  Designing optimal structure favorable to diffusion and effectively
controlling the trapping process are crucial in the study of trapping
problem---random walks with a single trap. In this paper, we study the trapping
problem occurring on unweighted and weighted networks, respectively. The
networks under consideration display the striking scale-free, small-world, and
modular properties, as observed in diverse real-world systems. For binary
networks, we concentrate on three cases of trapping problems with the trap
located at a peripheral node, a neighbor of the root with the least
connectivity, and a farthest node, respectively. For weighted networks with
edge weights controlled by a parameter, we also study three trapping problems,
in which the trap is placed separately at the root, a neighbor of the root with
the least degree, and a farthest node. For all the trapping problems, we obtain
the analytical formulas for the average trapping time (ATT) measuring the
efficiency of the trapping process, as well as the leading scaling of ATT. We
show that for all the trapping problems in the binary networks with a trap
located at different nodes, the dominating scalings of ATT reach the possible
minimum scalings, implying that the networks have optimal structure that is
advantageous to efficient trapping. Furthermore, we show that for trapping in
the weighted networks, the ATT is controlled by the weight parameter, through
modifying which, the ATT can behave superlinealy, linearly, sublinearly, or
logarithmically with the system size. This work could help improving the design
of systems with efficient trapping process and offers new insight into control
of trapping in complex systems.
