  In the study of small and large networks it is customary to perform a simple
random walk, where the random walker jumps from one node to one of its
neighbours with uniform probability. The properties of this random walk are
intimately related to the combinatorial properties of the network. In this
paper we propose to use the Ruelle-Bowens random walk instead, whose
probability transitions are chosen in order to maximise the entropy rate of the
walk on an unweighted graph. If the graph is weighted, then a free energy is
optimised instead of entropy rate.
  Specifically, we introduce a centrality measure for large networks, which is
the stationary distribution attained by the the Ruelle-Bowens random walk; we
name it Entropy Rank. We introduce a more general version, able to deal with
disconnected networks, under the name of Free Energy Rank. We compare the
properties of those centrality measures with the classic PageRank and HITS on
both toy and real-life examples, in particular their robustness to small
modifications of the network. It is observed that our centrality measures have
a better discriminating power than PageRank, being able to distinguish clearly
pages that PageRank holds for almost equally interesting, and is more sensitive
to the medium-scale details of the graph.
