Santini M, Vigna S (2009) Permuting web and social graphs. Internet Math 6(3):257–283

    Article  MathSciNet  Google Scholar 

  • Boldi P, Rosa M, Santini M, Vigna S (2011) Layered label propagation: a multiresolution coordinate-free ordering for compressing social networks. In: Proceedings of the 20th international conference on World Wide Web, ACM, WWW ’11, pp 587–596

  • Boldi P, Vigna S (2004) The webgraph framework i: compression techniques. In: Proceedings of the 13th international conference on World Wide Web, ACM, WWW ’04, pp 595–602

  • Bonchi F, García-Soriano D, Kutzkov K (2013) Local correlation clustering. arXiv preprint arXiv:1312.5105v1

  • Campan A, Truta TM (2009) Data and structural k-anonymity in social networks. Privacy, security, and trust in KDD. Springer, Berlin, pp 33–54

    Chapter  Google Scholar 

  • Conlon D, Fox J (2012) Bounds for graph regularity and removal lemmas. Geom Funct Anal 22(5):1191–1256

    Article  MathSciNet  Google Scholar 

  • Cormode G, Srivastava D, Yu T, Zhang Q (2010) Anonymizing bipartite graph data using safe groupings. VLDB J 19(1):115–139

    Article  Google Scholar 

  • Dasgupta S (2008) The hardness of \(k\)-means clustering. Tech. Rep. 09-16. University of California, San Diego

  • Dellamonica DJ, Kalyanasundaram S, Martin DM, Rödl V, Shapira A (2012) A deterministic algorithm for the Frieze-Kannan regularity lemma. SIAM J Discret Math 26(1):15–29

    Article  MathSciNet  Google Scholar 

  • Dellamonica DJ, Kalyanasundaram S, Martin DM, Rödl V, Shapira A (2015) An optimal algorithm for finding Frieze-Kannan regular partitions. Comb Prob Comput 24(02):407–437

    Article  MathSciNet  Google Scholar 

  • Fan W, Li J, Wang X, Wu Y (2012) Query preserving graph compression. In: Proceedings of the 2012 ACM SIGMOD international conference on management of data, ACM, SIGMOD ’12, pp 157–168

  • Frieze A, Kannan R (1999) Quick approximation to matrices and applications. Combinatorica 19(2):175–220

    Article  MathSciNet  Google Scholar 

  • Gowers WT (1997) Lower bounds of tower type for Szemerédi’s uniformity lemma. Geom Funct Anal 7(2):322–337

    Article  MathSciNet  Google Scholar 

  • Hay M, Miklau G, Jensen D, Towsley D, Li C (2010) Resisting structural re-identification in anonymized social networks. VLDB J 19(6):797–823

    Article  Google Scholar 

  • Hernández C, Navarro G (2011) Compression of web and social graphs supporting neighbor and community queries. In: Proceedings of the 6th ACM workshop on social network mining and analysis, ACM, SNAKDD ’11

  • Indyk P (2006) Stable distributions, pseudorandom generators, embeddings, and data stream computation. J ACM 53(3):307–323

    Article  MathSciNet  Google Scholar 

  • Jain K, Vazirani VV (2001) Approximation algorithms for metric facility location and \(k\)-median problems using the primal-dual schema and Lagrangian relaxation. J ACM 48(2):274–296

    Article  MathSciNet  Google Scholar 

  • Johnson WB, Lindenstrauss J (1984) Extensions of Lipschitz mappings into a Hilbert space. Contemp Math 26:189–206

    Article  MathSciNet  Google Scholar 

  • LeFevre K, Terzi E (2010) GraSS: graph structure summarization. In: Proceedings of the 2010 SIAM international conference on data mining, SIAM, SDM ’10, pp 454–465

    Chapter  Google Scholar 

  • Liu Z, Yu JX, Cheng H (2012) Approximate homogeneous graph summarization. Inf Media Technol 7(1):32–43

    Google Scholar 

  • Lloyd S (1982) Least squares quantization in PCM. IEEE Trans Inf Theory 28(2):129–137

    Article  MathSciNet  Google Scholar 

  • Lovász L (2012) Large networks and graph limits. American Mathematical Society, Providence

    Book  Google Scholar 

  • Maserrat H, Pei J (2010) Neighbor query friendly compression of social networks. In: Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, KDD ’10, pp 533–542

  • Megiddo N, Supowit KJ (1984) On the complexity of some common geometric location problems. SIAM J Comput 13(1):182–196

    Article  MathSciNet  Google Scholar 

  • Mettu RR, Plaxton CG (2003) The online median problem. SIAM J Comput 32(3):816–832

    Article  MathSciNet  Google Scholar 

  • Navlakha S, Rastogi R, Shrivastava N (2008) Graph summarization with bounded error. In: Proceedings of the 2008 ACM SIGMOD international conference on Management of data, ACM, SIGMOD ’08, pp 419–432

  • Riondato M, García-Soriano D, Bonchi F (2014) Graph summarization with quality guarantees. In: 2014 IEEE international conference on data mining, IEEE, ICDM ’14, pp 947–952

  • Schaeffer SE (2007) Graph clustering. Comput Sci Rev 1(1):27–64

    Article  Google Scholar 

  • Szemerédi E (1976) Regular partitions of graphs. In: Problèmes Combinatoires et Théorie des Graphes, Colloq. Internat. CNRS, Univ. Orsay., pp 399–401

  • Tassa T, Cohen DJ (2013) Anonymization of centralized and distributed social networks by sequential clustering. IEEE Trans Knowl Data Eng 25(2):311–324

    Article  Google Scholar 

  • Tian Y, Hankins RA, Patel JM (2008) Efficient aggregation for graph summarization. In: Proceedings of the 2008 ACM SIGMOD international conference on management of data, ACM, SIGMOD ’08, pp 567–580

  • Toivonen H, Zhou F, Hartikainen A, Hinkka A (2011) Compression of weighted graphs. In: Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, KDD ’11, pp 965–973

  • Tsourakakis CE (2008) Fast counting of triangles in large real networks without counting: algorithms and laws. In: 2008 IEEE international conference on data mining, IEEE, ICDM ’08, pp 608–617

  • Vassilevska Williams V (2011) Breaking the Coppersmith–Winograd barrier, unpublished manuscript

  • Ward JH (1963) Hierarchical grouping to optimize an objective function. J Am Stat Assoc 58(301):236–244

    Article  MathSciNet  Google Scholar 

  • Williams D (1991) Probability with Martingales. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Zheleva E, Getoor L (2008) Preserving the privacy of sensitive relationships in graph data. In: Privacy, security, and trust in KDD, Springer, pp 153–171

Download references

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.

Author information

Authors and Affiliations

  1. Two Sigma Investments LP, New York, NY, USA

    Matteo Riondato

  2. Eurecat, Barcelona, Spain

    David García-Soriano

  3. ISI Foundation, Turin, Italy

    Francesco Bonchi

Authors
  1. Matteo Riondato
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. David García-Soriano
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Francesco Bonchi
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Matteo Riondato.

Additional information

Responsible editor: G. Karypis.

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords