Trapping in complex networks

We investigate the trapping problem in Erds-Rényi (ER) and scale-free (SF) networks. We calculate the evolution of the particle density ρ(t) of random walkers in the presence of one or multiple traps with concentration c. We show using theory and simulations that in ER networks, while for short times ρ(t)∝exp(-Act), for longer times ρ(t) exhibits a more complex behavior, with explicit dependence on both the number of traps and the size of the network. In SF networks we reveal the significant impact of the trap's location: ρ(t) is drastically different when a trap is placed on a random node compared to the case of the trap being on the node with the maximum connectivity. For the latter case we find ρ(t) exp [-At/N ^γ-2/γ-1 k] for all γ> 2, where γ is the exponent of the degree distribution P(k)∝k^{- γ}.