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PRL 107, 078103 (2011)
PHYSICAL REVIEW LETTERS
week ending
12 AUGUST 2011Criticality and Transmission of Information in a Swarm of Cooperative Units
Fabio Vanni,1 Mirko Lukovid,2 and Paolo Grigolini1
'Center for Nonlinear Science, University of North Texas, PO. Box 311427, Denton, Texas, 76203, USA
2Max Planck Institute for Dynamics and Self-Organization, 37073 Gottingen, Germany
(Received 19 May 2011; published 12 August 2011)
We show that the intelligence of a swarm of cooperative units (birds) emerges at criticality, as an effect
of the joint action of frequent organizational collapses and of spatial correlation as extended as the flock
size. The organizational collapses make the birds become independent of one another, thereby allowing
the flock to follow the direction of the lookout birds. Long-range correlation violates the principle of
locality, making the lookout birds transmit information on either danger or resources with a time delay
determined by the time distance between two consecutive collapses.DOI: 10 1103/PhysRevLett 107 078103
In the recent few years, there has been intense activity to
explain why a swarm of birds behaves as a single individ
ual [1,2]. How is it possible that when a predator comes,
the swarm changes direction to escape from danger? How
is it possible that a subset of a swarm becoming aware of
the right direction toward a resource [3] convinces the
whole flock to pursue that specific course? In which sense,
using a metaphor made popular by Couzin [4], can we
interpret the swarm as a cognitive mind?
The main purpose of this Letter is to prove that this form
of intelligence is the effect of the joint action of frequent
organizational collapses, allowing the single birds to re
cover independence of the others, and of a correlation
length as extended as the flock size. Although the environ
ment perceiving units are a small fraction of the total
number of units, they exert a determinant action on the
swarm during the short rearrangement phase after an or
ganizational collapse, which makes each unit free to select
a new direction. This freewill condition allows the swarm
to select the new flying directions that are transmitted to all
the units by the few danger- or resource perceiving birds,
thanks to the criticality induced long range correlation.
The connection between dynamic instability and informa
tion transfer has been recently observed in locust nymphs
[3]. The results of this Letter confirm the importance of this
observation, establishing at the same time that the infor
mation transfer is made possible by the nonlocal nature of
the criticality condition, with a time delay depending on
the time distance 7 between two consecutive organiza
tional collapses. The correlation length between birds be
comes as extended as the finite swarm size, thereby
allowing the lookout birds to transmit their flying direction
to the whole swarm. Thus, the mean value (7), proportional
to the correlation length, remains finite.
To afford a convincing proof that the swarm's intelli
gence is determined by the joint action of organizational
collapses and criticality induced nonlocality, we proceed
in two main steps. In the first step we use a model of bird
organization, referred to in this Letter as the bird model, toPACS numbers: 87 23 Cc, 6460 fd, 8970 a, 8975 He
illustrate the concept of temporal complexity. The second
step is based on a simpler model, where the relative posi
tions of the birds are fixed and they have only to choose
between either flying to the right or the left.
We use the occurrence of organizational collapses to
define temporal complexity [5] as follows. Let us set the
origin of time at the moment of a failure occurrence, and let
us consider the probability that another failure occurs in the
small time interval [7, 7 + d r]:
dp = 0(7)dr. (1)
Temporal complexity [5] is established by examining the
asymptotic time limit of (7), which is expected [6] to be
the same as that of
()= ( 1) (2)
Note that the parameter T is the minimal recovery time and
that Eq. (2) is a generic expression for a system in a critical
state. The mean time distance between two consecutive
collapses is given by
(7) 2) (3)
for [t > 2. The ideal condition of temporal complexity
corresponds to t < 2, making (7) diverge, thereby gener
ating the nonergodic condition [7]. There are reasons to
believe that this condition may be a general property of
physiological systems, playing an important role for the
transfer of information from one to another complex sys
tem [8,9].
To prove that a flock of birds at criticality generates
temporal complexity, let us adopt the algorithm proposed
by Vicsek and co workers [10]. This model gives the
prescription for each single bird to take a direction equal
to the average of its nearest neighbors' orientations, inside
a given circle of interaction of radius r, with some uncer
tainty represented by a white noise of intensity i. The
signature of cooperative behavior is given by the intensity
of the global speed, defined by2011 American Physical Society
0031-9007/11/107(7)/078103(4)
078103-1
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Vanni, Fabio; Lukovic, Mirko & Grigolini, Paolo. Criticality and Transmission of Information in a Swarm of Cooperative Units, article, August 12, 2011; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc40392/m1/1/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.