# Electrical Networks with Prescribed Current and Applications to Random Walks on Graphs

## TL;DR

This paper addresses inverse problems in electrical networks and random walks on graphs, focusing on recovering conductivities, transition probabilities, and flows from partial measurements, with practical algorithms provided.

## Contribution

It introduces new methods for uniquely recovering network conductivities, transition probabilities, and flows from magnitude and boundary data, with convergence guarantees for algorithms.

## Key findings

- Unique recovery of conductivities from current magnitudes and boundary data.
- Transition probabilities of random walks can be determined from edge passage counts.
- Mass-preserving flows can be reconstructed from flow magnitudes and boundary fluxes.

## Abstract

We study the inverse problem of determining the conductivity matrix of an electrical network from the prescribed knowledge of the magnitude of the induced current along the edges coupled with the imposed voltage or injected current on the boundary nodes. This problem leads to a weighted $l^1$ minimization problem for the corresponding voltage potential. We also investigate the problem of determining the transition probabilities of random walks on graphs from the prescribed net number of times the walker passes along the edges of the graph. We also show that a mass preserving flow $J=(J_{i.j})$ on a network can be uniquely recovered from the knowledge of $|J|=(|J_{i,j}|)$ and the flux of the flow on the boundary nodes, where $J_{i,j}$ is the flow from node $i$ to node $j$ and $J_{i,j}=-J_{j,i}$. Convergent numerical algorithms for solving such problems are also presented.