We investigate the trapping problem in Erdős-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}.

We report the critical point for site percolation for the ``explosive'' type for two-dimensional square lattices using Monte Carlo simulations and compare it to the classical well-known percolation. We use similar algorithms as have been recently reported for bond percolation and networks. We calculate the explosive site percolation threshold as pc=0.695 and we find evidence that explosive site percolation surprisingly may belong to a different universality class than bond percolation on lattices, providing that the transitions (a) are continuous and (b) obey the conventional finite size scaling forms. Finally, we study and compare the direct and reverse processes, showing that while the reverse process is different from the direct process for finite size systems, the two cases become equivalent in the thermodynamic limit of large L.