# Structure from Voltage

arXiv: https://arxiv.org/abs/2203.00063 · HTML: https://mlsanity.com/paper/2203.00063/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 31 references that are in the map.
- First posted on arXiv 28 Feb 2022.

## Builds on (most cited first, 7 of 7)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [t-SNE, Forceful Colorings and Mean Field Limits](https://mlsanity.com/paper/2102.13009/) (2021)
- [Graph Algorithms for Topology Identification using Power Grid Probing](https://mlsanity.com/paper/1803.04506/) (2018)
- [The Impact of the Topology on Cascading Failures in Electric Power Grids](https://mlsanity.com/paper/1401.4473/) (2013)
- [A network approach for power grid robustness against cascading failures](https://mlsanity.com/paper/1505.06312/) (2015)
- [Phylogenetic networks as circuits with resistance distance](https://mlsanity.com/paper/2007.13574/) (2020)

## Abstract (arXiv)

Effective resistance (ER) is an attractive way to interrogate the structure of graphs. It is an alternative to computing the eigen-vectors of the graph Laplacian. Graph laplacians are used to find low dimensional structures in high dimensional data. Here too, ER based analysis has advantages over eign-vector based methods. Unfortunately Von Luxburg et al. (2010) show that, when vertices correspond to a sample from a distribution over a metric space, the limit of the ER between distant points converges to a trivial quantity that holds no information about the structure of the graph. We show that by using scaling resistances in a graph with n vertices by n2, one gets a meaningful limit of the voltages and of effective resistances. We also show that by adding a "ground" node to a metric graph one gets a simple and natural way to compute all of the distances from a chosen point to all other points.