Discrete differential manifolds and dynamics on networks

A discrete differential manifold is a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time. Several examples are presented and a notion of differentiability of maps between discrete differential manifolds is introduced. 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.