  We introduce a novel formulation for geometry on discrete points. It is based
on a universal differential calculus, which gives a geometric description of a
discrete set by the algebra of functions. We expand this mathematical framework
so that it is consistent with differential geometry, and works on spectral
graph theory and random walks. Consequently, our formulation comprehensively
demonstrates many discrete frameworks in probability theory, physics, applied
harmonic analysis, and machine learning. Our approach would suggest the
existence of an intrinsic theory and a unified picture of those discrete
frameworks.
