  Extensive empirical investigation has shown that a plethora of real networks
synchronously exhibit scale-free and modular structure, and it is thus of great
importance to uncover the effects of these two striking properties on various
dynamical processes occurring on such networks. In this paper, we examine two
cases of random walks performed on a class of modular scale-free networks with
multiple traps located at several given nodes. We first derive a formula of the
mean first-passage time (MFPT) for a general network, which is the mean of the
expected time to absorption originating from a specific node, averaged over all
non-trap starting nodes. Although the computation is complex, the expression of
the formula is exact; moreover, the computational approach and procedure are
independent of the number and position of the traps. We then determine
analytically the MFPT for the two random walks being considered. The obtained
analytical results are in complete agreement with the numerical ones. Our
results show that the number and location of traps play an important role in
the behavior of the MFPT, since for both cases the MFPT grows as a power-law
function of the number of nodes, but their exponents are quite different. We
demonstrate that the root of the difference in the behavior of MFPT is
attributed to the modular and scale-free topologies of the networks. This work
can deepen the understanding of diffusion on networks with modular and
scale-free architecture and motivate relevant studies for random walks running
on complex random networks with multiple traps.
