The space of circular planar electrical networks

TL;DR

This paper explores parametrizations of circular planar electrical networks, introducing a canonical minimal network called a 'standard' network, and provides methods to compute conductances and test network connectivity efficiently.

Contribution

It introduces a canonical 'standard' network with conductances as Pfaffian ratios, and offers efficient criteria for testing network connectivity through positivity conditions.

Findings

Conductances in standard networks are computed as ratios of Pfaffians.

Connectivity can be tested by checking positivity of specific minors.

Positivity of conductances can be verified via Pfaffians, reducing computational complexity.

Abstract

We discuss several parametrizations of the space of circular planar electrical networks. For any circular planar network we associate a canonical minimal network with the same response matrix, called a "standard" network. The conductances of edges in a standard network can be computed as a biratio of Pfaffians constructed from the response matrix. The conductances serve as coordinates that are compatible with the cell structure of circular planar networks in the sense that one conductance degenerates to 0 or infinity when moving from a cell to a boundary cell. We also show how to test if a network with n nodes is well-connected by checking that $\binom{n}{2}$ minors of the $n\times n$ response matrix are positive; Colin de Verdi\`ere had previously shown that it was sufficient to check the positivity of exponentially many minors. For standard networks with m edges, positivity of the conductances can be tested by checking the positivity of m+1 Pfaffians.