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PageRank solves the dominant eigenvector problem by iteratively finding the steady-state discrete flow condition of the network.
If NxM matrix A describes the link weight (amount of flow) from node n to node m, then
p_{n+1} = A . p_{n}
In the limit where p has converged to a steady state (p_n+1 = p_n), this is an eigenvector problem with eigenvalue 1.
The PageRank algorithm doesn't require the matrix to be held in memory, but is inefficient on dense (non-sparse) matrices. For dense matrices, MapReduce is the wrong solution -- you need locality and broad exchange among nodes -- and you should instead look at LaPACK and MPI and friends.
You can see a working pagerank implementation in the wukong library (hadoop streaming for ruby) or in the Heretrix pagerank submodule. (The heretrix code runs independently of Heretrix)
(disclaimer: I am an author of wukong.)