Why self-organize to criticality

Research summary

One paper, summarised for reference. Why collective behaviours self-organize to criticality: a primer on information-theoretic and thermodynamic utility measures — Qianyang Chen & Mikhail Prokopenko, R. Soc. Open Sci. (2025) 12, 241655. doi:10.1098/rsos.241655.

The question: why, not how

Self-organized criticality (SOC) explains how a system can end up at a critical point (a slow driving force balanced by fast dissipation), but not why being there benefits the collective. This primer asks the complementary question: is there an intrinsic utility — a task-independent benefit a system gains from its own organization — that is optimized at criticality? It compares four candidate utilities on one common example.

The common example: an Ising perception–action loop

The 2D Ising model is read as a perception–action loop: each site is an agent that senses its neighbourhood energy (S), acts by flipping or holding its spin (A), against a world = the lattice magnetization (W); the coupling strength $J$ is its “agency/embodiment,” and the spin-flip rule (Glauber vs. Metropolis) is a variant of that embodiment. Computing each utility as a function of $J$ shows where each would drive the collective if $J$ evolved to optimize it.

Four utilities, four different optima

• Predictive information $\mathcal I = I(S_{\text{past}}; S_{\text{future}}) = H(S_{\text{fut}}) - H(S_{\text{fut}}\mid S_{\text{past}})$ (diversity minus unpredictability) — peaks at weak/sub-critical coupling (Metropolis) or near-critical (Glauber); driven by sensory diversity (exploration).
• Empowerment = the channel capacity from an agent’s actions to its future sensations — peaks at strong/super-critical coupling, where an aligned neighbourhood makes each action maximally perceivable.
• Active inference / variational free energy (intrinsic part) = minimize surprise between local model and global state — also super-critical (alignment lets local sensing predict the global spin).
• Thermodynamic efficiency $\eta = \dfrac{-\,d\mathbb S/d\theta}{d\langle\beta\mathbb W_{\text{gen}}\rangle/d\theta}$ — the entropy reduction (predictability gained) per unit generalized work (equivalently, per integrated Fisher information) — peaks at the critical regime ($J_c\approx0.4407$), the same for both dynamics.

The punchline: only the measure that weighs informational benefit against energy cost — thermodynamic efficiency — is optimized at criticality; the purely informational utilities are maximized to one side or the other. Analytically, $\eta$ diverges at the critical point, $\eta \propto |J - J_c|^{-1}$ (and $\propto|\theta-\theta_c|^{-1}$ for the Curie–Weiss model), with the finite-size peak approaching $J_c$ as the lattice grows.

The Principle of Super-efficiency

They propose: self-organizing collective systems maximize the thermodynamic efficiency of interactions — maximal predictability of collective behaviour per unit of expended work — and this efficiency is maximal at the critical regime. Read either way: for a fixed energy budget, criticality maximizes the predictability gain; for a required predictability gain, it minimizes the energy cost. The authors marshal support from prior thermodynamic-efficiency studies (self-propelled particles, urban transformation, epidemic contagion — all peaking at their transitions) and empirical systems (starling flocks combining scale-free correlations with metabolic savings in formation; ant colonies with order transitions and sublinear energy scaling; energy-constrained brain-network formation).

Why it’s collected here

Two hooks for this lab. First, the Ising-as-perception-action-loop framing (each unit an embodied sense→act agent) is the same move as treating a reservoir-in-a-body as an agent, and it makes “coupling strength = agency” explicit. Second, it puts efficiency — benefit per unit cost — at the centre of why criticality, complementing Chis-Ciure & Levin’s search-efficiency view of intelligence and the free-energy thread, and offering a concrete, computable quantity ($\eta$, or the branching/Fisher-info machinery behind it) rather than another phenomenological signature.

Caveat

The analysis is at equilibrium; real biological systems are driven far from it (where, e.g., entropic forces can tear a flock apart absent a cohesive counterforce). Extending super-efficiency to non-equilibrium settings is the flagged open direction.

References

Chen & Prokopenko (2025), R. Soc. Open Sci. 12, 241655 · Nigmatullin & Prokopenko (2021), Entropy 23, 757 · Crosato et al. (2018), Phys. Rev. E 97, 012120 · Bak, Tang & Wiesenfeld (1987), Phys. Rev. Lett. 59, 381 · Friston (2010), Nat. Rev. Neurosci. 11, 127 · Klyubin, Polani & Nehaniv (2005), IEEE CEC (empowerment) · Ay et al. (2008), Eur. Phys. J. B 63, 329 (predictive information) · Onsager (1944), Phys. Rev. 65, 117 · Mora & Bialek (2011), J. Stat. Phys. 144, 268 · Cavagna et al. (2010), PNAS 107, 11865.