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

Thorsten Emmerich, Armin Bunde, Shlomo Havlin
2012 Physical Review E  
We consider Erdös-Rényi-type networks embedded in one-dimensional (d e = 1) and two-dimensional (d e = 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
more » ... tudy, as a function of the exponent δ and the embedding dimension d e , the average distance r ∼ t 1/dw a random walker has traveled after t time steps as well as the probability of the random walker's return to the origin P 0 (t). From these quantities we determine the network dimension d and the dimension d w of the random walk as a function of δ. We find that the fraction d/d w 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.
doi:10.1103/physreve.86.046103 pmid:23214648 fatcat:ri3hca27yvchtpavj6ihaw72dm