  The study of very large graphs is a prominent theme in modern-day
mathematics. In this paper we develop a rigorous foundation for studying the
space of finite labelled graphs and their limits. These limiting objects are
naturally countable graphs, and the completed graph space $\mathscr{G}(V)$ is
identified with the 2-adic integers as well as the Cantor set. The goal of this
paper is to develop a model for differentiation on graph space in the spirit of
the Newton-Leibnitz calculus. To this end, we first study the space of all
finite labelled graphs and their limiting objects, and establish analogues of
left-convergence, homomorphism densities, a Counting Lemma, and a large family
of topologically equivalent metrics on labelled graph space. We then establish
results akin to the First and Second Derivative Tests for real-valued functions
on countable graphs, and completely classify the permutation automorphisms of
graph space that preserve its topological and differential structures.
