Document Open Access Logo

Effective Resistances in Non-Expander Graphs

Authors Dongrun Cai, Xue Chen , Pan Peng

PDF
Thumbnail PDF

File

LIPIcs.ESA.2023.29.pdf
  • Filesize: 0.87 MB
  • 18 pages

Document Identifiers

Author Details

Dongrun Cai
  • School of Computer Science and Technology, University of Science and Technology of China, Hefei, China
Xue Chen
  • School of Computer Science and Technology, University of Science and Technology of China, Hefei, China
Pan Peng
  • School of Computer Science and Technology, University of Science and Technology of China, Hefei, China

Funding

  • Peng, Pan: Supported in part by NSFC grant 62272431 and "the Fundamental Research Funds for the Central Universities".

Acknowledgements

We thank Jingcheng Liu (Nanjing University) for suggesting the proof of Lemma 5 using the characteristic polynomial method. Also, we thank all anonymous reviewers for the helpful comments.

Cite As Get BibTex

Dongrun Cai, Xue Chen, and Pan Peng. Effective Resistances in Non-Expander Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.29

Abstract

Effective resistances are ubiquitous in graph algorithms and network analysis. For an undirected graph G, its effective resistance R_G(s,t) between two vertices s and t is defined as the equivalent resistance between s and t if G is thought of as an electrical network with unit resistance on each edge. If we use L_G to denote the Laplacian matrix of G and L_G^† to denote its pseudo-inverse, we have R_G(s,t) = (𝟏_s-𝟏_t)^⊤ L^† (𝟏_s-𝟏_t) such that classical Laplacian solvers [Daniel A. Spielman and Shang{-}Hua Teng, 2014] provide almost-linear time algorithms to approximate R_G(s,t).
In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair s and t. We consider the classical adjacency list model [Ron, 2019] for local algorithms. While recent works [Andoni et al., 2018; Peng et al., 2021; Li and Sachdeva, 2023] have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of n vertices and m edges:  
1) It needs Ω(n) queries to obtain 1.01-approximations of the effective resistance of an adjacent pair s and t, even for graphs of degree at most 3 except s and t.
2) For graphs of degree at most d and any parameter 𝓁, it needs Ω(m/𝓁) queries to obtain c ⋅ min{d,𝓁}-approximations where c > 0 is a universal constant.  Moreover, we supplement the first lower bound by providing a sublinear time (1+ε)-approximation algorithm for graphs of degree 2 except the pair s and t.
One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Sublinear Time Algorithm
  • Effective Resistance
  • Leverage Scores
  • Matrix Perturbation

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. In Proc. of the 57th FOCS, pages 335-344, 2016. Google Scholar
  2. Tasweer Ahmad, Lianwen Jin, Luojun Lin, and GuoZhi Tang. Skeleton-based action recognition using sparse spatio-temporal gcn with edge effective resistance. Neurocomputing, 423:389-398, 2021. Google Scholar
  3. Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 522-539. SIAM, 2021. Google Scholar
  4. Alexandr Andoni, Robert Krauthgamer, and Yosef Pogrow. On solving linear systems in sublinear time. In 10th Innovations in Theoretical Computer Science Conference (ITCS), 2018. Google Scholar
  5. Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM J. Comput., 41(6):1704-1721, 2012. URL: https://doi.org/10.1137/090772873.
  6. Serge Bernstein. Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné, volume 4. Hayez, imprimeur des académies royales, 1912. Google Scholar
  7. Vladimir Braverman, Aditya Krishnan, and Christopher Musco. Sublinear time spectral density estimation. In STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1144-1157. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520009.
  8. James R Bunch and John E Hopcroft. Triangular factorization and inversion by fast matrix multiplication. Mathematics of Computation, 28(125):231-236, 1974. Google Scholar
  9. Li Chen, Gramoz Goranci, Monika Henzinger, Richard Peng, and Thatchaphol Saranurak. Fast dynamic cuts, distances and effective resistances via vertex sparsifiers. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, pages 1135-1146. IEEE, 2020. Google Scholar
  10. Paul Christiano, Jonathan A. Kelner, Aleksander Madry, Daniel A. Spielman, and Shang-Hua Teng. Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs. In Proc. of the 43rd STOC, pages 273-282, 2011. Google Scholar
  11. Timothy Chu, Yu Gao, Richard Peng, Sushant Sachdeva, Saurabh Sawlani, and Junxing Wang. Graph sparsification, spectral sketches, and faster resistance computation, via short cycle decompositions. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, pages 361-372. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00042.
  12. Kenneth L. Clarkson and David P. Woodruff. Low rank approximation and regression in input sparsity time. In Symposium on Theory of Computing Conference, STOC'13, pages 81-90. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488620.
  13. Michael B. Cohen, Rasmus Kyng, Gary L. Miller, Jakub W. Pachocki, Richard Peng, Anup B. Rao, and Shen Chen Xu. Solving SDD linear systems in nearly mlog^1/2n time. In Proc. of the 46th STOC, pages 343-352, 2014. Google Scholar
  14. David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant. Approximating the spectrum of a graph. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD 2018, pages 1263-1271. ACM, 2018. URL: https://doi.org/10.1145/3219819.3220119.
  15. Michael Dinitz, Robert Krauthgamer, and Tal Wagner. Towards resistance sparsifiers. In Proc. of the 18th APPROX, pages 738-755, 2015. Google Scholar
  16. Petros Drineas and Michael W. Mahoney. Effective resistances, statistical leverage, and applications to linear equation solving. Technical report, available at https://arxiv.org/abs/1005.3097, 2010.
  17. David Durfee, Yu Gao, Gramoz Goranci, and Richard Peng. Fully dynamic effective resistances. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 914-925. ACM, 2019. Google Scholar
  18. David Durfee, Rasmus Kyng, John Peebles, Anup B Rao, and Sushant Sachdeva. Sampling random spanning trees faster than matrix multiplication. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 730-742. ACM, 2017. Google Scholar
  19. Wendy Ellens, FM Spieksma, P Van Mieghem, A Jamakovic, and RE Kooij. Effective graph resistance. Linear algebra and its applications, 435(10):2491-2506, 2011. Google Scholar
  20. Arpita Ghosh, Stephen Boyd, and Amin Saberi. Minimizing effective resistance of a graph. SIAM review, 50(1):37-66, 2008. Google Scholar
  21. Gramoz Goranci, Monika Henzinger, and Pan Peng. The power of vertex sparsifiers in dynamic graph algorithms. In Proc. of the 25th ESA, volume 87, pages 45:1-45:14, 2017. Google Scholar
  22. Gramoz Goranci, Monika Henzinger, and Pan Peng. Dynamic effective resistances and approximate schur complement on separable graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  23. John T Holodnak, Ilse CF Ipsen, and Thomas Wentworth. Conditioning of leverage scores and computation by qr decomposition. SIAM Journal on Matrix Analysis and Applications, 36(3):1143-1163, 2015. Google Scholar
  24. Oscar H Ibarra, Shlomo Moran, and Roger Hui. A generalization of the fast lup matrix decomposition algorithm and applications. Journal of Algorithms, 3(1):45-56, 1982. Google Scholar
  25. Ilse Ipsen and Thomas Wentworth. Sensitivity of leverage scores and coherence for randomized matrix algorithms. In Workshop on Advances in Matrix Functions and Matrix Equations, Manchester, UK, 2013. Google Scholar
  26. Ilse C. F. Ipsen. Relative perturbation results for matrix eigenvalues and singular values. Acta Numerica, 7:151-201, 1998. URL: https://doi.org/10.1017/S0962492900002828.
  27. Arun Jambulapati and Aaron Sidford. Efficient Õ(n/ε) spectral sketches for the laplacian and its pseudoinverse. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2487-2503. SIAM, 2018. Google Scholar
  28. Jonathan A Kelner, Gary L Miller, and Richard Peng. Faster approximate multicommodity flow using quadratically coupled flows. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 1-18. ACM, 2012. Google Scholar
  29. Yin Tat Lee. Probabilistic spectral sparsification in sublinear time. arXiv preprint arXiv:1401.0085, 2013. Google Scholar
  30. Huan Li and Zhongzhi Zhang. Kirchhoff index as a measure of edge centrality in weighted networks: Nearly linear time algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2377-2396. SIAM, 2018. Google Scholar
  31. Lawrence Li and Sushant Sachdeva. A new approach to estimating effective resistances and counting spanning trees in expander graphs. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2728-2745. SIAM, 2023. Google Scholar
  32. Richard J. Lipton, Donald J. Rose, and Robert Endre Tarjan. Generalized nested dissection. SIAM Journal on Numerical Analysis, 16(2):346-358, 1979. Google Scholar
  33. László Lovász et al. Random walks on graphs: A survey. Combinatorics, Paul erdos is eighty, 2(1):1-46, 1993. Google Scholar
  34. Aleksander Madry. Computing maximum flow with augmenting electrical flows. In Proc. of the 57th FOCS, pages 593-602, 2016. Google Scholar
  35. Aleksander Madry, Damian Straszak, and Jakub Tarnawski. Fast generation of random spanning trees and the effective resistance metric. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms, pages 2019-2036. Society for Industrial and Applied Mathematics, 2015. Google Scholar
  36. Grigorii Aleksandrovich Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problemy peredachi informatsii, 24(1):51-60, 1988. Google Scholar
  37. Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes. Explicit near-ramanujan graphs of every degree. SIAM Journal on Computing, 51(3):1-23, 2022. URL: https://doi.org/10.1137/20M1342112.
  38. M. Morgenstern. Existence and explicit constructions of q + 1 regular ramanujan graphs for every prime power q. Journal of Combinatorial Theory, Series B, 62(1):44-62, 1994. Google Scholar
  39. Cameron Musco, Praneeth Netrapalli, Aaron Sidford, Shashanka Ubaru, and David P. Woodruff. Spectrum approximation beyond fast matrix multiplication: Algorithms and hardness. In 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, volume 94 of LIPIcs, pages 8:1-8:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.8.
  40. Pan Peng, Daniel Lopatta, Yuichi Yoshida, and Gramoz Goranci. Local algorithms for estimating effective resistance. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pages 1329-1338, 2021. Google Scholar
  41. Dana Ron. Sublinear-time algorithms for approximating graph parameters. In Computing and Software Science, pages 105-122. Springer, 2019. Google Scholar
  42. Piotr Sankowski. Dynamic transitive closure via dynamic matrix inverse. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 509-517. IEEE Computer Society, 2004. Google Scholar
  43. Aaron Schild. An almost-linear time algorithm for uniform random spanning tree generation. Proc. of the 50th STOC, 2018. Google Scholar
  44. Daniel A. Spielman. Spectral and algebraic graph theory. Draft of textbook, available at http://cs-www.cs.yale.edu/homes/spielman/sagt/, 2019.
  45. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM J. Comput., 40(6):1913-1926, 2011. Google Scholar
  46. Daniel A. Spielman and Shang-Hua Teng. Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM J. Matrix Analysis Applications, 35(3):835-885, 2014. Google Scholar
  47. David P. Woodruff. Sketching as a tool for numerical linear algebra. Found. Trends Theor. Comput. Sci., 10(1-2):1-157, 2014. URL: https://doi.org/10.1561/0400000060.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact