Effect of trap position on the efficiency of trapping in treelike scale-free networks

Abstract

The conventional wisdom is that the role and impact of nodes on dynamical processes in scale-free networks are not homogenous, because of the presence of highly connected nodes at the tail of their power-law degree distribution. In this paper, we explore the influence of different nodes as traps on the trapping efficiency of the trapping problem taking place on scale-free networks. To this end, we study in detail the trapping problem in two families of deterministically growing scale-free networks with treelike structure: one family is non-fractal, the other is fractal. In the first part of this work, we attack a special case of random walks on the two network families with a perfect trap located at a hub, i.e. node with the highest degree. The second study addresses the case with trap distributed uniformly over all nodes in the networks. For these two cases, we compute analytically the mean trapping time (MTT), a quantitative indicator characterizing the trapping efficiency of the trapping process. We show that in the non-fractal scale-free networks the MTT for both cases follows different scalings with the network order (number of network nodes), implying that trap's position has a significant effect on the trapping efficiency. In contrast, it is presented that for both cases in the fractal scale-free networks, the two leading scalings exhibit the same dependence on the network order, suggesting that the location of trap has no essential impact on the trapping efficiency. We also show that for both cases of the trapping problem, the trapping efficiency is more efficient in the non-fractal scale-free networks than in their fractal counterparts.