Trapping in complex networks

Abstract

We investigate the trapping problem in Erdos-Renyi (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 $ρ(t) \propto \exp(-Act)$, 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 $ρ(t)\propto\exp\left[-At/N^\frac{γ-2}{γ-1}\av{k}\right]$ for all $γ>2$, where $γ$ is the exponent of the degree distribution $P(k)\propto k^{-γ}$.