Criticality in collective behavior

The thesis

A review of the criticality hypothesis for animal groups — that collectives processing information in a distributed way should sit near a (pseudo-)critical point separating qualitatively different aggregate behaviours, where susceptibility and information transmission are maximal. Its central move is the emerging, more nuanced view: de-emphasise being exactly at a critical point, and instead ask what is gained by a system tuning an optimal distance from criticality to manage competing trade-offs.

Applying phase-transition physics to biology (the caveats)

Far-from-equilibrium. Animal groups are non-equilibrium; the relevant theory is that of non-equilibrium phase transitions.
Finite $N$ . Groups are $N\sim10$ – $10^3$ (rarely $10^6$ ), not the thermodynamic $N\to\infty$ in which transitions are rigorously defined. Quasi-critical benefits (correlation length, transmission, susceptibility maxima) still peak in finite systems — but universality is tied to $N\to\infty$ , so sorting biological transitions into universality classes may be impractical, and empirical scaling laws in small groups must be read cautiously.
Boundaries dominate. With small $N$ , boundary conditions are non-negligible — biologically a feature (edge individuals gather most of the environmental information).

Four transitions in collective behaviour

Flocking — orientational order by spontaneous symmetry breaking (Vicsek’s self-propelled particles; a non-equilibrium XY-model whose long-range order evades Mermin–Wagner via Toner–Tu). Once thought continuous, now understood to be discontinuous through a density–order coupling that forms high-density bands; topological (k-nearest-neighbour) and variable-speed variants refine this (scale-free speed correlations require variable-speed models at criticality).
Collective decision-making — majority commitment as symmetry breaking (Ising/Potts, quorum/threshold rules), usually on well-mixed or network topologies. Criticality is under-discussed here (often framed as bifurcations), but the speed–accuracy trade-off is modulated by distance from the critical point.
Behavioural contagion — cascades as percolation / epidemic spreading, not symmetry breaking. Simple (pairwise) vs. complex (higher-order/threshold) contagion; the Dodds–Watts model gives continuous and discontinuous onsets depending on the threshold distribution.
Synchronization — the Kuramoto model’s continuous transition from incoherence to a common frequency, with the mean-field amplitude as order parameter.

Quantifying distance from criticality

Susceptibility — peak in the order parameter’s sensitivity to perturbation defines an effective critical point in a finite system; Fisher information generalises this; correlation length in spatial cases.
For cascades — a local-amplification / reproduction number $R_0$ (with $R_0=1$ critical in the infinite limit), and cascade-size distributions that go power-law at the transition (deviation from a power law as a criticality indicator, as with neural avalanches).
Discontinuous transitions — detected via bistability / hysteresis.
The payoff: near a continuous transition, collective behaviour is dominated by the distance from the transition, simplifying how individual-scale parameters must be tuned.

Function near a transition — the trade-off view

Far from a transition, collective behaviour is “boring” (weak coupling = sum of individuals; strong coupling = unresponsive/stuck); only near it can individual interactions shape the macroscopic state (the “edge of chaos”). But these collective effects are not always beneficial, motivating a zoom-in:

Maximal susceptibility helps when shared individual information is accurate and aligned, but hurts when it is noisy or motives conflict — so near-but-not-at the transition, tunable by context (fish startle spread depends on predation threat — Poel et al.’s subcritical escape waves; macaque conflict — Daniels et al.).
Critical slowing down manufactures long timescales from short-memory components (integrating noisy evidence; the speed–accuracy trade-off).
Spontaneous symmetry breaking yields consensus among equal options, but staying at the transition means large slow fluctuations — committing may require moving beyond it.
Hysteresis (discontinuous) gives switch-like commitment (a bee swarm locking onto one nest site), costly when it traps the group in a suboptimal state.

The crux: self-organization toward criticality

The hard problem: a phase transition is a macroscopic, group-level property, but adaptation is done by individuals with only local information — how can locals tune the global distance to criticality? A neuroscience mechanism (Bornholdt–Rohlf: too-silent/too-active nodes adjust their thresholds from local time-averages) relies on ergodicity and time-scale separation, both likely violated in animal groups. Candidate macroscopic knobs: group size $N$ (finite-size scaling locates the susceptibility peak — midge swarms may tune via $N$ ), density (the prime knob for fish escape tuning via perceived risk, with individual-threshold adaptation negligible there), and composition/heterogeneity (which can open Griffiths phases — extended quasi-critical regions).

Evolution runs into a social dilemma. Reaching criticality by individual-level selection requires that it be beneficial across contexts and that the group optimum coincide with the individual ESS — but group optimum ≠ individual ESS is the rule, and multi-level selection cannot generally rescue it. Hidalgo et al.’s agents evolve to criticality in variable environments; but Klamser & Romanczuk find the flocking critical point maximally evolutionarily unstable in spatial predator-prey, because it is exactly where collective dynamics is most sensitive to individual heterogeneity — steep fitness gradients push evolution away into the ordered phase.

Open challenges

Modern tracking enables real tests, but the inverse problem (inferring interaction models from data) is hard, with spurious criticality a known pitfall (Zipf’s law without fine-tuning). The fundamental open questions are the distributed self-tuning mechanisms and the evolutionary social dilemma — and even negative results (failing to find, or disproving, criticality) sharpen the understanding of collective behaviour, provided physics is kept in dialogue with evolutionary biology.

References

Romanczuk & Daniels (2022), arXiv:2211.03879 · Muñoz (2018), Rev. Mod. Phys. 90, 031001 · Vicsek et al. (1995), Phys. Rev. Lett. 75, 1226 · Toner & Tu (1995), Phys. Rev. Lett. 75, 4326 · Poel et al. (2022), Sci. Adv. 8, eabm6385 · Klamser & Romanczuk (2021), PLoS Comput. Biol. 17, e1008832 · Daniels, Krakauer & Flack (2017), Nat. Commun. 8, 14301 · Hidalgo et al. (2014), PNAS 111, 10095 · Mora & Bialek (2011), J. Stat. Phys. 144, 268 · Schwab, Nemenman & Mehta (2014), Phys. Rev. Lett. 113, 068102.