PageRank for a Non-Random Searcher I'm looking to adapt the PageRank algorithm as a centrality measure in a network. This network however, unlike the "random surfer" of the original paper on PageRank, or the random library browser for Eigenfactor.org, doesn't have a random browser who can leave and jump off to some other network. The theoretical reader is reading only this literature, and reading it completely.

As I understand it, the damping factor in the usual implementation of PageRank is 1 - probability of the random surfer jumping to a different site, and is usually set at 0.85. Is it reasonable then, in an entirely closed network, to set this value = 1.0, or is there something I'm not seeing?

Some details of the network, which would probably be helpful:

All the networks I will be looking at are fairly small, less than 1000 nodes, and directed. They're citation networks - with papers as nodes and edges as citations between papers, so inherently there are no isolated nodes not connected to any other nodes, as their inclusion in the network is conditional on there being a link to or from the network. There's no reason to believe the network is strongly connected - indeed, I'm pretty sure they're inherently not.
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.