Trapping in scale-free networks with hierarchical organization of modularity

A wide variety of real-life networks share two remarkable generic topological properties: scale-free behavior and modular organization, and it is natural and important to study how these two features affect the dynamical processes taking place on such networks. In this paper, we investigate a simple stochastic process--trapping problem, a random walk with a perfect trap fixed at a given location, performed on a family of hierarchical networks that exhibit simultaneously striking scale-free and modular structure. We focus on a particular case with the immobile trap positioned at the hub node having the largest degree. Using a method based on generating functions, we determine explicitly the mean first-passage time (MFPT) for the trapping problem, which is the mean of the node-to-trap first-passage time over the entire network. The exact expression for the MFPT is calculated through the recurrence relations derived from the special construction of the hierarchical networks. The obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the MFPT grows algebraically with the network order. Concretely, the MFPT increases as a power-law function of the number of nodes with the exponent much less than 1. We demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale-free networks. We argue that the scale-free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks.