脉冲图：有向图上的素数激活布尔动力学
Pulse Graphs: Prime-Activated Boolean Dynamics on Directed Graphs
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中文总结 AI 辅助
研究有限无环有向图上素数激活的布尔动力学即脉冲图，通过特定方法确定最大吸引子周期\(L(n)\)的指数阶，并对完全有向图和稀疏随机有向图进行分析，推导相关公式、分类吸引子及得出激活概率等。
中文摘要 AI 辅助
我们研究有限无环有向图上的同步布尔动力学，其中一个顶点在下一个时间步恰好当其入邻接点的活跃数量为素数时才活跃，我们称这些系统为脉冲图。用\(L(n)\)表示在\(n\)个顶点上可实现的最大吸引子周期。穷举得出\(L(1),\ldots,L(5)=1,1,1,3,9\)。主要结果确定了最大周期的指数阶：\(2^{n - 3}-1\leq L(n)\leq2^n - 1\)（\(n\geq5\)）。通过使用素数计数逻辑门实现最大长度仿射反馈寄存器获得下界。对于\(n\geq6\)，构造无环，最大入度为五，仅使用\(O(n)\)条边。对于完全有向图，推导了精确更新公式，将所有吸引子分类为不动点或互补二周期，证明每个轨道在三次更新内到达其最终吸引子并明确计数吸引子。还推导了独立随机输入下的激活概率。对于稀疏随机有向图，相关的素数 - 泊松平均场映射在\(c_*\approx3.824963\)，\(\rho_*\approx0.368241\)处经历非退化折叠，阈值以上立即出现局部双稳性。
英文摘要
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: \[ 2^{n-3}-1\leq L(n)\leq2^n-n-1 \qquad(n\geq5). \] 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 odd $n\geq7$, a period-3 control module improves the lower bound to \[ 3(2^{n-4}-1). \] 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.
