Connectivity-dependent effects of depth-1 canalisation on attractor dynamics in small-world Boolean networks

Boolean networks provide a framework for studying how network structure and local rules shape long-term dynamics. We studied synchronously updated, directed, signed Watts–Strogatz Boolean networks with depth-1 canalising threshold rules, using 825,000 graph–rule replicates across five controlled parameters. Connectivity reversed the adjusted period effect of full canalisation relative to threshold-only rules. At network size N = 100, full canalisation increased the adjusted geometric mean period by 22.1–25.8% at neighbourhood parameter κ = 1, but shortened it at κ = 2–5 across all five inhibitoryedge probabilities, with contrasts averaged equally over simulated rewiring probabilities. The magnitude depended on inhibitory-edge probability and rewiring. Increasing rewiring from p = 0.01 to p = 0.60 shortened adjusted periods after averaging over other simulated settings. Full canalisation lowered, rather than increased, the fixed-point rate. A separate design of 7,200 fixed Boolean systems at inhibitory-edge probabilities 0.30 and 0.50 examined sampled basins and one-bit perturbations. Full canalisation reduced the number of attractors reached from sampled initial states, increased return probability within 2000 updates, and reduced restricted mean return time and mean Hamming fraction after 25 updates. These findings identify a connectivity-dependent period response within the tested model and sampling design.