Critical Line in Random Threshold Networks with Inhomogeneous Thresholds

We calculate analytically the critical connectivity $K_c$ of Random Threshold Networks (RTN) for homogeneous and inhomogeneous thresholds, and confirm the results by numerical simulations. We find a super-linear increase of $K_c$ with the (average) absolute threshold $|h|$, which approaches $K_c(|h|) \sim h^2/(2\ln{|h|})$ for large $|h|$, and show that this asymptotic scaling is universal for RTN with Poissonian distributed connectivity and threshold distributions with a variance that grows slower than $h^2$. Interestingly, we find that inhomogeneous distribution of thresholds leads to increased propagation of perturbations for sparsely connected networks, while for densely connected networks damage is reduced; the cross-over point yields a novel, characteristic connectivity $K_d$, that has no counterpart in Boolean networks. Last, local correlations between node thresholds and in-degree are introduced. Here, numerical simulations show that even weak (anti-)correlations can lead to a transition from ordered to chaotic dynamics, and vice versa. It is shown that the naive mean-field assumption typical for the annealed approximation leads to false predictions in this case, since correlations between thresholds and out-degree that emerge as a side-effect strongly modify damage propagation behavior.

💡 Research Summary

The paper presents a comprehensive analytical and numerical study of Random Threshold Networks (RTNs), focusing on the critical connectivity Kc that separates ordered from chaotic dynamics. An RTN consists of N binary nodes (spins σi = ±1) updated synchronously by a sign function of a weighted sum of inputs plus a node‑specific threshold hi (hi ≤ 0). Interaction weights cij are drawn from {−1, 0, +1} with equal probability, and the in‑degree distribution is assumed Poissonian with average (\bar K) (sparse regime (\bar K\ll N)).

First, the authors derive the exact probability ps(k,|h|) that a node with k inputs changes its state when one of its inputs is flipped, for the case of homogeneous thresholds (hi ≡ h). Using combinatorial arguments they obtain closed‑form expressions (Eqs. 6 and 8) that display an oscillatory dependence on k−|h| and asymptotically decay as ps ∼ k⁻¹ᐟ² for large k. Averaging over the Poisson degree distribution yields the mean damage propagation probability (\langle p_s\rangle). Within the annealed approximation the expected damage after one update step is (\bar d(t+1)=\langle p_s\rangle \bar K). The critical line is defined by (\bar d=1). Solving this condition numerically for arbitrary h gives Kc(|h|).

A key result is that Kc grows super‑linearly with the absolute threshold |h|. For large |h| the authors show analytically that
\