Dimension Theory of Graphs and Networks

TL;DR

This paper develops a notion of dimension for irregular structures like graphs and networks, exploring its properties, stability, and construction of graphs with various fractal dimensions, relevant for physics and mathematics at the Planck scale.

Contribution

It introduces a new dimension concept for graphs and networks, analyzes its stability, and provides methods to construct graphs with specific fractal dimensions.

Findings

Dimension is stable under many perturbations.

Graphs with nearly arbitrary fractal dimensions can be systematically constructed.

The framework may aid in understanding dimensional phase transitions and renormalization.

Abstract

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of continuum physics and mathematics. A core concept is the notion of dimension. In the following we develop such a notion for irregular structures like (large) graphs and networks and derive a number of its properties. Among other things we show its stability under a wide class of perturbations which is important if one has 'dimensional phase transitions' in mind. Furthermore we systematically construct graphs with almost arbitrary 'fractal dimension' which may be of some use in the context of 'dimensional renormalization' or statistical mechanics on irregular sets.