  Entropy production (EP) is known as a fundamental quantity for measuring the
irreversibility of processes in thermal equilibrium and states far from
equilibrium. In stochastic thermodynamics, the EP becomes more visible in terms
of the probability density functions of the trajectories of a particle in the
state space. Inspired by a previous result that complex networks can serve as
state spaces, we consider a data packet transport problem on complex networks.
Entropy is produced owing to the complexity of pathways as the packet travels
back and forth between two nodes. The EPs are exactly enumerated along the
shortest paths between every pair of nodes, and the functional form of the EP
distribution is determined by extreme value analysis. The asymptote of the
accumulated EP distribution is found to follow the Gumbel distribution.
