Measuring Shape with Topology

TL;DR

This paper introduces a novel topological measure of shape that captures complex geometric structures and includes a new fractal dimension, demonstrated on various stochastic and polymer models.

Contribution

It presents a new topological shape measure based on neighborhood topology, including a novel fractal dimension, applicable to complex structures.

Findings

Effective in analyzing branched polymers

Applicable to Brownian trees and self-avoiding walks

Demonstrates computability and utility of the measure

Abstract

We propose a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure. One aspect of this measure gives a new notion of fractal dimension. We demonstrate the utility and computability of this measure by applying it to branched polymers, Brownian trees, and self-avoiding random walks.