Statistial Physis of Random Searhes
G. M. Viswanathan 1;2
, V. Afanasyev 3
,Sergey V. Buldyrev 2
,Shlomo Havlin 2;4
,
M. G.E. daLuz 5
, E.P. Raposo 6;7
, and H. EugeneStanley 2
1
Departamento deFsia,UniversidadeFederaldeAlagoas,
57072-970, Maeio{AL, Brazil
2
Center forPolymerStudiesandDepartmentofPhysis,
BostonUniversity,Boston,MA02215,USA
3
BritishAntarti Survey,NaturalEnvironmentResearhCounil,
HighCross,Madingley Road, CB30ET,Cambridge,UK
4
Gonda-GoldshmiedCenterandDepartmentofPhysis,
BarIlanUniversity,RamatGan,Israel
5
Departamento deFsia,UniversidadeFederaldoParana,
81531-970, Curitiba{PR,Brazil
6
Laboratorio deFsiaTeoriaeComputaional,
Departamento deFsia,UniversidadeFederaldePernambuo,
50670-901,Reife{PE,Brazil
7
Lyman Laboratory ofPhysis,HarvardUniversity,
Cambridge,MA02138
Reeived29August,2000
We apply the theory of random walks to quantitatively desribe the general problem of how to
searheÆientlyfor randomlyloated objetsthatanonlybedetetedinthe limitedviinity of
asearherwhotypiallyhasanitedegreeof\freewill"tomoveandsearhatwill. Weillustrate
Levy ight searhproessesby omparisonto Brownian random walks anddisuss experimental
observationsofLevyightsinthespeialaseofbiologialorganismsthatsearhforfoodsites. We
reviewreentndingsindiatingthataninversesquareprobabilitydensitydistributionP(`)` 2
of steplengths ` an lead to optimal searhes. Finally we survey the explanationsput forth to
aountforthesesurprisingndings.
I Introdution
What is the most eÆient strategy for searhing
ran-domly loated objets whose exat loations are not
knownapriori? This questionhasbeenreently
stud-iedbyphysiists[1,2℄. Thegeneralproblemofhowto
searh eÆiently is ahallengingone, beause on the
onehand thesearherstypiallyhaveaertain degree
of \free will" to move and searh aording to their
hoie. Ontheotherhand,theyaresubjettoertain
physial andbiologialonstraintswhih restrittheir
behavior.
A lassi example of eÆient searh strategies
re-lates to animal foraging. On the one hand, the
an-imal's brain is suÆiently omplex to allow a broad
rangeofbehavioralhoiesand\freedom," butonthe
other handthe animalmust adapt andrestrit its
be-havior to inrease the hanes of survival, e.g., if an
thenitwilldie.
Therihnessoftheproblemstemspartiallybeause
ofthe\ignorane"oftheloationsoftherandomly
lo-ated \targetsites." However,evenif thepositions of
all target sites were ompletely known in advane by
a \demon" as resoureful as Laplae's [3℄, the
prob-lem of what sequential order to visit the sites in
or-der to redue the energy osts of loomotion is itself
rather hallenging: the famous \travelling salesman"
optimization problem [4℄. The ignorane of the
tar-getsiteloations,however,introduesyetanotherlevel
of diÆulty and renders the problem unsuited to
de-terministi searhalgorithms that donotuse some
el-ement of randomness. Indeed, only a statistial
ap-proahtothesearhproblemandealadequatelywith
astatistial mehanialentropyanadequatelymodel
the ignorane involved in the relationship between a
single marosopi (\thermodynami") state and the
very large number of orresponding mirostates of a
systemin thermodynamiequilibrium[5℄. Similarly, it
hasbeenarguedthatstatistialphysisisideallysuited
to thestudy ofomplexphenomena ofthis nature [6℄.
Indeed,thegeneralproblemofhowtosearheÆiently
forrandomly-loatedtargetsitesanbequantitatively
desribed[1,2,6℄usingideasdevelopedinthestudyof
randomwalks[7,8,9℄.
Quantifyingthestatistialpropertiesofsearh
pat-terns is of pratialrelevane not only in physis but
also in theoretial eology, industry, and oneivably
even to problems suh as the searh for missing
hil-dren. Veryreently,suhoneptshaveevenfound
ap-pliation in information tehnology (e.g., information
foraging theory[10℄).
II Random walks and random
searhes
Whenasearherwandersinsearhof thetargetsites,
the resulting motion an be desribed quantitatively
as a list of the visited sites, in sequential (temporal)
order. Suh motionis typially random. Speially,
the motion has some degree of stohasti noise, just
likearandomwalkreatedbyahypothetial\drunk"
whotakesstepsforwardsandbakwardsrandomlywith
equalprobability.
Brownianmotionanbethoughtofasakindof
ran-dom walk, but the partile an move ona ontinuous
sale. Thename \Brownian"refers to Robert Brown
who, in 1827, observedthe irregular motion of pollen
grains suspended in water[11℄. Brownianmotion was
notsatisfatorilyexplaineduntil1905whenAlbert
Ein-stein publishedhis lassipaper[12℄. Although
Brow-nian randomwalkswere therst to be studied, there
alsoexist non-Brownianrandom-walks.
Speially,randomwalksanbelassiedeitheras
Brownian(B)randomwalksorLevy(L)walks:
(B) Thestep lengths `
j
havea
harater-istisale,usually dened bytherst
orseondmoment(meanandvariane
respetively)ofthesteplengthdensity
distribution P(`). An essential
fea-tureofsuhrandomwalksisthattheir
squaredisplaementinreases linearly
withthenumberofstepstaken.
(L) The step lengths have no
hara-teristi sale, by whih we mean
that the moments diverge and the
P(`)
P(`): The square
dis-plaementofLevy randomwalks,also
alledLevyights,angrow
quadrati-allywiththenumberofsteps,sotheir
behavior is dominated by extremely
longbut raresteplengths.
The majority of studied probability distributions
leadto Brownianmotionasaonsequeneofthe
Cen-tralLimitTheorem(CLT).Ifthestepsare(i)largein
number(10 2
)and(ii)independent, i.e.,freeof
or-relations,then anyprobability distribution with nite
momentswillleadto Brownianmotion. Levy
distribu-tionshavediverginglowermoments,thereforetheCLT
is not appliableand superdiusive behavioris
possi-ble. Moreover,Levywalksresultinasetofvisitedsites
thatformafratal.
It is often possible to estimate experimentally the
probabilitydensitydistributionP(`)ofthesteporight
lengths` takenbyasearher. Until reentlyithas
of-ten been assumed [7, 8, 9℄ that suh a histogram of
ightlengthsP(`
j
)hasawelldened seondmoment.
Hene arise Gaussian,Poisson and other lassial
dis-tributions that leadto Brownian behavior. Indeed, it
hasgenerallybeenassumedapriorithatsearhers
per-formmovementsintheirenvironmentsthatorrespond
tonormaldiusion.
Reently,however,ithasbeenquestionedifthis
as-sumption is unneessarily restritive, and whether its
preditionsanbesupported by existingexperimental
data [1,2,13,14,15℄. Toaddress this question, one an
assumethemoregeneralLevydistribution[8,9,21℄,
P(`
j )`
j
; (1)
with 1 < 3 where, in fat, Gaussian behavior is
a speial ase for 3 [16℄. Values 1 do not
orrespond to normalizable probability distributions.
Apart from its intrinsi mathemati merit, as being
thelargestlassof stable distributions,Levy
distribu-tionshavein thelast deadefound usefulappliations
in biology [8, 9℄, and studies of biologialsearh
pro-essesspeially[2,13,14,15℄. (NotethatP.Levy
origi-nallyhadstudiedLevystatistissine1937,seeref.[8℄.)
WhereasBrownianmotionorrespondstonormal
diu-sion,Levyights,inontrast,orrespondtoanomalous
super-diusivemotion [8, 9, 17℄ (Fig. 1). Levy ights
µ=2.5
µ=2.0
µ=1.5
Figure1. 2-D randomwalks for =2:5; 2:0; and 1:5
re-spetivelywithidentialtotallengthsof 10 3
units.
Miro-organisms, mammals, birds, and insets show episodes of
approximatelystraightloomotionrandomlyinterruptedby
re-orientationevents.
Searhproessesanbefoundinbiologial
phenom-ena[19℄. Extensiveexperimentaldataexistforthe
spe-ialaseofanimalsearhproesses,inwhihananimal
optimizes its searh for, say, food [2,13,14,15℄.
Evo-lution has through natural seletion led over time to
highlyeÆient|evenoptimal|biologialsearh
strate-gies. Aordingtooptimalforagingtheory,animalsseek
tomaximizethereturns(inalories,nutrientset.) on
their labor in deiding how best to forage [20℄. Sine
physialaswellasneurophysiologialandevolutionary
fatorsomeintoplay,searhingisarihproblemthat
ontinuestopresentmulti-faetedandinterdisiplinary
hallenges.
Miroorganisms,insets,birds,and mammals have
beenfoundtofollowaLevydistributionofightlengths
or times (assumed to be proportional orat least
or-related statistially) [1,2,13,14,15℄(Fig. 2). Moreover,
the exponent appears to be the same in many
in-stanes [1℄. When the netar onentration is
nor-mal (low), the ight length distribution of bumble
bees[1,22℄ deayslikeEq. (1)with 2(Fig. 2(a)).
Similarly, thevalue2isalsofound forthe
searh-ing time distribution of the Wandering Albatross [2℄
(Fig.2(b))anddeer(Fig.2())inbothwildandfened
areas [1, 23℄. Even the value 2 2:5 found for
amoebas [14℄ supports the hypothesis that
opt = 2
mightbeauniversalvalueoftheexponentinLevyight
searhes. What, mightweask, drives animals to this
typeofbehaviorandwhatbenets,ifany,dotheythus
1.5
2.0
2.5
3.0
3.5
log
10
x
−1.0
0.0
1.0
2.0
log
10
N(x)
High Food
Low Food
(a)
µ
=2
µ
=3.5
0.0
0.5
1.0
1.5
2.0
2.5
log
10
t
−2.0
−1.0
0.0
1.0
2.0
3.0
log
10
N(t)
(b)
µ
=2
1.0
1.5
2.0
2.5
log
10
t
0.0
0.5
1.0
1.5
2.0
fenced
free
(c)
µ
=2.0
log
10
N(t)
Figure 2. (a)Doublelog plot ofthe ight length
perent-agedistributionsfor searhing bumblebees, digitizedfrom
ref.[22℄. Notethevalue 2fornormal (low)netar
on-entration. The value 3:5 for ( 10) higher netar
onentrations inwhihlong ightsbeome veryrare(see
text)isalsoonsistentwiththetheory.(b)doublelogplot
of the histograms of ight times (in 1h intervals) for the
WanderingAlbatross[2℄.()Doublelogplotofthesearhing
time (ses)perentage distributions for deer inwild areas
III Levy ight searh patterns
Why ight lengths might follow a Levy distribution
rather than a Gaussian or Poisson distribution is of
general interest. The reasons behind the
experimen-tally observed Levy ights in biologial searhes have
neverbeen fully understood, but a numberof studies
haveshed somelight. Levandowskyet al.[13, 14℄ have
suggested reasons why miroorganisms may perform
Levy ights in three dimensions (3-D), showing that
aLevydistributionisadvantageoussinethe
probabil-ity of returning to a previously-visited site is smaller
than for a Gaussian distribution, irrespetive of the
valueofhosen[24℄. Arelatedexplanationproposed
by M.F.Shlesinger (see[2℄)argues thatforagersmay
perform Levy ights beause the number of new
vis-ited sitesis muh largerfornLevywalkersthanfor n
Brownian walkers[25℄. Then Levywalkersdiuse so
rapidly that the ompetition for theresoures (target
sites) among themselves is greatlyredued relativeto
the ompetitionenounteredbythe nBrownian
walk-ers, whotypially remainlose tothe origin,heneto
eah other. A Levyight strategy is also a good
so-lution for therelatedproblem where N radar stations
searh for M targets [26℄. Yet another proposed
hy-pothesisisthatthefratalpropertiesofthesetofsites
visited byaLevy walkerare relatedto sale invariant
propertiesoftheunderlyingeosystem[2℄. Speially,
afrataldistributionoftargetsitesmayexplainthe
ob-servedLevyights[2℄. Veryreently,therehasbeena
studyofhowthesearheÆienydependsonthevalue
oftheLevyexponent[1℄. Thisstudyndsthatthere
is an optimum value
opt
=2whih an leadto
opti-mal searhes when the target sites are randomly and
sparsely distributed. Below,wedisuss this latest
de-velopmentin greaterdetail.
BystudyinghowthesearheÆienyvarieswith,
one an ompare dierent lasses of searh strategies
haraterized by unique values of : In the rst ase
of \non-destrutive searh", the forager an visit the
sametargetsitemanytimes. Nondestrutivesearhis
more realisti (see below) and an our in either of
twoases: (i)ifthetargetsitesbeometemporarily
de-pleted,or(ii)iftheforagerbeomessatiatedandleaves
thearea. Intheseondaseof\destrutivesearh",the
target sitefound bytheforagerbeomes undetetable
in subsequent ights. Considerthefollowingidealized
modelthat apturessomeoftheessentialdynamisof
searhesinthelimitingaseinwhihpredator-prey
re-lationshipsareignored,andlearningisminimized.
As-sume that target sites are distributed randomly, and
theforagerbehavesas follows(seeFig.3):
(1) Ifthere is atargetsiteloated within
a\diretvision"distaner
v
,thenthe
nearesttargetsite.
(2) If there is notarget site within a
dis-tane r
v
, then the forager hooses a
diretion at random and a distane
`
j
from the probability distribution,
Eq. (1). It then inrementally moves
to the new point, onstantly looking
for a target within a radius r
v along
itsway. Ifitdoesnotdetetatarget,
itstopsaftertraversingthedistane`
j
andhoosesanewdiretionandanew
distane`
j+1
,otherwiseitproeedsto
thetarget asinstep(1).
2 r
v
l
j
(a)
(b)
Figure3. Searhstrategy: (a)Ifthereisatargetsite(full
square) loated within a \diret vision" distane rv; then
theforagermovesonastraightline toit. (b)Ifthereisno
targetsitewithin adistanerv,thentheforagerhoosesa
randomdiretionandarandomdistane `
j
fromtheLevy
probabilitydistributionP(`
j )`
j
,andthenproeedsas
explainedinthetext.
One an solve this model as follows: let be the
meanfreepathoftheforagerbetweensuessivetarget
sites (for2-D, (2r
v )
1
where is the targetsite
areadensity). Themeanightdistane is
h`i R
rv dxx
1
+ R
1
x
dx
R
1
rv x
dx
=
1
2
2
r 2
v
r 1
v
+
2
r 1
v
: (2)
Theseondtermof this\mean eld"alulationisan
approximation beause it assumes that the distanes
between suessive sites are identially equal to ; so
that there are no ights longer than : A new target
siteisalwaysenounteredamaximumdistaneaway
from the previous target site, eetively resulting in
a trunated Levy distribution [27℄. A more rigorous
treatment that onsidersnot onlythe meanvaluebut
also a Poisson distribution of the free paths does not
dis-One denes the searh eÆieny funtion () to
betheratioofthenumberoftargetsitesvisitedtothe
totaldistanetraversed bytheforager. Sinethetotal
distane travelled is approximatelyequalto the
prod-ut ofthe total numberof ightsand the meanight
lengthh`i,therefore
= 1
Nh`i
; (3)
whereN isthemeannumberofightstakenbyaLevy
foragerinordertotravelbetweentwosuessivetarget
sites.
Considerrst the ase of destrutive searh, when
thetargetsiteis\eaten"ordestroyedbythesearhing
animal andbeomes unavailable in subsequent ights.
ThemeannumberofightsN
d
takentotravelan
aver-agedistanebetweentwosuessivetargetsitessales
as N d (=r v ) 1 (4)
for 1<3. Here 1 isthe fratal dimensionof
the set of sites visited by a Levy random walker[28℄.
Note that N
d
(=r
v )
2
for > 3 (Brownian ase).
Consider the ommon ase in whih the target sites
are \sparsely" distributed, dened by r
v .
Sub-stituting Eqs. (2) and (4) into (3) one nds that the
mean eÆieny has nomaximum, with lowervalues
ofleadingtomoreeÆientsearhes. Notethatwhen
= 1+ with ! 0 +
, the fration of ights with
`
j
< beomes negligible, and eetively the forager
movesalongstraightlines untilitdetetsatargetsite.
Consider next the ase of nondestrutive searh
for sparsely distributed target sites. Sine
previously-visitedsitesanthenberevisited,themeannumberN
d
ofightsbetweensuessivetargetsitesinEq.(4)
over-estimates the true number N
n
for the nondestrutive
ase. It anbeshownthatN
n N
1=2
d
holds generally,
soitfollowsthat
N n (=r v ) ( 1)=2 (5)
for 1 < 3: Notie that N
n
=r
v
for > 3
(Brownianase). Indeed,wehavereentlyprovedthat
Eq. 5 is in fat a rigorousresult [17℄. This result has
also been systematially tested using simulations and
foundto beomebetter andbetteras(=r
v
)inreases
(omparealsoFigs.4(a)and(b)). Notethat ifr
v
then N
d
N
n
: Substituting Eqs.(2) and (5)into (3)
anddierentiatingwithrespetto,onendsthatthe
optimaleÆieny=1=(N
n
h`i)isahievedat
opt
=2 Æ ; (6)
whereÆ1=[ln(=r
v )℄
2
:Sointheabseneofapriori
knowledgeaboutthedistributionoftargetsites,an
op-timalstrategyforaforageristohoose
opt
=2when
=r islargebutnotexatlyknown.
2.0
4.0
6.0
8.0
10.0
λ=10
λ=10
2
λ=10
3
λ=10
4
1.0
1.5
2.0
2.5
3.0
µ
0.0
2.0
4.0
6.0
8.0
(a)
1−D
(b)
λη
1.0
1.5
2.0
2.5
3.0
µ
m
3.8
4.0
4.2
4.4
4.6
η
x 10
4
1.0
1.5
2.0
2.5
3.0
µ
1.0
1.1
1.2
1.3
λη
(c)
2−D
Figure 4. The produt of the searheÆieny and the
meanfreepathvs.in1-Dfordierent;found(a)from
Eqs. (2) and (3) (rv = 1) for the ase of nondestrutive
searhand(b)fromsimulations. ()foundfrom
simula-tionsin2-D,with=5000(r
v
=1). Ineahase,
opt 2
emerges as anoptimal value of the Levy ight exponent.
Inset: the foodis distributed inpathes of food-rihareas
in anotherwise empty environment, obtained by
omput-ing
m
= dlogN(`)=dlog`from the histograms N(`)of
ightlengths. Onlyightswithlog
10
`<4.5areonsidered
inordertoeliminatetheeetsoftheperiodiboundaries.
Again,m2seemstooptimize thesearheÆieny.
IV Disussion
Theaboveresultsareindependentof thedimensionof
thesearhspae. This isanalogous to thebehaviorof
Brownian random walks whose mean square
displae-mentisproportionaltothenumberofstepsin any
di-mension [7℄. Furthermore, Eqs. (4) and (5) desribe
the orret saling properties even in the presene of
short-range orrelations in the diretions and lengths
of the ights. Short-range orrelations an alter the
so these ndings remain unhanged. Hene, learning,
predator-preyrelationships,andothershort-term
mem-ory eets beome unimportantin thelong-time
long-distane limit.
Note also that for both destrutive and
nonde-strutive searhes, Brownian behavior, orresponding
to 3; issigniantlylesseÆientthan Levyight
motion. This nding suggests that a power law
dis-tribution of ight lengths may be essential for
opti-mal searhes when the target sites are sparsely and
randomly distributed. (One may argue that animals
wouldpossiblystarvetodeathwhileadopting a
Gaus-sianstrategyofforaging.)
Forompleteness,onsideralsotheaseinwhihthe
targetsitesareplentiful,i.e.,r
v
. Thenh`iand
N
d N
n
1:Hene,beomesindependentof. This
behavior does not orrespond to Levy ight searhes
but is moresimilar to aBrownianrandom walk. The
independene of on is adiret onsequeneof the
extremerarityoflongightswith`
j >r
v :
Thesetheoretialresultshavebeensupported with
numerialsimulationswhihdonotdependon
approx-imations. Indeed,1-D and 2-Dsimulationshavebeen
performed of the above model to study how varies
withfortheaseofnondestrutivesearhes. For the
aseofnondestrutivesearhes,Fig.4(a)showsthe
sim-ulation resultsfor variousvalues of andr
v
=1: For
1-D, theposition ofthemaximumin for the
simula-tion agreeswiththeanalytialresults(Fig. 4(b)),and
approahes
opt
=2as!1. Thesimulationresults
for 2-D nondestrutive searh also showmaxima near
opt
=2:Fig.4()showssimulatedsearhinasystem
of size 10 4
10 4
withr
v
=1;periodi boundary
on-ditions, and=r
v
=510 3
. Moreover,fordestrutive
searhwith r
v
, simulationsshowthat !1
op-timizes the eÆieny as predited. In ontrast, if the
target sites are densely distributed suh that r
v ;
then,asexpeted,wendnosignianteetofvarying
:Thesendingsagreewiththetheoretialpreditions
and raisethepossibilitythat Levyightsearheswith
<3maybeonned to instanes of low global
tar-getsiteonentration,sinetheprinipaladvantageof
hoosing small | longights| beomesnegligible
whenthereareampletargetsites. Indeed,beesappear
nottoapplyLevyightforagingforartiallyhigh
ne-tar onentrations(seealsoFigs.2(a)and2(b)).
Wealsonotethatnondestrutivesearhismore
re-alisti thandestrutivesearhbeausein nature,
ow-ers,berries,krill,sh,et. areusuallyfoundinpathes
or lumpswhihare rarelyompletely depleted. Thus
ananimalanrevisitthesamefoodpathmanytimes,
and a path an restore itself by regrowth.
Simula-tions of destrutive searhes in pathy target site
dis-tributions give results onsistent with nondestrutive
searhesfor uniformly distributed target sites. As an
illustrative example, Fig.4 (inset) shows () found
aremanysmallrandomlydistributedfood-rihregions,
eah with radius R , outsideof whih there is no food
to be found. Tospeed up the simulations, it was
as-sumed that the forager performs a Levy Flight only
outsidetheregionofradiusR ;andthatitinstead
per-forms Brownian motion within, nding food at eah
site along the way separated by the loal mean free
path = R =3. The system size used was10 5
10 5
:
A low path area to system area ratio of 10 6
is ahieved using a path radius of R = 10; and 10 4
pathes. Here,
m
dlogN(`)=dlog`wasmeasured
fromthehistogramN(`)ofightlengthsinsteadof
us-ingtheparameterfromthemodel. Suhahistogram
representsanexperimentallyobservabledistributionof
ightlengths. Notethat
m
isonsistentwiththe
the-oretiallypreditedvalue
opt :
Aknowledgements
Wethank CNPq, NIH, and NSF for nanial
sup-port.
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