research-article

Open access

Testing Cluster Structure of Graphs

Authors:

Christian SohlerAuthors Info & Claims

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

Pages 723 - 732

https://doi.org/10.1145/2746539.2746618

Published: 14 June 2015 Publication History

Abstract

We study the problem of recognizing the cluster structure of a graph in the framework of property testing in the bounded degree model. Given a parameter ε, a d-bounded degree graph is defined to be (k, φ)-clusterable, if it can be partitioned into no more than k parts, such that the (inner) conductance of the induced subgraph on each part is at least φ and the (outer) conductance of each part is at most c_d,kε⁴φ², where c_d,k depends only on d,k. Our main result is a sublinear algorithm with the running time ~O(√n ⋅ poly(φ,k,1/ε)) that takes as input a graph with maximum degree bounded by d, parameters k, φ, ε, and with probability at least 2/3, accepts the graph if it is (k,φ)-clusterable and rejects the graph if it is ε-far from (k, φ*)-clusterable for φ* = c'_d,kφ² ε⁴}/log n, where c'_d,k depends only on d,k. By the lower bound of Ω(√n) on the number of queries needed for testing graph expansion, which corresponds to k=1 in our problem, our algorithm is asymptotically optimal up to polylogarithmic factors.

References

[1]

N. Alon, S. Dar, M. Parnas, and D. Ron. Testing of clustering. SIAM Journal on Discrete Mathematics, 16(3):393--417, 2003.

Digital Library

[2]

T. Batu, L. Fortnow, R. Rubinfeld, W. D. Smith, and P. White. Testing that distributions are close. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 259--269, 2000.

Digital Library

[3]

T. Batu, L. Fortnow, R. Rubinfeld, W. D. Smith, and P. White. Testing closeness of discrete distributions. Journal of the ACM, 60(1):4, 2013.

Digital Library

[4]

U. Brandes, M. Gaertler, and D. Wagner. Engineering graph clustering: Models and experimental evaluation. ACM Journal of Experimental Algorithmics, 12, 2007.

Digital Library

[5]

S.-O. Chan, I. Diakonikolas, G. Valiant, and P. Valiant. Optimal algorithms for testing closeness of discrete distributions. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1193--1203, 2014.

Digital Library

[6]

F. R. Chung. Spectral Graph Theory. American Mathematical Society, 1997.

[7]

A. Czumaj, O. Goldreich, D. Ron, C. Seshadhri, A. Shapira, and C. Sohler. Finding cycles and trees in sublinear time. Random Structures and Algorithms, 45(2):139--184, Sept. 2014.

Digital Library

[8]

A. Czumaj and C. Sohler. Abstract combinatorial programs and efficient property testers. SIAM Journal on Computing, 34(3):813--842, 2005.

Digital Library

[9]

A. Czumaj and C. Sohler. Testing expansion in bounded-degree graphs. Combinatorics, Probability and Computing, 19(5--6):693--709, 2010.

Digital Library

[10]

T. K. Dey, A. Rossi, and A. Sidiropoulos. Spectral concentration, robust k-center, and simple clustering. arXiv preprint arXiv:1404.1008, 2014.

[11]

S. Fortunato. Community detection in graphs. Physics Reports, 486, 2010.

[12]

O. Goldreich. Introduction to testing graph properties. In O. Goldreich, editor, Property Testing -- Current Research and Surveys, pages 105--141. Springer Verlag, 2011.

Digital Library

[13]

O. Goldreich and D. Ron. A sublinear bipartiteness tester for bounded degree graphs. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC), pages 289--298, 1998.

Digital Library

[14]

O. Goldreich and D. Ron. On testing expansion in bounded-degree graphs. Electronic Colloquium on Computational Complexity (ECCC), 7(20), 2000.

[15]

O. Goldreich and D. Ron. Property testing in bounded degree graphs. Algorithmica, 32:302--343, 2002.

Digital Library

[16]

S. Kale, Y. Peres, and C. Seshadhri. Noise tolerance of expanders and sublinear expansion reconstruction. SIAM Journal on Computing, 42(1):305--323, 1013.

[17]

S. Kale and C. Seshadhri. An expansion tester for bounded degree graphs. SIAM Journal on Computing, 40(3):709--720, 2011.

Digital Library

[18]

R. Kannan, S. Vempala, and A. Vetta. On clusterings: Good, bad and spectral. Journal of the ACM, 51(3):497--515, 2004.

Digital Library

[19]

J. R. Lee, S. Oveis Gharan, and L. Trevisan. Multi-way spectral partitioning and higher-order cheeger inequalities. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), pages 1117--1130, 2012.

Digital Library

[20]

R. Levi, D. Ron, and R. Rubinfeld. Testing properties of collections of distributions. Theory of Computing, 9(8):295--347, 2013.

[21]

A. Louis, P. Raghavendra, P. Tetali, and S. Vempala. Many sparse cuts via higher eigenvalues. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), pages 1131--1140, 2012.

Digital Library

[22]

K. Makarychev, Y. Makarychev, and A. Vijayaraghavan. Approximation algorithms for semi-random partitioning problems. In Proceedings of the 44th ACM Symposium on Theory of Computing (STOC), pages 367--384, 2012.

Digital Library

[23]

A. Nachmias and A. Shapira. Testing the expansion of a graph. Information and Computation, 208(4):309--314, 2010.

Digital Library

[24]

I. Newman and C. Sohler. Every property of hyperfinite graphs is testable. SIAM Journal on Computing, 42(3):1095--1112, 2013.

[25]

A. Y. Ng, M. I. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm. In Advances in Neural Information Processing Systems, pages 849--856, 2001.

Digital Library

[26]

L. Orecchia and Z. A. Zhu. Flow-based algorithms for local graph clustering. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1267--1286, 2014.

Digital Library

[27]

S. Oveis Gharan and L. Trevisan. Partitioning into expanders. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1256--1266, 2014.

Digital Library

[28]

R. Peng, H. Sun, and L. Zanetti. Partitioning well-clustered graphs with k-means and heat kernel. arXiv preprint arXiv:1411.2021, 2014.

[29]

M. A. Porter, J.-P. Onnela, and P. J. Mucha. Communities in networks. Notices of the AMS, 56(9):1082--1097, 2009.

[30]

S. Raskhodnikova, D. Ron, A. Shpilka, and A. Smith. Strong lower bounds for approximating distribution support size and the distinct elements problem. SIAM Journal on Computing, 39(3):813--842, 2009.

Digital Library

[31]

D. Ron. Algorithmic and analysis techniques in property testing. Foundations and Trends in Theoretical Computer Science, 5(2):73--205, 2010.

Digital Library

[32]

R. Rubinfeld. Taming big probability distributions. XRDS: Crossroads, The ACM Magazine for Students, 19(1):24--28, 2012.

Digital Library

[33]

S. E. Schaeffer. Graph clustering. Computer Science Review, 1(1):27--64, 2007.

Digital Library

[34]

J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8):888--905, 2000.

Digital Library

[35]

M. Tanaka. Multi-way expansion constants and partitions of a graph. arXiv:1112.3434, 2013.

[36]

G. Valiant and P. Valiant. The power of linear estimators. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 403--412, 2011.

Digital Library

[37]

Z. A. Zhu, S. Lattanzi, and V. Mirrokni. A local algorithm for finding well-connected clusters. In Proceedings of the 30th International Conference on Machine Learning (ICML 2013), pages 396--404, 2013.

Cited By

Wang HWei ZWen JYang MMohar BShinkar IO'Donnell R(2024)Revisiting Local Computation of PageRank: Simple and OptimalProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649661(911-922)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649661
Wang TWang Z(2024)A community scale test for node affiliation based on network sampling and wavelet analysisPhysica A: Statistical Mechanics and its Applications10.1016/j.physa.2024.129778(129778)Online publication date: May-2024
https://doi.org/10.1016/j.physa.2024.129778
Shen RPeng POh ANaumann TGloberson ASaenko KHardt MLevine S(2023)A sublinear-time spectral clustering oracle with improved preprocessing timeProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3667934(41809-41828)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3667934
Show More Cited By

Index Terms

Testing Cluster Structure of Graphs
1. Theory of computation
  1. Design and analysis of algorithms

Recommendations

On the Cheeger constant for distance-regular graphs
Abstract
The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least λ 1 2, where λ 1 is the smallest positive eigenvalue of ...
Testing Cluster Properties of Signed Graphs
WWW '23: Proceedings of the ACM Web Conference 2023

This work initiates the study of property testing in signed graphs, where every edge has either a positive or a negative sign. We show that there exist sublinear query and time algorithms for testing three key properties of signed graphs: balance (or 2-...
Random Walks on Regular and Irregular Graphs

For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list until it completes a full cycle. The main result of this ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

June 2015

916 pages

ISBN:9781450335362

DOI:10.1145/2746539

General Chair:
Rocco Servedio
Columbia University
,
Program Chair:
Ronitt Rubinfeld
MIT and Tel Aviv University

Copyright © 2015 Owner/Author.

Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 14 June 2015

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Engineering and Physical Sciences Research Council
European Research Council
DIMAP, University of Warwick

Conference

STOC '15

Sponsor:

SIGACT

STOC '15: Symposium on Theory of Computing

June 14 - 17, 2015

Oregon, Portland, USA

Acceptance Rates

STOC '15 Paper Acceptance Rate 93 of 347 submissions, 27%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Upcoming Conference

STOC '25

Sponsor:
sigact

57th Annual ACM Symposium on Theory of Computing (STOC 2025)

June 23 - 27, 2025

Prague , Czech Republic

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

18
Total Citations
View Citations
777
Total Downloads

Downloads (Last 12 months)89
Downloads (Last 6 weeks)13

Reflects downloads up to 01 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Wang HWei ZWen JYang MMohar BShinkar IO'Donnell R(2024)Revisiting Local Computation of PageRank: Simple and OptimalProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649661(911-922)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649661
Wang TWang Z(2024)A community scale test for node affiliation based on network sampling and wavelet analysisPhysica A: Statistical Mechanics and its Applications10.1016/j.physa.2024.129778(129778)Online publication date: May-2024
https://doi.org/10.1016/j.physa.2024.129778
Shen RPeng POh ANaumann TGloberson ASaenko KHardt MLevine S(2023)A sublinear-time spectral clustering oracle with improved preprocessing timeProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3667934(41809-41828)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3667934
Miasnikof PShestopaloff ARaigorodskii A(2023)Statistical power, accuracy, reproducibility and robustness of a graph clusterability testInternational Journal of Data Science and Analytics10.1007/s41060-023-00389-615:4(379-390)Online publication date: 16-Apr-2023
https://doi.org/10.1007/s41060-023-00389-6
Li YYang DLi J(2023)Testing Higher-Order Clusterability on GraphsCombinatorial Optimization and Applications10.1007/978-3-031-49614-1_15(203-214)Online publication date: 9-Dec-2023
https://doi.org/10.1007/978-3-031-49614-1_15
Gluch GKapralov MLattanzi SMousavifar ASohler CMarx D(2021)Spectral clustering oracles in sublinear timeProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458161(1598-1617)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458161
Manghiuc B(2021)Distributed Detection of Clusters of Arbitrary SizeStructural Information and Communication Complexity10.1007/978-3-030-79527-6_21(370-387)Online publication date: 20-Jun-2021
https://doi.org/10.1007/978-3-030-79527-6_21
Apers S(2020)Expansion Testing using Quantum Fast-Forwarding and Seed SetsQuantum10.22331/q-2020-09-16-3234(323)Online publication date: 16-Sep-2020
https://doi.org/10.22331/q-2020-09-16-323
Łącki JMitrović SOnak KSankowski PMakarychev KMakarychev YTulsiani MKamath GChuzhoy J(2020)Walking randomly, massively, and efficientlyProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384303(364-377)Online publication date: 22-Jun-2020
https://dl.acm.org/doi/10.1145/3357713.3384303
Kumar ASeshadhri CStolman A(2020)Random Walks and Forbidden Minors I: An $n^{1/2+o(1)}$-Query One-Sided Tester for Minor Closed Properties on Bounded Degree GraphsSIAM Journal on Computing10.1137/19M124546352:6(FOCS18-216-FOCS18-249)Online publication date: 18-Dec-2020
https://doi.org/10.1137/19M1245463
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Table of Contents