Swarm criticality & transmission

The claim

A swarm behaves “as a single cognitive mind” — a small informed subset (lookouts sensing a predator or resource) steering the whole flock — as an effect located at a critical point, from the joint action of two ingredients:

Frequent organizational collapses that momentarily make each bird independent of the others, opening the short “free-will” window in which the informed few can set a new direction;
a correlation length as extended as the flock size, so that choice, once made, is felt everywhere (a violation of locality).

The information the lookouts hold is transmitted to the whole swarm with a delay set by the time between consecutive collapses. The argument is carried by two models — a Vicsek flock (used to establish temporal complexity) and a decision-making lattice (used to establish transmission).

Temporal complexity

Collapses are treated as renewal events: the probability of the next collapse falling in $[\tau, \tau+d\tau]$ is $dp = \psi(\tau)\,d\tau$ , with the generic critical-state waiting-time density

\psi(\tau) = \frac{\mu-1}{T}\,\frac{1}{\left(1+\tau/T\right)^{\mu}},

where $T$ is the minimal recovery time. The mean inter-collapse time,

\langle\tau\rangle = \frac{T}{\mu-2}\quad(\mu>2),

is finite for $\mu>2$ but diverges for $\mu<2$ — the non-ergodic, “ideal temporal complexity” condition, conjectured to be a general property of physiological systems and important for transfer of information between complex systems. At criticality the tail is a pure power law, $\psi(\tau)\sim\tau^{-\mu}$ , with $\mu \approx 1.35$ .

Model 1 — the Vicsek flock

Each bird adopts the average heading of neighbours within an interaction radius $r$ , plus white noise of intensity $\eta$ (the knob tuned to criticality). Cooperation is read off the global speed / polarization order parameter

\Phi_a(t) = \frac{1}{N v_0}\left|\sum_{k=1}^{N} v_0\, e^{i\theta_k(t)}\right|.

A collapse is registered when $\Phi_a(t)$ drops below an (arbitrary) threshold — $\Phi_a=0$ never actually occurs, but near-vanishing marks the organized→disorganized transition. The survival function $\Psi(t)=\int_t^\infty \psi(\tau')\,d\tau'$ recovers $\mu\approx 1.35$ at criticality.

1D version: abrupt left/right flips, as in migrating locust bands (Buhl 2006); each flip is an unambiguous collapse with negligible reorganization time, and the same $\mu\approx 1.35$ recurs — suggesting universality.
Finite size: the power law is truncated by an exponential drop at long times, and the truncation moves later as the swarm grows. Both the correlation length and $\langle\tau\rangle$ scale with the linear size $L$ :

\frac{\langle\tau_1\rangle}{\langle\tau_2\rangle} = \frac{L_1}{L_2}.

Model 2 — the decision-making lattice (transmission)

Each bird is a node on an $L\times L$ periodic lattice with a binary choice $s=\pm 1$ and base decision rate $g$ . With cooperation strength $K$ , each flip is biased toward the majority of its four neighbours ( $p_{2\to1}=g\,e^{K[M(1)-M(2)]/M}$ , $M=4$ ); $K>0$ lengthens the mean state duration, which grows exponentially at criticality. A small corner cluster of lookouts is driven by a regular alternating “environmental” signal, and its transmission across the lattice is tracked.

Sub-critical $K$ : the swarm ignores the lookouts (or does not depart from its unperturbed behaviour at all). Super-critical $K$ : the swarm is too rigid to re-synchronise. At criticality ( $K_c\approx 1.61$ ): the whole swarm tracks the lookouts.
Information view: the lookout motion is a transmitted bit sequence $t(n)\in\{-1,+1\}$ ; a distant receiver time-averages a node’s signal into $b(n)$ over windows $\Delta t$ and takes $r(n)=\mathrm{sgn}[b(n)]$ . The lookout–node correlation

C = \frac{1}{B}\sum_{n=1}^{B} r(n)\,t(n)

is bell-shaped in $K$ , maximal at criticality, and $\to 1$ (perfect, and independent of distance from the lookouts) for large $\Delta t$ . There is no direct transmitter–receiver link: the message rides the criticality-induced long-range correlation, with a delay equal to the inter-collapse time.

The cost of size

Because $\langle\tau\rangle\sim L$ , the transmission delay grows with the swarm — larger flocks transmit more slowly (less efficiently). The mechanism is criticality-induced nonlocality rather than travelling waves; the authors flag communication-algorithm / engineering implications and the link to intelligence-as-criticality (Chialvo 2010).

References

Vanni, Lukovic & Grigolini (2011), Phys. Rev. Lett. 107, 078103 · Vicsek et al. (1995), Phys. Rev. Lett. 75, 1226 · Cavagna et al. (2010), PNAS 107, 11865 · Turalska et al. (2009), Phys. Rev. E 80, 021110; Turalska, West & Grigolini (2011), Phys. Rev. E 83, 061142 · Buhl et al. (2006), Science 312, 1402 · Couzin (2007), Nature 445, 715 · Chialvo (2010), Nat. Phys. 6, 744 · Scher & Montroll (1975), Phys. Rev. B 12, 2455.