Discrete Differential Manifolds and Dynamics on Networks

TL;DR

This paper introduces the concept of discrete differential manifolds, providing a framework for modeling dynamics on networks and discrete physical theories, with new notions of differentiability and continuity.

Contribution

It defines discrete differential manifolds, explores differentiability of maps between them, and establishes a topology where differentiability implies continuity.

Findings

Discrete differential manifolds form a useful framework for network dynamics.

Differentiability of maps implies continuity in these spaces.

Examples illustrate the application of the theory.

Abstract

A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time. We present several examples and introduce a notion of differentiability of maps between discrete differential manifolds. Particular attention is given to differentiable curves in such spaces. Every discrete differentiable manifold carries a topology and we show that differentiability of a map implies continuity.