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

This paper presents a thorough analytical and numerical investigation of the critical connectivity K_c in Random Threshold Networks (RTNs), extending the classic annealed‑approximation framework to account for both homogeneous and heterogeneous node thresholds, as well as correlations between thresholds and node degree. The authors begin by revisiting the standard RTN model: each node i updates its binary state σ_i(t+1) = sign(∑j w{ij}σ_j(t) – h_i), where w_{ij}=±1 are random couplings and h_i is the activation threshold. Under the annealed approximation, the probability that a single-bit perturbation spreads to a neighbor (damage propagation probability p_d) can be expressed in terms of the distribution of the weighted input sum. For homogeneous thresholds (h_i = h for all i) the authors derive an exact expression for p_d and impose the criticality condition K·p_d = 1, obtaining a closed‑form relationship K_c(|h|). They show that K_c grows super‑linearly with the absolute threshold and, using asymptotic expansion for large |h|, arrive at the universal scaling law

  K_c(|h|) ≃ h² / (2 ln|h|).

This result holds for any Poissonian degree distribution and for any threshold distribution whose variance grows slower than h², establishing a broad universality class for RTNs.

The analysis is then generalized to heterogeneous thresholds. By allowing the thresholds to be drawn from a distribution with mean ⟨|h|⟩ and variance σ_h², the authors recompute p_d and find that the mean threshold still dominates K_c, while the variance introduces a subtle correction that becomes significant only when the network is sparsely connected. Numerical simulations confirm that for low average degree K the presence of threshold heterogeneity actually enhances damage spreading, lowering the effective K_c, whereas for dense networks the same heterogeneity suppresses damage, raising K_c. The crossover between these regimes defines a novel characteristic connectivity K_d, which has no analogue in traditional Boolean network theory. K_d depends on both ⟨|h|⟩ and σ_h² and marks the point where the sign of the heterogeneity effect flips.

A further contribution of the paper is the systematic study of local correlations between node thresholds and in‑/out‑degree. The authors construct ensembles where high‑threshold nodes preferentially have low degree (negative correlation) or high degree (positive correlation). Even weak anti‑correlations are shown to shift the system from an ordered to a chaotic phase, and conversely, weak positive correlations can restore order in otherwise chaotic regimes. This phenomenon arises because the annealed approximation implicitly assumes independence between thresholds and degree; when this assumption is violated, the effective out‑degree distribution of perturbed nodes changes, altering the branching factor that governs damage propagation. Consequently, naïve mean‑field predictions become quantitatively and qualitatively incorrect.

All analytical predictions are validated by extensive Monte‑Carlo simulations. The authors sample Poissonian networks with average degrees ranging from 1 to 10, use several threshold distributions (Gaussian, uniform, exponential), and measure the Hamming distance evolution after a single‑bit flip. The simulated K_c values collapse onto the theoretical curves, and the asymptotic h²/(2 ln h) scaling is clearly observed for |h|≫1. Moreover, the simulated K_d matches the analytically derived crossover point, confirming the robustness of the new characteristic connectivity.

In summary, the paper makes three major advances: (1) it provides an exact analytical expression for the critical connectivity of RTNs with arbitrary (but not too broad) threshold distributions, revealing a universal h²/(2 ln h) scaling; (2) it uncovers the dual role of threshold heterogeneity, introducing the novel crossover connectivity K_d that separates regimes of enhanced versus suppressed damage spreading; and (3) it demonstrates that even modest correlations between thresholds and node degree profoundly affect network dynamics, invalidating the standard annealed‑approximation assumption of independence. These insights deepen our understanding of how microscopic node properties shape macroscopic dynamical phases in threshold‑type complex systems, with potential implications for neural networks, gene‑regulatory circuits, and engineered Boolean‑logic hardware where threshold variability and degree‑threshold correlations are unavoidable.