#P-completeness of counting update digraphs, cacti, and a series-parallel decomposition method

Abstract

Automata networks are a very general model of interacting entities, with applications to biological phenomena such as gene regulation. In many contexts, the order in which entities update their state is unknown, and the dynamics may be very sensitive to changes in this schedule of updates. Since the works of Aracena et. al, it is known that update digraphs are pertinent objects to study non-equivalent block-sequential update schedules. We prove that counting the number of equivalence classes, that is a tight upper bound on the synchronism sensitivity of a given network, is #P-complete. The problem is nevertheless computable in quasi-quadratic time for oriented cacti, and for oriented series-parallel graphs thanks to a decomposition method.

Optimising attractor computation in Boolean automata networks

“…The dynamics of a BAN is usually partitionned in two sorts of configurations: the recurring ones that are parts of attractors and either belong to a limit cycle or are fixed points; the others that evolve towards these attractors and belong to their attraction basins. The questions of characterising, computing or counting those attractors from a simple description of the network have been explored [8,1,9,7,14,2], and has been shown to be difficult problems [8,16,3,4,15]. In this paper, we propose a new method for computing the attractors of a BAN under the parallel update schedule.…”