TITLE: Continuous notions with compelling discrete analogues
QUESTION [12 upvotes]: Following up on the previous MO question "Are there any important mathematical concepts without discrete analogue?", I'd like to ask the opposite: what are examples of notions in math that were not originally discrete, but have good discrete analogues?  While a few examples arose in the answers to that earlier MO question, this wasn't what that question was asking, so I'm sure there are many more examples not mentioned there or at least not really explained there.  What reminded me of this older MO question was seeing an MO question "Why is the Laplacian ubiquitous?", since that is an instance of an important notion which has a discrete analgoue. 
In an answer, it would be interesting to hear about the relationship between the continuous and discrete versions of the notion, if possible, and references could also be helpful.  Thanks!

REPLY [7 votes]: Finite graphs are a rich source of discrete analogues (I will be partially repeating the OP and some other answers here): 

The Laplacian on a finite graph is a discrete analogue of the Laplacian on a Riemannian manifold. In particular, it is possible to formulate the heat equation, the wave equation, and the Schrödinger equation on a finite graph. There are actually two Laplacians, a vertex Laplacian and an edge Laplacian, which give a discrete analogue of Hodge theory. 
The Ihara zeta function of a finite graph is a discrete analogue of the Selberg zeta function of a Riemannian manifold. A regular graph satisfies an analogue of the Riemann hypothesis if and only if it is a Ramanujan graph. There is also an analogue of the Selberg trace formula in this setting; Terras has written extensively about this kind of thing. 
The Picard group (or critical group, or sandpile group) of a finite graph is a discrete analogue of the Picard group of an algebraic curve. More generally a lot of the theory of algebraic curves can be transported to this setting, e.g. the Riemann-Roch theorem. 

(Finite graphs are also a rich source of other kinds of analogues; for example the Ihara zeta function is also analogous to the Dedekind zeta function of a number field, with coverings of graphs analogous to extensions of number fields and the Picard group analogous to the class group. There is even an analogue of the analytic class number formula in this setting although I have forgotten the reference.)

REPLY [2 votes]: Continuous-time random walks on graphs are in some sense a discrete analogue of diffusions on a Riemannian manifold (of course, the reverse can be argued, but I think that diffusions play a more central role in modern probability theory).  Of course, the most important diffusion is Brownian motion, i.e., the Markov process associated with the Laplace-Beltrami operator.  From my perspective, the natural analogue of Brownian motion is the operator $\mathcal{L}_V$ given by (we use unweighted graphs for simplicity)
\begin{equation*}
(\mathcal{L}_Vf)(x) := \sum\_{y\sim x}(f(y)-f(x)).
\end{equation*}
A more 'common' choice might be the rate-1 continuous time random walk with generator $\mathcal{L}_C$ given by
\begin{equation*}
(\mathcal{L}_Cf)(x) := \frac{1}{\deg(x)}\sum\_{y\sim x}(f(y)-f(x)).
\end{equation*}
However, this choice of generator has several 'bad' properties if you want to view it as an analogue of Brownian motion -- for example, the generator is always bounded on $L^2(\deg)$, it cannot have discrete spectrum, and the associated random walk cannot explode; in contrast, the operator $\mathcal{L}_V$ may be unbounded, and discrete spectrum and explosiveness are possible.
Once you have this discrete (space) analogue of Brownian motion on a Riemannian manifold, a natural question is to ask what the discrete analogue of the Riemannian metric should be for this process.  It is not too hard to find examples that show that the graph metric is a bad analogue, since the Riemannian metric governs heat flow (in some sense) on a Riemannian manifold (see e.g. here), but Gaussian heat kernel estimates do not hold for the random walk associated with $\mathcal{L}_V$ if you take the manifold heat kernel estimates and replace the distance function with the graph metric.  A reasonable analogue has been formulated recently, see e.g. here and here.