Another interesting mapping is that a vector is (or can be thought of as) a discrete function (f(x) = ....) over an interval, a dot product of two vectors is a discrete integral product, and a matrix is a discrete scalar field.
I wonder what the continuous form of a graph is... Some sort of a manifold perhaps?
> I wonder what the continuous form of a graph is... Some sort of a manifold perhaps?
Exactly!
The correspondence between manifolds and graphs is very beautiful. What many folks call today "graph signal processing" has traditionally been called "discrete differential geometry". Scalar fields are functions defined on vertices, vector fields are functions defined on edges, the incidence matrix is the gradient operator, its transpose is the divergence, the Laplacian is the divergence of the gradient, integrals and fluxes are scalar products by indicator functions, the boundary operator is minus the gradient, Green's formula is just matrix transposition, etc.
You can even go further in the analogy and define p-forms as functions defined on the p-cliques of the graph, and from that rebuild a whole discrete Hodge theory. The correspondence is almost perfect, except for the fact that you cannot write easily the product rule for derivatives (because you cannot multiply pointwise scalar fields with vector fields).
And electromagnetism seems to be a by product of discrete differential geometry. Such a fascinating subject. Makes continuous treatment look like a mess.
I wonder what the continuous form of a graph is... Some sort of a manifold perhaps?