Abstract Phase-space Networks Describing Reactive Dynamics

An abstract network approach is proposed for the description of the dynamics in reactive processes. The phase space of the variables (concentrations in reactive systems) is partitioned into a finite number of segments, which constitute the nodes of the abstract network. Transitions between the nodes are dictated by the dynamics of the reactive process and provide the links between the nodes. These are weighted networks, since each link weight reflects the transition rate between the corresponding states-nodes. With this construction the network properties mirror the dynamics of the underlying process and one can investigate the system properties by studying the corresponding abstract network. As a working example the Lattice Limit Cycle (LLC) model is used. Its corresponding abstract network is constructed and the transition matrix elements are computed via Kinetic (Dynamic) Monte Carlo simulations. For this model it is shown that the degree distribution follows a power law with exponent -1, while the average clustering coefficient $c(N)$ scales with the network size (number of nodes) $N$ as $c(N)\sim N^{-\nu}, \nu \simeq 1.46$. The computed exponents classify the LLC abstract reactive network into the scale-free networks. This conclusion corroborates earlier investigations demonstrating the formation of fractal spatial patterns in LLC reactive dynamics due to stochasticity and to the clustering of homologous species. The present construction of abstract networks (based on the partition of the phase space) is generic and can be implemented with appropriate adjustments in many dynamical systems and in time series analysis.