Prime-Graph with Boolean Dynamics on Directed Networks: Periodic Orbits, Attractor Structure, and Mean-Field graphical Analysis

The prime-count update rule produces markedly different behaviors depending on the underlying graph structure. Exhaustive enumeration shows that nontrivial periodic at-tractors first occur on four vertices, while a period-9 attractor already appears on five vertices. These examples demonstrate that a simple arithmetic activation rule can gen-erate rich finite-state dynamics. On general directed graphs, Pulse Graphs support information propagation, Boolean logic, and feedback mechanisms. The rotating-register construction gives attractors of linear length, while the affine feedback construction yields exponential periods of order λ(2n). Thus the dynamical complexity of Pulse Graphs grows exponentially with the number of vertices. In contrast, the complete directed graph exhibits highly constrained behavior. Every trajectory reaches either a fixed point or a complement two-cycle after at most three updates. This shows that the complexity of Pulse Graphs depends not only on the prime activation rule but also on the underlying network topology.  

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