Criticality Without Fine-Tuning
Self-organizing criticality is often treated as a fragile phenomenon requiring precise tuning of a control parameter. We show that in sufficiently heterogeneous networks, critical-like scaling emerges robustly across a finite region of parameter space — and propose a practical detector that does not require direct access to the underlying dynamics.
Motivation
Power-law signatures in natural data are routinely attributed to self-organized criticality. The usual objection is that criticality sits on a knife-edge; only delicate tuning can keep a system there. We test whether that intuition survives in heterogeneous, high-dimensional networks.
Setup
We sweep three classes of systems — neuronal cultures, simulated regulatory networks, and an engineered Boolean ensemble — over a 7-dimensional parameter grid. For each, we estimate the branching parameter ρ and the avalanche exponents τ, σ.
Result
All three classes exhibit a finite *critical region* — not a point — where avalanches obey scale-free statistics. The width of the region grows linearly with the log of network heterogeneity.
Implication
If criticality in nature is heterogeneous by default, we may have been hunting for the wrong thing. The detectable signature is not a single exponent but a *band* of exponents, with width predicted by the heterogeneity of the substrate.