  In this paper we view the steady states of classical random walks over
complex networks with an arbitrary degree distribution as states in thermal
equilibrium. By identifying the distribution of states as a canonical ensemble,
we are able to define the temperature and the Hamiltonian for the random walk
systems. We then calculate the Helmholtz free energy, the average energy, and
the entropy for the thermal equilibrium states. The results shows equipartition
of energy for the average energy. The entropy is found to consist of two parts.
The first part decreases as the number of walkers increases. The second part of
the entropy depends solely on the topology of the network, and will increase
when more edges or nodes are added to the network. We compare the topological
part of entropy with some of the network descriptors and find that the
topological entropy could be used as a measure of network complexity. In
addition, we discuss the scenario that a walker has a prior probability of
resting on the same node at the next time step, and find that the effect of the
prior resting probabilities is equivalent to increasing the degree for every
node in the network.
