  In this paper, we explore the reduction of functionality in a complex system
as a consequence of cumulative random damage and imperfect reparation, a
phenomenon modeled as a dynamical process on networks. We analyze the global
characteristics of the diffusive movement of random walkers on networks where
the walkers hop considering the capacity of transport of each link. The links
are susceptible to damage that generates bias and aging. We describe the
algorithm for the generation of damage and the bias in the transport producing
complex eigenvalues of the transition matrix that defines the random walker for
different types of graphs, including regular, deterministic, and random
networks. The evolution of the asymmetry of the transport is quantified with
local information in the links and further with non-local information
associated with the transport on a global scale such as the matrix of the mean
first passage times and the fractional Laplacian matrix. Our findings suggest
that systems with greater complexity live longer.
