PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
Signa u es o c i icali y in u ning a alanches o schooling ish
And eu Puy ,1,*Elisabe Gimeno ,1,2Da id Ma ch-Pons ,1,2M. Ca men Miguel ,2,3and Romualdo Pas o -Sa o as 1
1Depa amen de Física, Uni e si a Poli ècnica de Ca alunya, Campus No d B4, 08034 Ba celona, Spain
2Depa amen de Física de la Ma è ia Condensada, Uni e si a de Ba celona, Ma í i F anquès 1, 08028 Ba celona, Spain
3Ins i u e o Complex Sys em (UBICS), Uni e si a de Ba celona, 08028 Ba celona, Spain
(Recei ed 29 Sep embe 2023; accep ed 27 July 2024; published 9 Sep embe 2024)
Mo ing animal g oups ansmi in o ma ion h ough p opaga ing wa es o beha io al cascades, exhibi ing
cha ac e is ics akin o sys ems nea a c i ical poin om s a is ical physics. Using da a om eely swimming
schooling ish in an expe imen al ank, we in es iga e spon aneous beha io al cascades in ol ing u ning
a alanches, whe e la ge di ec ional shi s p opaga e ac oss he g oup. We analyze se e al a alanche me ics
and p o ide a de ailed pic u e o he dynamics associa ed wi h u ning a alanches, employing ools om
a alanche beha io in condensed-ma e physics and seismology. Ou esul s iden i y powe -law dis ibu ions
and obus scale- ee beha io h ough da a collapses and scaling ela ionships, con i ming a necessa y condi ion
o c i icali y in ish schools. We explo e he biological unc ion o u ning a alanches and link hem o collec i e
decision-making p ocesses in selec ing a new mo emen di ec ion o he school. We epo ele an bounda y
e ec s a ising om in e ac ions wi h he ank walls and in luen ial oles o bounda y indi iduals. Finally, spa ial
and empo al co ela ions in a alanches a e explo ed using he concep o a e shocks om seismology, e ealing
clus e ing o a alanche e en s below a designa ed imescale and an Omo i law wi h a as e decay a e han
obse ed in ea hquakes.
DOI: 10.1103/PhysRe Resea ch.6.033270
I. INTRODUCTION
A ascina ing and con o e sial hypo hesis in biology is
ha some sys ems may ope a e close o a c i ical poin om
s a is ical physics, sepa a ing an o de ed s a e o he sys em
om a diso de ed one [1–3]. Biological sys ems a a c i ical
poin a e belie ed o possess unc ional ad an ages such as
op imali y in signal de ec ion, s o ing, and p ocessing; la ge
co ela ions in coo dina ed beha io ; and a wide spec um
o possible esponses [4–6]. C i icali y is o en associa ed
wi h scale in a iance, exempli ied by powe -law dis ibu ions
lacking ele an cha ac e is ic scales besides na u al cu o s
[1,2,7]. In pa icula , his is obse ed o sys ems exhibi ing
spa io empo al ac i i y in he o m o cascades o a alanches
wi h a iable du a ion and size, which a he c i ical poin a e
dis ibu ed as powe laws wi h anomalously la ge a iance.
The e has been e idence o c i icali y signa u es in many di -
e en biological sys ems, including neu al ac i i y and b ain
ne wo ks, gene egula o y ne wo ks, collec i e beha io o
cells, o collec i e mo ion [4,5,8].
The ield o collec i e mo ion, in pa icula , s udies he
g oup mo emen pa e ns exhibi ed by social o ganisms, such
as locks o bi ds, ish schools, insec swa ms, he ds o
*Con ac au ho : [email p o ec ed]
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he
dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)
and he published a icle’s i le, jou nal ci a ion, and DOI.
mammals, and human c owds [9,10]. In his con ex , ana-
ly ical and expe imen al s udies o mo ing animal g oups
sugges he exis ence o phase ansi ions be ween phases o
cohe en and incohe en mo ion [11–14]. Mo eo e , g oups
o animals can ansmi in o ma ion ac oss he g oup in he
o m o p opaga ing wa es o a alanches o beha io , as
occu s in ish schools [15–20], honeybees [21], bi d locks
[22–24], sheep he ds [25], o macaque monkeys [26]. Models
o collec i e mo ion ha e also ep oduced ea u es o hese
phenomena [27–30]. These beha io al cascades a e ypically
ep esen ed by beha io al shi s in he speed, accele a ion, o
heading o indi iduals and can a ise ei he spon aneously o
om esponses o en i onmen al cues, such as he p esence
o p eda o s, ood sou ces, o obs acles. F om a biological
poin o iew, hey can occu when indi iduals ollow he
beha io o o he s wi hou ega ding hei own in o ma ion
[31]. F om a physical pe spec i e, beha io al cascades can
show signa u es ypical om sys ems loca ed nea a c i ical
poin . Mainly, hese signa u es include la ge suscep ibili y
o sensi i i y o pe u ba ions [20,26,30,32,33], scale- ee
co ela ions [33,34], and possible indica ions o powe -law
beha io in he a alanche size dis ibu ion [16,19,20,25]. In
addi ion, he e is some e idence ha he s a e o c i icali y
can be egula ed by mo ing animal g oups depending on hei
needs [26,30,35], whe e he a alanche dynamics may ansi-
ion om being supe c i ical wi h local changes p opaga ing
h ough he en i e g oup o c i ical wi h changes p opaga ing
a all possible scales o he sys em o o subc i ical wi h
changes emaining local [36].
In his s udy we ocus on analyzing he p ope ies
o u ning a alanches in eely mo ing ish [19]. These
2643-1564/2024/6(3)/033270(13) 033270-1 Published by he Ame ican Physical Socie y
ANDREU PUY e al. PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
beha io al cascades in ol e he p opaga ion o la ge changes
in he heading di ec ion o indi iduals wi hin a g oup, o en
esul ing in a eo ien a ion o he g oup’s global ajec o y.
Speci ically, we examine spon aneous u ning a alanches o
schooling ish eely swimming in a ank. Ou in es iga ion
un eils scale- ee p ope ies in he s a is ical dis ibu ions o
di e en a alanche me ics and hei dependence on he num-
be o indi iduals in he school. Addi ionally, we in es iga e
he o igins o hese a alanches, analyzing hei igge ing wi h
espec o space, ime, and indi idual ini ia o s. Ou indings
e eal he ele ance o in e ac ions wi h ank walls and a
la ge in luence o bounda y indi iduals. We also explo e
he dynamical e olu ion o a alanches and i s ela ion wi h
he s a e o he school, as well as hei spa ial and empo al
co ela ions. Wi hin he limi s o ou expe imen al se up, ou
esul s s ongly sugges he p esence o a scale- ee a alanche
dynamics, which could be compa ible wi h he school ope a -
ing in he icini y o a c i ical poin .
II. AVALANCHE DEFINITION AND BASIC OBSERVABLES
Beha io al cascades in ish ha e been de ined measu ing
changes o di e en quan i ies. He e we ocus on a alanches
de ined in e ms o la ge changes in he heading o indi idu-
als, gi en by hei eloci y ec o [19]. As an expe imen al
subjec , we conside he mo ion o N=8, 16, 32, and 50
indi iduals o he species o black neon e a Hyphessob ycon
he be axel odi, a social ish ha ends o o m pola ized,
compac , and plana schools, eely swimming in an app oxi-
ma ely wo-dimensional expe imen al ank. We eco ded and
digi ized indi idual ish ajec o ies and calcula ed he co e-
sponding eloci ies and accele a ions ( e e o Appendix A
o expe imen al and da a acquisi ion de ails). In o de o
emo e he dependence on he expe imen al ame a e o he
eco dings, we measu e he changes in ime o he heading in
e ms o he u ning a e ω, de ined as he absolu e alue o
he angula eloci y, i.e.,
ω=|
×
a|
2,(1)
whe e
and
aa e he ins an aneous eloci y and accele a ion
o an indi idual, espec i ely, and is he modulus o he in-
s an aneous eloci y. (See Appendix B o a de i a ion o his
exp ession.) We conside he absolu e alue due o symme y
in he u ning di ec ion.
In Fig. 1(a) we show he p obabili y densi y unc ion (PDF)
o he u ning a e P(ω) o schools o di e en numbe s o
indi iduals N. He e and in he ollowing, we wo k in na u al
uni s o pixels and ames o dis ance and ime, espec i ely.
In addi ion, e o bands in he PDF plo s a e calcula ed om
he s anda d de ia ion o a Be noulli dis ibu ion wi h he
p obabili y gi en by he ac ion o coun s in each bin o
he nume ical PDF [37]. As we can see, schools o di e en
numbe s o indi iduals show essen ially he same beha io in
hei u ning a e dis ibu ions. Mos o he ime, he u ning
a e is e y small and uni o mly dis ibu ed, co esponding
o ish swimming locally in a s aigh ajec o y. In some
ins ances, howe e , la ge u ning a es can be obse ed, in
which indi iduals swi ly ea ange hei headings and hus
eo ien hei di ec ion o mo ion.
(a)
10−610−410−2100
ω
10−7
10−5
10−3
10−1
101
103
P(ω)
N=8
N=16
N=32
N=50
(b)
0.05 0.10 0.15 0.20 0.25 0.30
ω h
0.0
0.2
0.4
0.6
0.8
1.0
FIG. 1. (a) PDF o he u ning a e ωand (b) ac i i y a e
o u ning a alanches as a unc ion o he u ning h eshold ω h.
The di e en cu es co espond o expe imen al da a om schools
wi h di e en numbe s o indi iduals N. Quan i ies a e exp essed in
na u al uni s o ames and pixels.
Inspi ed by a alanche beha io in condensed-ma e
physics [38], we de ine a alanches by in oducing a u ning
h eshold ω h sepa a ing small u ns om la ge ones [19].
Conside ing an ac i e ish as one wi h a u ning a e ω>ω
h,
we in oduce he dynamical a iable n de ined as he numbe
o ac i e ish obse ed a ame . Then sequences o consec-
u i e ames in which n >0 (i.e., in which he e is a leas
one ac i e ish) de ine a u ning a alanche. In ideo S1 o he
Supplemen al Ma e ial [39] we show some examples o la ge
u ning a alanches o a school o N=50 ish.
The mos basic cha ac e iza ion o u ning a alanches is
gi en by he du a ion Tand size So a alanches and by
hei in e e en ime i. An a alanche s a ing a ame 0has
du a ion Ti he sequence o dynamic a iables n ul ills
n 0−1=0, n >0 o = 0,..., 0+T−1, and n 0+T=0.
The size So an a alanche is gi en by he o al numbe o
ac i e ish in he whole du a ion o he a alanche, i.e., S=
0+T−1
= 0n . The in e e en ime ibe ween wo consecu i e
a alanches is gi en by he numbe o ames be ween he end
o one a alanche and he s a o he nex one, ha is, by a
sequence ul illing n >0, n =0 o = +1,..., + i,
and n + i+1>0, whe e indica es he las ame o he i s
a alanche [40].
The e ec s o he u ning h eshold in a alanches can be
assessed wi h he ac i i y a e , de ined as he p obabili y
ha a andomly chosen ame belongs o an a alanche. We
033270-2
SIGNATURES OF CRITICALITY IN TURNING … PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
(a)
100101102
T
10−6
10−5
10−4
10−3
10−2
10−1
P(T)
N=8
N=16
N=32
N=50
(b)
100101102103
S
10−6
10−5
10−4
10−3
10−2
10−1
P(S)
(c)
100101102
T
100
101
102
103
ST
(d)
100101102
i
10−6
10−5
10−4
10−3
10−2
10−1
P( i)
FIG. 2. (a) PDF o he du a ion T, (b) PDF o he size S,(c)a -
e age size STas a unc ion o he du a ion T, and (d) PDF o
he in e e en ime i o ω h =0.1. The di e en cu es co espond
o schools o di e en numbe s o indi iduals N. The exponen s
om he g een dashed powe laws a e (a) α=2.4±0.2, (b) τ=
1.97 ±0.14, (c) m=1.41 ±0.06, and (d) γ=1.62 ±0.08.
compu e i as he a io be ween he numbe o ames wi h
ac i i y n >0 and he o al numbe o ames in he expe i-
men al se ies. As we can see om Fig. 1(b), o ixed N he
ac i i y a e dec eases wi h he u ning h eshold ω h, since
by inc easing ω h we a e dec easing he u ning a es ha we
conside la ge and we ind ewe ames wi h n >0. On he
o he hand, inc easing he numbe o indi iduals Na ixed ω h
esul s in an inc ease o he ac i i y a e. We can in e p e his
as a school wi h a la ge numbe o indi iduals has a highe
p obabili y o any o hem o display a la ge u ning a e.
Realis ic alues o ω h used o compu e a alanches a e
es ima ed o lie wi hin he ange ω h ∈[0.01,0.3]. Smalle
alues esul in in ini e a alanches ha span he en i e du a ion
o he expe imen , whe eas la ge alues p oduce e y ew
a alanches.
III. STATISTICAL DISTRIBUTIONS
In Figs. 2(a) and 2(b) we show he dis ibu ions o he
du a ion Tand size S, espec i ely, ob ained o a ixed u n-
ing h eshold ω h =0.1 and o schools o di e en numbe s
o indi iduals N. We ind ha bo h PDFs show a powe -law
scaling egion o he o m
P(T)∼T−α,P(S)∼S−τ,(2)
limi ed by a peak o low alues and a shoulde o bump wi h
a as decaying (exponen ial) ail o high alues. The cha ac-
e is ic exponen s αand τ, ob ained om a linea eg ession
in double-loga i hmic scale in he scaling egion, ake he
alues α=2.4±0.2 and τ=1.97 ±0.14, whe e he e o
ba s ep esen 95% con idence in e als. Di e en alues o
ω h lead o simila a e age exponen s, e.g., α=2.9±0.8 and
τ=2.4±0.4 o ω h =0.15 (see Fig. S1 in he Supplemen al
Ma e ial [39]). These exponen s align wi h p e ious es ima es
de i ed om smalle s a is ics and using a di e en de ini ion
o u ning ish [19]. In e es ingly, dis ibu ions o schools
o di e en numbe s o indi iduals collapse on o he same
unc ional o m wi h he excep ion o he ail, which can be
in e p e ed in e ms o ini e-size e ec s, as la ge schools end
o c ea e a alanches o la ge du a ion and size.
The du a ion and size o indi idual a alanches a e no
independen , as we can check by plo ing he a e age size
STo a alanches o du a ion T[see Fig. 2(c)]. F om his
igu e we can obse e a supe linea beha io
ST∼Tm,(3)
wi h m=1.41 ±0.06. The alue o mcan be ela ed o he
exponen s o he du a ion and size dis ibu ions as [19,36]
m=α−1
τ−1.
Ou expe imen al alue mis ully compa ible wi h he heo-
e ical p edic ion m=1.4±0.3 o ω h =0.1 [expe imen al
m=1.35 ±0.16 and heo e ical p edic ion m=1.4±0.7 o
ω h =0.15 (see Fig. S1c in [39])].
In Fig. 2(d) we show he PDF o he in e e en ime i o
ω h =0.1 and o schools o di e en numbe s o indi iduals
N. We ind again an in e media e scale- ee egion, limi ed
be ween he small- ime beha io and a shoulde wi h an expo-
nen ially dec easing ail. He e also plo s o di e en numbe s
o indi iduals Ncollapse on he same unc ional o m, wi h
he excep ion o he ail. A i o he o m
P( i)∼ −γ
i
in he scaling egion leads o an a e age exponen γ=1.62 ±
0.08 [γ=1.63 ±0.04 o ω h =0.15 (see Fig. S1d in [39])].
I is no ewo hy ha he beha io o he decaying ails wi h N
is e e sed wi h espec o he du a ion and size PDFs, wi h
a la ge numbe o indi iduals leading o smalle in e e en
imes. This obse a ion is consis en wi h he beha io o he
ac i i y a e , as schools wi h a la ge numbe o indi iduals
ha e a highe p obabili y o be in an a alanche.
IV. DATA COLLAPSE
The dependence o he ails on he du a ion and size dis-
ibu ions wi h he school size Nobse ed abo e and wi h he
u ning h eshold ω h epo ed in [19] sugges s he possibili y
o a ela ionship be ween ω h and N esul ing in a alanches
wi h collapsing dis ibu ions. In o de o es o his hy-
po hesis, we selec he h eshold ω h ha , o each alue o
N, leads o a ixed ac i i y a e = 0.F omFig.1(b) we
es ima e, o 0=0.4, ω h =0.055,0.076,0.11,0.13 o N=
8,16,32,50, espec i ely. We plo he esul ing dis ibu ions
in Figs. 3(a),3(b), and 3(c) o he du a ion T,sizeS, and
in e e en ime i, espec i ely. In a sys em wi h no empo al
co ela ions in he ac i i y o indi iduals, a ixed ac i i y a e
esul s in du a ion and in e e en ime dis ibu ions collapsing
on o he same unc ional exponen ial o ms (see Appendix C).
Su p isingly, e en i his is no he case o empi ical u ning
a alanches in schooling ish, bo h he du a ion and in e e en
ime dis ibu ions achie e a da a collapse a ixed .On he
o he hand, he size dis ibu ions do no collapse pe ec ly,
033270-3
ANDREU PUY e al. PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
(a)
100101102
T
10−5
10−4
10−3
10−2
10−1
P(T)
N=8
N=16
N=32
N=50
(b)
100101102103
S
10−6
10−5
10−4
10−3
10−2
10−1
P(S)
(c)
100101102
i
10−6
10−5
10−4
10−3
10−2
10−1
P( i)
(d)
10−1100101
i/ i
10−4
10−3
10−2
10−1
100
P( i) i
FIG. 3. Da a collapse o he PDFs o (a) he du a ion T, (b) he
size S, and (c) he in e e en ime i o schools o di e en num-
be s o indi iduals Nconside ing a alanches wi h a ixed ac i i y
a e =0.4 (co esponding o ω h =0.055,0.076,0.11,0.13 o
N=8,16,32,50, espec i ely). (d) Da a collapse o he in e e en
ime gi en by Eq. (4) o ω h =0.1.
possibly because o co ela ions in he u ning a es o indi id-
uals a a gi en ame, which esul s in mo e ac i e indi iduals
in an a alanche ame o schools o la ge numbe s o indi-
iduals. In e es ingly, also in he unco ela ed case, he size
dis ibu ions a e no expec ed o collapse (see Appendix C).
On a simila no e, o a alanches o sel -o ganized c i ical
phenomena ac oss di e en con ex s, i has been ound ha
he in e e en ime dis ibu ions can be collapsed in o he
scaling o m [41,42]
P( i)=1
i i
i,(4)
whe e (x) is a uni e sal scaling unc ion and he only cha -
ac e is ic scale is he a e age in e e en ime i.InFig.3(d)
we show his so o collapse o a u ning h eshold ω h =
0.1; as we can see, i also applies o u ning a alanches in
schooling ish. This e eals sel -simila beha io , wi h he
in e e en ime dis ibu ions only di e ing in hei a e age
alue o schools o di e en numbe s o indi iduals. In he
unco ela ed case, his collapse is also eco e ed, bu now
only in he limi o a la ge a e age in e e en ime (see
Appendix C).
As a inal check o he scale- ee na u e o u ning
a alanches, we conside he scaling o he a alanche shape
n , de ined by he numbe o ac i e indi iduals o a u ning
a alanche a he ame o i s du a ion [43]. Many scale- ee
a alanche sys ems exhibi a collapse beha io in he a alanche
shape gi en by he scaling ela ion
n =Tm−1( /T),
whe e m=(α−1)/(τ−1) is he exponen ela ing he a e -
age a alanche size STwi h he du a ion T[Eq. (3)] and (z)
0.0 0.2 0.4 0.6 0.8 1.0
/T
0.3
0.4
0.5
0.6
T1−mn
S∈[30,45)
S∈[45,67)
S∈[67,100)
FIG. 4. Rescaled a alanche shape T1−mn as a unc ion o he
no malized ime /T. A alanche shapes a e a e aged o e simila
sizes Swi hin he powe -law scaling egion o he size dis ibu ion.
is a uni e sal scaling unc ion [6,43–45]. In he case o u ning
a alanches, his scaling beha io is eco e ed in a alanches
wi hin he powe -law scaling egime o he size dis ibu ion, as
shown in Fig. 4. In his plo he a alanche shape is compu ed
by no malizing he a alanche ime ame by i s du a ion T
and a e aging o e a alanches in a gi en size ange. We use
he alue m=1.41 ob ained in he nume ical analysis o he
du a ion and size dis ibu ions.
V. AVALANCHE TRIGGERING
In his sec ion we explo e whe he a alanches a e igge ed
in some p e e en ial poin s in space o ime, as well as by
pa icula indi iduals in he g oup. He e and in he ollowing
sec ions we show esul s o a alanches in a school o N=50
indi iduals, which ha e he longes eco ding ime, and a
u ning h eshold ω h =0.1.
A plausible hypo hesis is ha a alanches a e mo e e-
quen ly igge ed nea he ank walls due o bounda y e ec s.
These could a ise when ish a e app oaching a wall and need
o pe o m a la ge u n in o de o a oid colliding wi h i . To
check his hypo hesis we conside he posi ion o he cen e
o mass (cm)
xc.m. o he school, de ined as
xc.m. ≡1
N
i
xi,
whe e
xia e he posi ions o he ish a a gi en ins an o
ime. We de ine he igge ing loca ion o an a alanche as he
posi ion o he c.m. a he i s ame 0o he a alanche. We
s udy he dis ibu ion o igge ing loca ions on he su ace
o he ank. Because ish do no swim uni o mly all a ound
he ank, in o de o ex ac a s a is ically signi ican densi y
o igge ing loca ions, we no malize hei coun s agains he
coun s o all obse ed posi ions o he cm along he ime
e olu ion o he school. We show his in Fig. 5(a), whe e
he axis o ien a ions co espond o he ank walls. The g ay
egion in he colo map, sepa a ing he low-densi y ( ed) and
high-densi y (blue) alues, co esponds o he expec ed den-
si y in he absence o co ela ions, which we calcula e om
he o al coun s o igge ing loca ions di ided by he o al
coun s o posi ions o he cm As we can see in his plo , he
dis ibu ion o a alanches in he ank is qui e homogeneous,
al hough he e is a sligh endency o a alanches o occu
033270-4
SIGNATURES OF CRITICALITY IN TURNING … PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
(a)
01000 2000
xc.m.
0
500
1000
1500
2000
2500
yc.m.
0.000
0.025
0.050
0.075
0.100
0.125
T igge ing densi y
(b)
0 1000 2000
xc.m.
0
500
1000
1500
2000
2500
yc.m.
101
102
S
(c)
0 2000 4000 6000 8000 10000 12000 14000
0
5
10
15
c.m.
101
102
S
(d)
0 1000 2000
x
0
500
1000
1500
2000
2500
y
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Ini ia o s densi y
(e)
−1000 0 1000
x−xc.m.
−1000
−500
0
500
1000
y−yc.m.
0.00
0.01
0.02
0.03
0.04
Cen e ed ini ia o s densi y
FIG. 5. A alanche igge ing in space, in ime, and wi hin he
g oup. (a) Densi y o he posi ion o he cen e o mass
xcm a he
s a 0o an a alanche ( he igge ing loca ion) no malized agains
all ajec o ies o he cen e o mass, (b) a e age size S o igge ing
loca ions o a alanches, (c) blue line showing he empo al e olu-
ion o he cen e o mass speed cm and do s showing a alanches
igge ed a he gi en speed cm and ime 0and colo ed by hei
size S, (d) and (e) densi y o he posi ion o ini ia o s no malized
agains he posi ions o all indi iduals a he s a 0o an a alanche
o (d) he labo a o y e e ence ame and (e) he cen e -o -mass
e e ence ame and only o cen e ed indi iduals. In (a), (d), and (e)
he g ay colo in he colo map co esponds o he expec ed densi y
in he absence o co ela ions, gi en by he o al coun s o he quan-
i y conside ed di ided by he o al coun s o he no maliza ion. In
(c) we only plo a alanches ha p opaga ed o indi iduals o he han
he ones ac i e in he i s ame o he a alanche. In (e) he y
coo dina e is o ien ed in he di ec ion o mo ion o he g oup gi en
by he cen e -o -mass eloci y.
away om he walls. Howe e , i we display he a e age size
So a alanches gene a ed a he di e en igge ing loca ions,
we ob ain a di e en pic u e [Fig. 5(b)], in which a alanches
o la ge sizes end o occu mo e equen ly nea he ank
co ne s. This obse a ion sugges s ha in e ac ions wi h he
ank walls indeed p omo e he eme gence o la ge u ning
a alanches, esul ing in impo an o ien a ion ea angemen s
o he school.
Since la ge a alanches seem o be o igina ing om
in e ac ions wi h he walls, we in es iga e whe he hese
in e ac ions a e esponsible o he shoulde o bump obse ed
in he he ails o he du a ion and size dis ibu ions. They
a e pa icula ly no iceable in g oups wi h la ge numbe s o
indi iduals (N=32 and 50), which a e expec ed o ha e mo e
equen in e ac ions wi h he ank walls. This ea u e, known
as d agon kings, b eaks he powe -law pa adigm by displaying
o e ep esen ed ex eme e en s [46–48]. D agon kings a e
ypically gene a ed by mechanisms di e en om hose o
smalle e en s, which, in his case, may be wall in e ac ions.
To explo e his, we analyze he s a is ical dis ibu ions o
a alanches wi h igge ing loca ions away om he walls,
which we es ic o occu inside he squa e posi ioned a he
cen e o he ank wi h side L/3, whe e Lis he side o he
ank ( e e o Fig. S2 in [39]). We expec hese a alanches o
a ise spon aneously and no be p omo ed by in e ac ions wi h
he ank walls. Despi e limi ed s a is ics, d agon kings a e no
longe obse ed, and he dis ibu ions now showcase ex ended
powe -law egions wi h he same cha ac e is ic exponen s
as p e iously measu ed. We quan i y he p esence o d agon
kings in he size dis ibu ion o N=50 wi h a s a is ical
d agon king de ec ion es [48,49] (see Appendix D). Em-
ploying a signi icance le el α=0.05 o he null hypo hesis
ha he e a e no d agon kings, he es con i ms d agon kings
(p alue p<10−15) o he o al size dis ibu ion [Fig. 2(b)]
and ejec s hei p esence (p=0.1) o he size dis ibu ion
es ic ed o he cen al egion o he ank (Fig. S2 in [39]).
To unde s and empo al igge ing o a alanches, we s udy
how he a alanche s a ing ime 0 ela es o he g oup dy-
namics ep esen ed by he cen e -o -mass speed c.m., which
is de ined as
c.m. ≡
1
N
i
i
.(5)
The cen e -o -mass speed is cha ac e ized by ha ing oscilla-
ions due o a bu s -and-coas mechanism o he indi iduals
[50–52], wi h inc eases associa ed wi h an ac i e phase pow-
e ed by he ish muscles and dec eases coming om a passi e
gliding phase. In Fig. 5(c) we plo , o a ime window o
5 min om a single eco ding, he empo al e olu ion o he
cen e -o -mass speed as he blue line. We ma k wi h do s
a alanches igge ed a he co esponding ime 0and speed
c.m., colo coded by hei size S. We only conside a alanches
ha p opaga ed o indi iduals o he han he ones ac i e in
he i s ame o he a alanche. As we can obse e, while
small-size a alanches end o be andomly dis ibu ed o e
di e en alues o c.m., la ge a alanches a e mo e o en lo-
ca ed nea he minima o he speed, e en when he minimum
changes ac oss ime. We no ice ha his beha io does no
o igina e om small speeds being ela ed o la ge u ning
a es, because we ind he u ning a e is in e sely ela ed
o he speed only o cm <4 and appea s o be independen
o la ge speeds (see Fig. S3 in [39]). Ins ead, his sugges s
ha la ge a alanches may eme ge om u nings ela ed o
decision-making p ocesses occu ing a he onse o he ac i e
phase o he bu s -and-coas mechanism [51,53,54].
Apa om he spa io empo al igge ing o a alanches, we
can s udy how a alanches a e igge ed a he indi idual le el
wi hin he school conside ing a alanche ini ia o s, de ined
as he indi iduals ha a e ac i e in he i s ame o he
a alanche. P e iously, i was obse ed ha some indi iduals
ha e a p obabili y la ge han andom luc ua ions o be he
ini ia o s o beha io al cascades [19]. He e ins ead we ocus
033270-5
ANDREU PUY e al. PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
(a)
0.0 0.2 0.4 0.6 0.8 1.0
/T
4
6
8
10
12
14
c.m.
(b)
0.0 0.2 0.4 0.6 0.8 1.0
/T
0.4
0.5
0.6
0.7
0.8
0.9
φ
(c)
0.0 0.2 0.4 0.6 0.8 1.0
/T
800
1000
1200
1400
1600
1800
dw
S∈[5,9)
S∈[9,15)
S∈[15,27)
S∈[27,48)
S∈[48,85)
S∈[85,153)
S∈[153,274)
S∈[274,492)
S∈[492,885)
FIG. 6. Dynamics wi hin u ning a alanches o (a) he cen e -o -mass speed cm, (b) he pola iza ion φ, and (c) he di ec ed wall dis ance
d
wdepending on he no malized ime /Tand a e aged o simila sizes S. The g een dashed ho izon al line is he a e age o he gi en a iable
o e he whole expe imen .
on he loca ion o indi idual ini ia o s wi hin he expe imen al
ank and inside he school. Again, we ha e o keep in mind
ha indi iduals a e no loca ed uni o mly a ound he ank
a he s a o an a alanche. The e o e, in o de o ex ac a
s a is ically signi ican densi y o ini ia o loca ions wi hin he
g oup, we no malize hei coun s agains he coun s o he po-
si ions o all indi iduals a he onse ime 0o he a alanche.
We show he esul ing plo in Fig. 5(d). We ind ha ini ia o s
end o accumula e nea he ank walls and pa icula ly a he
co ne s. This is compa ible wi h he idea ha la ge u ning
a alanches a e p omo ed by in e ac ions wi h he ank walls.
In o de o explo e he na u al ela i e posi ion o a alanche
ini ia o s wi hin he school, we selec indi iduals ha do no
ha e ele an in e ac ions wi h he ank walls. We de ine cen-
e ed indi iduals as hose ha a e posi ioned in he cen al
squa e o he ank wi h side L/3, whe e Lis he side o
he ank. I we plo he densi y o he posi ions o cen e ed
ini ia o s wi hin he ank no malized by he posi ions o all
cen e ed indi iduals a he onse ime 0o an a alanche (see
Fig. S4 in [39]), indeed we see a uni o m pa e n ha con i ms
he idea ha cen e ed ini ia o s do no expe ience signi ican
in e ac ions wi h he ank walls. We s udy he ela i e posi ion
o cen e ed ini ia o s wi hin he school in Fig. 5(e), whe e we
plo he densi y o he posi ions o cen e ed ini ia o s no mal-
ized agains all cen e ed indi iduals a he igge ing ime 0
o he a alanche in he cen e -o -mass e e ence ame. In his
plo he ycoo dina e is di ec ed in he di ec ion o mo ion o
he cen e o mass. As we can see, ini ia o s o a alanches
away om he ank walls accumula e on he bounda y o he
school and wi hou any p e e ed di ec ion along he mo e-
men o he g oup.
VI. DYNAMICAL EVOLUTION OF AVALANCHES
In his sec ion we examine how an a alanche can a ec
he beha io o he whole g oup by measu ing se e al g oup
p ope ies along he e olu ion o he a alanche. In o de o
compa e a alanches wi h di e en sizes S, as in he case o
he a alanche shape discussed abo e, we i s no malize he
empo al e olu ion o he a alanche by i s du a ion Tand hen
a e age he dynamics o e g oups o a alanches wi h simila
sizes.
Fi s , we in es iga e he speed o he g oup gi en by he
cen e -o -mass speed cm, de ined in Eq. (5). We show how
i e ol es du ing a u ning a alanche, a e aged o di e en
sizes S,inFig.6(a). Fo compa ison, we plo he a e age alue
o e he whole expe imen as he g een dashed ho izon al line.
We obse e ha a alanches end o s a below he a e age cm
and ha a alanches o small size do no al e he school speed
no iceably. On he o he hand, la ge -size a alanches end o
o igina e a lowe alues o cm and inc ease he school speed
du ing hei e olu ion.
As a second cha ac e is ic o he school we conside he
global o de measu ed in e ms o he pola iza ion φ[12],
φ≡
1
N
i
i
i
,
which ends o 1 i he school is o de ed and all indi iduals
mo e in he same di ec ion and akes a alue close o ze o
i he school is diso de ed and ish mo e in andom and
independen di ec ions [12]. We show i s e olu ion wi hin an
a alanche in Fig. 6(b). Small-size a alanches end o s a
in highly pola ized con igu a ions and do no change signi -
ican ly he le el o o de . Con a ily, la ge a alanches end
o s a wi h less-o de ed con igu a ions han he a e age and
u he educe he o de as he a alanche sp eads. Howe e ,
a la e s ages his end is e e sed and he school eco e s a
highly o de ed s a e.
To gain u he in o ma ion abou he possible ole o he
walls, we s udy he dynamical e olu ion o a alanches wi h
espec o he dis ance o he ank walls. We de ine he di ec ed
wall dis ance d
was he dis ance om he cen e o mass o he
school o he ank walls in he di ec ion o he eloci y o he
cen e o mass. Fo a squa e ank, his dis ance is de ined as
d
w≡min1+ y
x2
[( x)(L−x)+(− x)x],
×1+ x
y2
[( y)(L−y)+(− y)y],
whe e he posi ions
xand eloci ies
e e o he cen e o
mass; (x) is he Hea iside s ep unc ion, which disc imi-
na es he o wa d and backwa d mo ion; Lis he side o he
ank; and he wo e ms in he min unc ion e e o he walls
on he xand ycoo dina es, espec i ely. We plo he e olu ion
o his quan i y du ing u ning a alanches in Fig. 6(c).Aswe
033270-6
SIGNATURES OF CRITICALITY IN TURNING … PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
can obse e, small-size a alanches do no al e he di ec ed
wall dis ance. On he o he hand, la ge a alanches end o
s a close o he wall and end a highe di ec ed dis ances.
This indica es ha la ge u ning a alanches ypically p oduce
a la ge change o he g oup o ien a ion om acing a nea by
wall o acing a a he away wall. We ha e also s udied he
e olu ion o he dis ance o he nea es wall, which we e e
as he minimum wall dis ance dw,
dw≡min(x,L−x,y,L−y).
We obse e (see Fig. S5 in [39]) ha his quan i y dec eases
and has a minimum o la ge a alanche sizes, indica ing ha
du ing he a alanche e olu ion he school ends o app oach
he closes wall, o la e mo e away om i .
VII. AVALANCHE CORRELATIONS
Ano he impo an aspec in a alanche beha io is he p es-
ence o co ela ions, namely, whe he he occu ence o an
a alanche induces he occu ence o o he a alanches such
ha hey appea clus e ed in space and/o ime [42]. The idea
o co ela ions and clus e ing in a alanches is closely linked
o he concep o main e en s and a e shocks in seismology
[55]. In his con ex , a e shocks a e ypically smalle e en s
ha occu a e a main e en in nea by loca ions and s and
ou om he backg ound noise. A ele an esul he e is he
obse a ion o he Omo i law, which s a es ha he p obabili y
o obse e an a e shock a a gi en ime a e a main e en
ollows he dis ibu ion
P( )=K
( +c)p,(6)
whe e K,c, and pa e cons an s, wi h p∼1[56].
In seismology, ea hquakes a e quan i ied by hei mag-
ni ude, which is a measu e ela ed o he loga i hm o he
ene gy eleased. Analogously, o u ning a alanches we can
in oduce he magni ude mas
m≡ln S,
whe e Sis he size o he a alanche. Conside ing he ob-
se ed size dis ibu ion om Eq. (2), magni udes o u ning
a alanches ollow he dis ibu ion
P(m)∼e−bm,(7)
wi h b=τ−1, which is analogous o he well-known
Gu enbe g-Rich e law o ea hquakes [57].
In o de o classi y e en s (ei he ea hquakes o
a alanches) in o main e en s and a e shocks, we conside
he me hod p oposed by Baiesi and Paczuski [58,59]. This
me hod is based on he de ini ion o he p oximi y ηij in he
space- ime-magni ude domain om an e en j o a p e ious
(in ime) e en i[58,60,61]. Assuming ha e en s a e o de ed
in ime 1< 2< 3<···, he p oximi y is de ined as
ηij ≡ ij d
ijP(mi)i i<j
∞o he wise,
whe e ij is he ime in e al be ween e en s iand j, ij is
he spa ial dis ance be ween he e en s loca ions, dis he
ac al dimension o he se o e en s posi ions, and P(mi)is
(a)
500 1000 1500 2000
xc.m.
500
1000
1500
2000
yc.m.
a=1
a=2
a=3
a=4
a≥5
(b)
100102104
Tj
10−1
101
103
105
Rj
42
21
1
20
40
60
80
100
120
Coun s
(c)
100101102
j
10−5
10−4
10−3
10−2
10−1
P( j)
FIG. 7. Co ela ion measu es o a e shocks: (a) numbe o a -
e shocks ape pa en depending on he igge ing loca ion o he
pa en , (b) coun s o he join dis ibu ion o he escaled space Rj
and ime Tj( he con ou plo co esponds o andomized a alanches,
in which a alanche posi ions, in e e en imes, and magni udes ha e
been shu led), and (c) PDF o he ime in e al jbe ween pa en s
and a e shocks o j<250. We only conside ed a alanches wi h
magni udes m⩾1.6. In (c) he ed dashed line co esponds o a i o
he Omo i law (6) wi h c=4.3±0.4andp=2.2±0.1.
he Gu enbe g-Rich e law o e en i, which in ou case is
gi en by Eq. (7). In he con ex o u ning a alanches, we ha e
o conside wo ac s. (i) A alanches ha e a ini e du a ion
ha is compa able o he in e e en ime be ween consecu i e
a alanches. We he e o e conside ij,i<j, as he numbe
o ames be ween he end o a alanche iand he s a o
a alanche j. (ii) Du ing an a alanche, he school mo es. We
hus conside he dis ance ij,i<j, as he dis ance be ween
he cen e o mass o he school a he end o a alanche iand
he cen e o mass o he school a he beginning o a alanche
j. Addi ionally, he dis ibu ion o he posi ions o he cen e
o mass a he s a o a alanches does no seem o show a
ac al s uc u e, so we use he e d=2.
The p oximi y ηij is a measu e o he expec ed numbe o
e en s o magni ude mi o occu , looking backwa d in ime
om e en jwi hin a ime in e al ij and dis ance ij,in he
absence o co ela ions, in such a way ha he ime and po-
si ion o p e ious a alanches beha e as independen Poisson
p ocesses [58]. The e o e, smalle alues o he p oximi y a e
associa ed wi h a la ge p obabili y ha he e en s iand ja e
ac ually co ela ed.
Using he p oximi y ηij, e e y e en jcan be associa ed
wi h a nea es neighbo o pa en pj, de ined as he e en
in he pas (pj<j) ha minimizes he p oximi y wi h j,
namely, ηpjj⩽ηij ∀i<j. This p oximi y is e e ed o as
he nea es -neighbo p oximi y ηj, i s ime in e al j, and he
spa ial dis ance j. The se o e en s wi h he same pa en
is conside ed he a e shocks o ha pa en . In Fig. 7(a) we
examine he dis ibu ion o he igge ing loca ions o pa en s,
033270-7
ANDREU PUY e al. PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
colo coded by hei numbe o a e shocks a. We ind a
possible in luence o he ank walls, as pa en s wi h a la ge
numbe o a e shocks end o be loca ed close o he co ne s.
In addi ion, we conside he measu e o clus e ing p o-
posed wi hin his amewo k in Re . [60]. This o malism is
based in he escaled ime Tjand escaled space Rj[60,61],
de ined as
Tj≡ jP(mpj),
Rj≡( j)dP(mpj)
such ha
ηj=TjRj.
In eal ea hquakes, i is obse ed ha he join dis ibu ion o
Tjand Rjis bimodal. One mode co esponds o backg ound
e en s and is compa ible wi h a andom (Poisson) dis ibu ion
o imes and posi ions o e en s. The o he mode, on he o he
hand, co esponds o clus e ed e en s, co ela ed in space and
ime [61].
In Fig. 7(b) we show he join dis ibu ion o Tjand Rj
o u ning a alanches in e ms o a colo densi y plo . In
he same igu e, we display in e ms o a con ou plo he
join dis ibu ion ob ained o andomized da a, in which
a alanche posi ions, in e e en imes, and magni udes ha e
been shu led. We ind ha he expe imen al da a show clea ly
wo modes in he dis ibu ion. In one mode, o la ge alues
o Tj, inc easing he escaled ime Tj esul s in a dec ease
o he escaled space Rj. This is almos iden ical o he
dis ibu ion ob ained o he shu led da a, indica ing ha i
co esponds essen ially o backg ound, unco ela ed noise.
The o he mode occu s o smalle alues o Tjand displays
he opposi e beha io , inc easing he escaled ime Tj esul s
in a highe escaled space Rj. This beha io is di e en om
he backg ound noise and co esponds o clus e ed (co e-
la ed) a alanches.
We can unde s and he imescale sepa a ion be ween he
modes aking in o accoun ha u ning a alanches ake place
inside a school ha is mo ing a ound he ank. The school
ypically pe o ms a ecu en mo emen on he ank, isi -
ing a gi en poin in he ank wi h some a e age pe iod. We
can quan i a i ely analyze his beha io looking a he mean-
squa e displacemen o he posi ion o he cen e o mass,
which measu es he a e age displaced dis ance o he g oup
in ime s a ing om any poin in he ajec o y (see Fig. S6 in
[39]). The i s maximum occu s a ound c=250 ames and
co esponds o he a e age ime he school needs o pe o m
a hal - u n a ound he ank and becomes maximally sepa a ed
om i s ini ial posi ion. A e shocks wi h a lowe ime in e al
end o inc ease hei spa ial dis ance as he school mo es
away om he pa en loca ion. A e his ime and up o e y
la ge ime in e als, he school may e u n owa ds he pa en
posi ion and we can ind a e shocks occu ing a lowe spa ial
dis ances. Howe e , hese end o occu a he andomly and
canno be dis inguished om andom e en s. This highligh s a
majo di e ence wi h ea hquakes, whe e signi ican co ela-
ions can occu in he same loca ion a widely sepa a ed ime
in e als.
Finally, we examine he Omo i law displaying he dis ibu-
ion o he ime in e al jbe ween pa en s and a e shocks
in Fig. 7(c). The dis ibu ion is compu ed conside ing he se-
quences o a e shocks o each pa en , shi ing he sequences
o se each pa en a a common ime ze o, and s acking
all sequences in a single common sequence [62]. F om he
abo e easons, we only conside ime in e als below c=
250 ha co espond o signi ican co ela ed a e shocks. A
leas -squa es i ing o he empi ical da a o he Omo i law
gi en by Eq. (6) (g een dashed line) yields he pa ame-
e s c=4.3±0.4 and p=2.2±0.1. This indica es a alue
p>1, implying a as e decay a e o a e shocks han in
ea hquakes.
VIII. CONCLUSION
In his pape we ha e p esen ed an empi ical analysis o
spon aneous beha io al cascades in schooling ish conside ing
u ning a alanches, whe e la ge u ns in he di ec ion o mo-
ion o indi iduals a e p opaga ed ac oss he g oup. This was
achie ed by collec ing ex ensi e s a e-o - he-a acking da a
o schooling ish, comp ising up o 1.8×105 ime samplings
a a esolu ion o 50 ames/s, o expe imen s in ol ing
a ying numbe s o ish, up o g oups o 50 indi iduals. This
da a se yielded o e 104a alanche e en s, ep esen ing a
signi ican ad ancemen compa ed o p e ious s udies on be-
ha io al cascades ( o e e ence, in [30] he au ho s epo ed
102a alanche e en s). We ha e analyzed di e en a alanche
me ics and p o ided a highly de ailed pic u e o he dy-
namics associa ed wi h beha io al cascades, employing ools
om a alanche beha io in condensed-ma e physics and
seismology.
We ha e unco e ed e idence o scale- ee beha io ac oss
a ious aspec s o u ning a alanches in schooling ish. Anal-
ysis o p obabili y dis ibu ions o undamen al obse ables,
such as he a alanche du a ion, size, and in e e en imes,
e ealed long ails compa ible wi h powe -law o ms. Ad-
jus ing o d agon king e en s, which a e disp opo iona ely
ep esen ed by ex eme e en s induced by in e ac ions wi h
ank walls, we ound he powe -law egion o a alanche size
ex ended up o wo decades. We also es ablished a scaling
ela ionship be ween he cha ac e is ic exponen s o he du-
a ion and size dis ibu ions. Fu he mo e, a da a collapse in
he dis ibu ions o he du a ion and in e e en imes a a ixed
ac i i y a e indica es a connec ion in a alanche dynamics
ac oss schools wi h a ying numbe s o indi iduals and he
u ning h eshold de ining he a alanche. We also con i med
wo p e iously obse ed da a collapses in c i ical a alanche
sys ems: in he in e e en imes dis ibu ion no malized by he
mean and in he a alanche shape o mean empo al p o ile ia
a scaling ela ionship wi h he du a ion.
While powe laws a e o en a ibu ed o c i ical phenom-
ena ela ed o phase ansi ions, al e na i e mechanisms can
also p oduce such dis ibu ions [4,63]. A igh e p edic ion o
c i icali y is mani es ed h ough da a collapses and ela ions
be ween scaling exponen s [4,44,45], indica ing quan i a i e
uni e sal a alanche dynamics ac oss scales. Ul ima ely, hese
indings a e insu icien o demons a e c i icali y, bu hey
cons i u e necessa y condi ions and embody a c ucial heo-
e ical aspec ha has ecei ed limi ed a en ion in beha io al
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SIGNATURES OF CRITICALITY IN TURNING … PHYSICAL REVIEW RESEARCH 6, 033270 (2024)
cascades o mo ing animal g oups. In ou wo k, we add essed
his gap and complemen ed exis ing s udies, p o iding e -
idence ha ish schools could ope a e in he icini y o a
c i ical poin . In pa icula , gi en he appa en lack o ex e -
nally uned pa ame e s in he sys em, hey would ep esen an
ins ance o sel -o ganized c i icali y [8,64]. While ou expe -
imen al se ings a e cu en ly limi ed o s udying small g oup
sizes o up o 50 indi iduals, u u e wo k should aim o es
o c i icali y e ec s in la ge g oups, in ol ing hund eds o
housands o indi iduals, o d aw adequa e compa isons wi h
s a is ical physics sys ems.
Being nea a c i ical poin can o e ad an ages such as
e icien collec i e decision-making and in o ma ion ans e
ac oss he g oup [4,5,8,20]. In his con ex , u ning a alanches
may a ise om sel -o ganized c i ical p ocesses ha acili a e
in o ma ion exchange among membe s o a social sys em,
compa able o a alanches seen in he social in e ac ions o
collec i e knowledge c ea ion [65,66]. Speci ically, u ning
a alanches allow ish o decide collec i ely on hei di ec-
ion o mo emen . Fo his eason, i is no su p ising ha
we obse e la ge a alanches occu ing a he onse o he
ac i e phase o he bu s -and-coas mechanism in ish locomo-
ion, whe e decision-making p ocesses o change indi idual
di ec ions a e belie ed o occu [51,53,54]. Du ing he p o-
cess o deciding a new collec i e di ec ion, coo dina ion
and g oup o de dec ease. Howe e , once a new di ec ion is
chosen, speed inc eases and coo dina ion eeme ges. A sim-
ila beha io was obse ed in he phenomenon o collec i e
U- u ns, in ol ing di ec ional swi ches o ish swimming
in a ing-shaped ank [18,67]. We a gue ha collec i e U-
u ns can be unde s ood as a speci ic example o u ning
a alanches.
Bounda y e ec s, a ising om in e ac ions wi h ank walls
o dis inc beha io s o indi iduals a he g oup’s bo de , a e
equen ly o e looked in he s udy o animal collec i e mo-
ion. This wo k highligh ed signi ican e ec s o ank walls on
a alanche beha io . While walls do no inc ease he numbe
o a alanches, hose in hei p oximi y o en exhibi la ge
sizes and mani es in co ela ed clus e s, esul ing in a highe
occu ence o a e shocks. Mo eo e , indi iduals ha a e ini-
ia o s o a alanches a e mo e equen ly ound nea walls.
This phenomenon can be a ibu ed o he ank walls ac ing
as obs acles, dis up ing he g oup’s mo emen and p omp ing
collec i e decisions o a subsequen di ec ion away om
he walls [68,69]. No ably, la ge a alanches induced by ank
walls p ima ily impac he ail o du a ion, size, and in-
e e en ime dis ibu ions, mani es ing as shoulde s o d agon
kings. The in e media e scale- ee beha io in hese dis ibu-
ions appea s o be in insic o spon aneous u ning a alanche
mechanisms, a he han being p omo ed by he walls. Ad-
di ionally, bounda y e ec s om indi iduals a he g oup’s
bo de play a ole, as hey a e o en ini ia o s o a alanches.
This aligns wi h p e ious indings associa ing hese posi ions
wi h highe social in luence [16,70]. An al e na i e explana-
ion is ha indi iduals a he g oup’s bo de may be mo e
exposed o isks [71], main aining a heigh ened ale s a e
and making hem mo e p one o ini ia ing a la ge change o
di ec ion.
We ha e examined he spa ial and empo al co ela ions in
u ning a alanches h ough he concep o a e shocks [42].
We obse ed ha u ning a alanches o schooling ish e-
eal signi ican clus e ed and co ela ed e en s below a ime
in e al co esponding o a hal - u n o he school a ound
he ank. This obse a ion poin s o a undamen al p op-
e y linked o he absence o collec i e memo y o la ge
imescales [13]. Fu he mo e, we ound ha he p obabil-
i y a e o obse ing co ela ed a e shocks a e a main
e en in u ning a alanches ollows an Omo i law wi h
a decay a e exponen p∼2, signi ican ly as e han in
seismology (p∼1).
We belie e his wo k makes a con ibu ion o he ongo-
ing inqui y in o c i icali y, pa icula ly wi hin he ealm o
animal collec i e mo ion and, mo e b oadly, in biological
sys ems. The limi ed numbe o analyses conduc ed on la ge
da a se s wi h expe imen al e idence o sel -simila beha io ,
a hallma k o c i ical sys ems, highligh s he need o u he
explo a ion and cla i ica ion in his a ea. Fu u e expe imen s
should aim o s udy la ge sys ems o e longe pe iods o ime
o deepen ou unde s anding o hese phenomena.
ACKNOWLEDGMENTS
We acknowledge inancial suppo om p ojec s PID2022-
137505NB-C21 and PID2022-137505NB-C22 unded by
MICIU/AEI/10.13039/501100011033, and by “ERDF: A way
o making Eu ope”. A.P. acknowledges suppo h ough a
ellowship om he Sec e a ia d’Uni e si a s i Rece ca o
he Depa amen d’Emp esa i Coneixemen , Gene ali a de
Ca alunya, Ca alonia, Spain. We hank P. Romanczuk, H. J.
He mann, and E. Vi es o help ul commen s.
APPENDIX A: EXPERIMENTAL DATA
We employ schooling ish o he species black neon e a
(Hyphessob ycon he be axel odi), a small eshwa e ish
wi h an a e age body leng h o 2.5 cm ha has a s ong
endency o o m cohesi e, highly pola ized, and plana
schools [72]. The expe imen s, pe o med a he Scien i ic and
Technological Cen e s UB, Uni e si y o Ba celona (Spain),
we e e iewed and app o ed by he E hics Commi ee o
he Uni e si y o Ba celona (P ojec No. 119/18). They in-
ol ed schools o N=8, 16, 32, and 50 indi iduals eely
swimming in a squa e ank o side L=100 cm wi h a
wa e column 5 cm deep, esul ing in an app oxima ely wo-
dimensional mo emen . Videos o he ish mo emen we e
eco ded wi h a digi al came a a 50 ames/s, wi h a es-
olu ion o 5312 ×2988 pixels pe ame, he side o he
ank measu ing L=2730 pixels. Digi ized indi idual a-
jec o ies we e ob ained om he ideo eco dings using he
open sou ce so wa e id acke .ai [73]. In alid alues e u ned
by he p og am caused by occlusions we e co ec ed in a
supe ised way, semiau oma ically in e pola ing wi h spline
unc ions (now inco po a ed in he Valida o ool om e -
sion 5 o id acke .ai). Fo be e accu acy, we p ojec ed he
ajec o ies in he plane o he ish mo emen , wa ping he
ank walls o he image in o a p ope squa e ( o de ails see
Re . [74]). We smoo hed he ajec o ies wi h a Gaussian
il e [75] wi h σ=2 and unca ing he il e a 5σ,em-
ploying he scipy.ndimage.gaussian_ il e 1d unc ion
om he SCIPY PYTHON scien i ic lib a y [76]. Indi idual e-
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