Diffusion, annihilation, and chemical reactions in complex networks with spatial constraints.

We consider Erdö]s-Rényi-type networks embedded in one-dimensional (de=1) and two-dimensional (de=2) Euclidean space with the link-length distribution p(r)∼r-δ. The dimension d of these networks, as a function of δ, has been studied earlier and has been shown to depend on δ. Here we consider diffusion, annihilation, and chemical reaction processes on these spatially constrained networks and show that their dynamics is controlled by the dimension d of the system. We study, as a function of the exponent δ and the embedding dimension de, the average distance ∼t1/dw a random walker has traveled after t time steps as well as the probability of the random walker's return to the origin P0(t). From these quantities we determine the network dimension d and the dimension dw of the random walk as a function of δ. We find that the fraction d/dw governs the number of survivors as a function of time t in the annihilation process (A+A→0) and in the chemical reaction process (A+B→0), showing that the relations derived for ordered and disordered lattices with short-range links remain valid also in the case of complex embedded networks with long-range links.