🤖 AI Summary
This study investigates synchronous Boolean dynamical systems on finite acyclic directed graphs governed by a prime activation rule—where a node becomes active at the next time step if and only if the number of its active in-neighbors is prime. Introducing prime-counting logic into Boolean networks for the first time, the work combines combinatorial constructions, maximal-length affine feedback shift registers, mean-field approximations, and a prime–Poisson mapping to characterize system dynamics. Key contributions include establishing that the maximum attractor period \(L(n)\) for an \(n\)-node system satisfies \(2^{n-3}-1 \leq L(n) \leq 2^n-1\), deriving exact update rules and attractor classifications for complete graphs, and uncovering a non-degenerate fold bifurcation in sparse random graphs at a critical average degree \(c^* \approx 3.825\), indicative of a pronounced phase transition.
📝 Abstract
We study synchronous Boolean dynamics on finite loopless directed graphs in which a vertex is active at the next time step exactly when its number of active in-neighbors is prime. We call these systems Pulse Graphs. Let $L(n)$ denote the largest attractor period realizable on $n$ vertices. Exhaustive enumeration gives \[ L(1),\ldots,L(5)=1,1,1,3,9. \] Our main result determines the exponential order of the maximum period: \[ 2^{n-3}-1\leq L(n)\leq2^n-1 \qquad(n\geq5). \] The lower bound is obtained by implementing a maximal-length affine feedback register using prime-count logic gates. For $n\geq6$, the construction is loopless, has maximum in-degree five, and uses only $O(n)$ edges.
For complete directed graphs, we derive an exact update formula, classify all attractors as fixed points or complement two-cycles, prove that every orbit reaches its eventual attractor within three updates, and count the attractors explicitly. We also derive the activation probability under independent random inputs. For sparse random directed graphs, the associated prime-Poisson mean-field map undergoes a nondegenerate fold at \[ c_\ast\approx3.824963, \qquad ρ_\ast\approx0.368241, \] with local bistability immediately above the threshold.