PageRank for a Non-Random Searcher
Let $\alpha$ be the probability of clicking through a link and let $1-\alpha$ be the probability of going to a random website. I think you're asking whether PageRank will work with $\alpha=1$. The answer is yes if and only if the network is strongly connected and the induced random walk is aperiodic. I'm not sure what "entirely closed" means.

Setting $\alpha = 1 $ gives intuition for how PageRank works, but there are very good reasons for setting $\alpha &lt; 1$. Namely, we want PageRank (1) to converge in every graph, (2) to converge to the same thing regardless of the initial rank distribution, (3) to converge as quickly as possible. PageRank is just a fancy name for how to set up a particular Markov Chain. For any $\alpha \in [0,1] $ you'll get some Markov Chain, you just need to make sure that it has properties 1-3.

This is the best PageRank survey that I've seen (better than Wikipedia or the original paper) and explains this: http://www.ams.org/samplings/feature-column/fcarc-pagerank

Edit: As per MCH's answer we also want the 4th condition that every vertex has positive rank, which is also ensured by the random surfer model.