Advances in the Simulation and Modeling of Complex Systems using Dynamical Graph Grammars

Abstract

The Dynamical Graph Grammar (DGG) formalism can describe complex system dynamics with graphs that are mapped into a master equation. An exact stochastic simulation algorithm may be used, but it is slow for large systems. To overcome this problem, an approximate spatial stochastic/deterministic simulation algorithm, which uses spatial decomposition of the system's time-evolution operator through an expanded cell complex (ECC), was previously developed and implemented for a cortical microtubule array (CMA) model. Here, computational efficiency is improved at the cost of introducing errors confined to interactions between adjacent subdomains of different dimensions, realized as some events occurring out of order. A rule instances to domains mapping function $ϕ$, ensures the errors are local. This approach has been further refined and generalized in this work. Additional efficiency is achieved by maintaining an incrementally updated match data structure for all possible rule matches. The API has been redesigned to support DGG rules in general, rather than for one specific model. To demonstrate these improvements in the algorithm, we have developed the Dynamical Graph Grammar Modeling Library (DGGML) and a DGG model for the periclinal face of the plant cell CMA. This model explores the effects of face shape and boundary conditions on local and global alignment. For a rectangular face, different boundary conditions reorient the array between the long and short axes. The periclinal CMA DGG demonstrates the flexibility and utility of DGGML, and these new methods highlight DGGs' potential for testing, screening, or generating hypotheses to explain emergent phenomena.