What are some applications of loops in real life? I am aware of the real world applications of simple graphs and graphs with multiple edges but how are loops used? For example, a few computers linked to each other would be an example of simple graphs or graphs with multiple edges, where multiple edges can represent links with different speeds of communication. But where would one see loops in real life?
 A: One can produce a broad class of examples via the graphs associated to Markov chains, which have applications in chemistry, queueing theory, genetics, board games, and more.
Consider a system with $n$ states, denoted $s^b$, that evolves in discrete steps: At each (discrete) time step, the system may change the state, say; if it is in state $s^b$, let $p^a{}_b$ be the probability that (at a particular time step) it changes to state $s^a$. In particular, in a Markov chain these probabilities are state functions they depend only on the current state, not on the previous history of the system, so the probabilities $p^a{}_b$ are the same for each time step; in turn, these probabilities together define the Markov chain completely.
A particularly simple example comes from a Poisson process: Suppose one buys a new pet crocodile, which has two states, $s_{\text{alive}}$ and $s_{\text{dead}}$. Each day it is alive, the crocodile has some small ($\epsilon \ll 1$) probability of dying, and a dead crocodile stays dead. Then, the probabilities are $p^{\text{alive}}{}_{\text{alive}} = 1 - \epsilon$, $p^{\text{dead}}{}_{\text{alive}} = \epsilon$, $p^{\text{alive}}{}_{\text{dead}} = 0$, and (sadly) $p^{\text{dead}}{}_{\text{dead}} = 1$.
We often represent a Markov chain as a matrix with $(a, b)$ entry $p^a{}_b$ (each row must, by definition of probability, sum to $1$), which facilitates easy computation of many interesting properties related to the underlying system. But we can also represent a Markov chain as a labeled digraph with no multiple edges, whose vertices are the states of the system, and whose (directed) edges represent transitions from one state to another; the edge from $s^b$ to $s^a$ is labeled with the probability $p^a{}_b$. (Sometimes one omits edges corresponding to transitions with probability zero, which makes it easier, e.g., to find sinks, that is, states from which the system does not escape.) In this case, a (directed) loop encodes the transition from a state to itself, like the crocodile remaining alive (or remaining dead).
Edit Wikipedia has a short article dedicated to examples of Markov chains, including some digraphs of the sort mentioned.
A: I guess you don't accept regular expression checking as a "real world example" (even though these are of utmost importance to computer programming), but it absolutely involves self-loops, just as the following example I gave: $x^*y$:

I'm guessing our difference here is in using graphs to describe finite state machines--which you don't (perhaps) see as a "real world example".
