Effective resistance in metric spaces (2306.15649v1)
Abstract: Effective resistance (ER) is an attractive way to interrogate the structure of graphs. It is an alternative to computing the eigenvectors of the graph Laplacian. One attractive application of ER is to point clouds, i.e. graphs whose vertices correspond to IID samples from a distribution over a metric space. Unfortunately, it was shown that the ER between any two points converges to a trivial quantity that holds no information about the graph's structure as the size of the sample increases to infinity. In this study, we show that this trivial solution can be circumvented by considering a region-based ER between pairs of small regions rather than pairs of points and by scaling the edge weights appropriately with respect to the underlying density in each region. By keeping the regions fixed, we show analytically that the region-based ER converges to a non-trivial limit as the number of points increases to infinity. Namely the ER on a metric space. We support our theoretical findings with numerical experiments.
- Graph sparsification by effective resistances. SIAM Journal on Computing, 40(6):1913–1926, 2011.
- Prediction on a graph with a perceptron. Advances in neural information processing systems, 19, 2006.
- Detecting community structure in complex networks via resistance distance. Physica A: Statistical Mechanics and its Applications, 526:120782, 2019.
- A kernel view of the dimensionality reduction of manifolds. In Proceedings of the twenty-first international conference on Machine learning, page 47, 2004.
- Phylogenetic networks as circuits with resistance distance. Frontiers in Genetics, 11:1177, 2020.
- Measuring cascade effects in interdependent networks by using effective graph resistance. In 2015 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS), pages 683–688, 2015.
- The impact of the topology on cascading failures in a power grid model. Physica A: Statistical Mechanics and its Applications, 402:169–179, 2014.
- A network approach for power grid robustness against cascading failures. In 2015 7th International Workshop on Reliable Networks Design and Modeling (RNDM), pages 208–214, 2015.
- Graph algorithms for topology identification using power grid probing. IEEE Control Systems Letters, 2(4):689–694, 2018.
- László Lovász. Random walks on graphs. Combinatorics, Paul erdos is eighty, 2(1-46):4, 1993.
- Mixing times for random walks on geometric random graphs. In ALENEX/ANALCO, pages 240–249, 2005.
- On the cover time and mixing time of random geometric graphs. Theoretical Computer Science, 380(1-2):2–22, 2007.
- Getting lost in space: Large sample analysis of the resistance distance. In Advances in Neural Information Processing Systems, volume 23. Curran Associates, Inc., 2010.
- Hitting and commute times in large random neighborhood graphs. The Journal of Machine Learning Research, 15(1):1751–1798, 2014.
- On extension of effective resistance with application to graph laplacian definiteness and power network stability. IEEE Transactions on Circuits and Systems I: Regular Papers, 66(11):4415–4428, 2019.
- Resistance distance. Journal of mathematical chemistry, 12(1):81–95, 1993.
- Operator theory and analysis of infinite networks. arXiv preprint arXiv:0806.3881, 3, 2008.
- Minimizing effective resistance of a graph. SIAM Review, 50(1):37–66, 2008.
- A queueing network-based distributed laplacian solver for directed graphs. Information Processing Letters, 166:106040, 2021.
- Cover trees for nearest neighbor. In Proceedings of the 23rd international conference on Machine learning, pages 97–104, 2006.
- Streamrak a streaming multi-resolution adaptive kernel algorithm. Applied Mathematics and Computation, 426:127112, 2022.
- S. Muthukrishnan. Data streams: Algorithms and applications. Foundations and Trends® in Theoretical Computer Science, 1(2):117–236, 2005.
- Advances in metric embedding theory. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 271–286, 2006.