# Laplacian networks: growth, local symmetry and shape optimization

## TL;DR

This paper investigates the growth of Laplacian networks using the Loewner equation, revealing equivalences between growth rules and exploring their impact on network shape and flux optimization.

## Contribution

It introduces a formalism linking three growth rules for Laplacian networks and analyzes their effects on network evolution and shape optimization.

## Key findings

- Growth rules are mathematically equivalent under certain conditions.
- Different growth rules can lead to distinct network configurations.
- Flux optimization does not always produce the static shape that maximizes tip flux.

## Abstract

Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three formulations of the direction rule, namely geodesic growth, growth that maintains local symmetry and growth that maximizes flux into tips for a given amount of growth. Surprisingly, we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips.