Mean-preserving spreads of node output biases suppress damage spreading in random Boolean networks

Random Boolean networks (RBNs) exhibit an order–chaos transition in which small perturbations either die out or spread across the network. Here we study this transition for the classical random truth-table ensemble when node output biases are heterogeneous. The perturbation-propagation phenomenon is summarized by the Derrida map and its slope at the origin (the Derrida slope). Treating node-dependent biases as quenched disorder in local sensitivities, we obtain under the classical annealed approximation λ≃1N∑iKi2pi(1−pi), where Ki is the in-degree of node i. Because the sensitivity kernel s(p)=2p(1−p) is concave on [0,1], any mean-preserving spread of the bias distribution lowers the mean sensitivity and therefore suppresses damage spreading at fixed μ=E[p]. For homogeneous in-degree K, this becomes the explicit variance law λ=2K{μ(1−μ)−Var[p]}. We emphasize that the algebra is simple; the contribution is instead to expose bias heterogeneity as a controllable stability axis, formalize the monotonicity by convex order, and verify the trend in quenched simulations. The numerical results quantify finite-size fluctuations, show robustness to structured directed topologies, and isolate the covariance correction induced by mild K–p correlations. Thus, within the present ensemble, one can stabilize perturbation propagation without changing either the mean activity or the mean degree—simply by broadening the bias distribution in a mean-preserving way.