Random processes with convex coordinates on triangular graphs

VOL. 20 NO. 2 (1997) 383-392

RESEARCH

NOTES

RANDOM

PROCESSES

WITH

CONVEX

COORDINATES

ON

TRIANGULAR

GRAPHS

J.N.BOYD P.N. RAYCHOWDHURY

Department

of Mathematical Sciences

Virginia CommonwealthUniversity Richmond Virginia 23284-2014 U.S.A.

(Received September 28, 1994)

ABSTRACT. Probabilitiesforreaching specifieddestinationsandexpectation valuesfor lengths for randomwalksontriangulararrays ofpoints andedges arecomputed. Probabilities and expectation values aregivenasfunctionsof theconvex(barycentric)coordinatesof the

starting point.

KEYWORDS AND PHRASES. Randomwalk,expectationvalue, trap,convex(barycentric)

coordinates.

1991AMSSUBJECT CLASSIFICATION CODES. Primary60J15, Secondary 60J05.0

1. INTRODUCTION.

Barycentric coordinateswereintroduced into mathematicsbythegeometerA.F. MSbius

(1790-1868).

Supposethat

Vz,V2,V3,...,Vn,V,+

arethelinearlyindependentverticesofan n-dimensional Euclidean simplex. The barycentric coordinates of point Pin the Euclidian space

n+l n+l

aregiven by theunique

(n

+ 1)-tupleof realnumberscqsuch that

P=

c,Viwith

y

c,=l.

[1]

==1

The points of the closedsimplex

S,

VzV2V

...VnV

+ arethose points for which all

barycentriccoordinatesarenonnegative. The barycentric coordinatesofvertex

V,

aregiven by a, 1,aj 0 for i. Inourpaper,weare concernedonlywith points of the closedsimplex, andwewillreferto

(a

z,o2,a3

a,)

astheconvexcoordinatesofPe

S,

witha’especttothe vertices intheorder

V,V2,V

3

Vn,V,

₊1"

However,

ourprimaryconcernwillbe withthe

closedtriangularregion

VzV:V

3.

Convex

(or

barycentric)coordinates admit ofphysicalinterpretation. The pointPis the center ofmassforthe discrete distribution ofpointmassesa,at

V,

for 1,2,3, n,n+l. Thenotionsof balanceand mechanicalequilibrium permitusto discover theconvexcoordinates

ofvariousspecialpointsof triangles and tetrahedra andtorelate the geometryoftrianglesto mechanics.

[2]

Convex

coordinatesaxequite versatile mathematical objects. Their invarianceunder

affine transformations has manyapplicationsingeometryand theirinterpretationaslengths,

areas,andvolumesleadstoawaytocount partitions of theintegers.

[3]

Since 0 < a, < for each pointof

S,,

ithasalsoseemednaturaltoseek probabilistic

interpretationsand applicationsfor convex coordinates. Suchinterpretations andapplications

A

RANDOM

WALK ON A

LINE

SEGMENT. Letus consideraline onwhichthereis

definedaCartesiancoordinatesystemsothat pointsV andV of have Cartesian coordinates x 0andx 1,respectively.

Next,

letusconsidern- points,

AI,A2,...,A

n 1, equallyspaced

betweenV and

V2

asshowninFigure1. For convenience in notation, letusgiveV and

V2

thealternatenamesA0and

A,,

respectively. The collection ofnsegments

A,A

+1, 0,1,2,...,

n-1, andn

+

1 points

As,

0,1,2 n,constitutes a onedimensionalgraph. Theconvex coordinatesof point

A,

withrespectto

VI=A

oand

V=A,

are

(al,

)=

(1-

,

i)for

0,1,2 n.

FIGURE1. TheOneDimensional Graph.

Nowletussupposethatarandom walk takesplaceontheone-dimensionalgraph. If the walk has reached

A,

1,2,...,n- 1, thenextstepmustbeeitherto

A,_

with transition

probability p(Ai-* Ai_

1)

orto

A,

+ alsowith transitionprobability p(A,

A,

+

1)

End points

V1=A

_oand

V=A,

serve astraps. That is,p(Ao

A)

p(A,

A,_

1)

0. The walk mustcontinueuntil itreacheseither

VI=A

oor

V=A,.

Letp(Ai) denote the probability thatawalk starting from

Ai

reaches endpoint

V.

Then p(Ai)mustsatisfy theequation

p(Ai) p(Ai Ai_

1) p(A) +

p(A

Ai

₊

1)P(A,

₊

1)

21-

((A,_,)

+

((A,+,))

()

for 1,2 n-1, subjecttotheboundaryconditionsp(Ao)=O, p(An)=l.

A

functionsuchas

p(A)is anexample ofadiscrete harmonicfunction;and,ifp(A) satisfiesboth1and the boundary conditions,itdoessouniquely.

[7,9]

Itissimpletoshow that p(A)=a2,theconvex coordinateof

A

withrespect to

V2=A,

meets the requirements given above. Thustheprobability thatawalk startingat

A

with terminate at

V

is

a.

Likewise, the probability that the walk willterminate at

V1

is

q(A)=l- p(A)

1-a

a1.

THE EXPECTED LENGTH OF THE

ONE

DIMENSIONAL RANDOM WALK.

Each walkbeginningat

A

eventuallyterminates atV

A0

or

V

An.

Let

E(A)

denote the expectedlengthofawalkasmeasuredby thenumberof steps taken from startingpoint

A

to

eitherendpoint. Sincethe walk beginswithasinglestep frominteriorpoint

A

toeither

A,_

or

A

₊1,wecanwrite

E(A)=

p(A-

A_I)-(1

+

+

p(A-,A+I).(1

+

E(A+))

(2)

1/2

(E(A_1)

+

E(A,

+

))

+

1.

Sinceawalkbeginningat eitherendpointwillhavezerolength,ourboundary conditionsare that

E(A0)

0and

E(A)

0.

Supposethatboth

E(A)

andE(A)satisfy 2 and the boundary conditions. Then F(A,)

E(A)

E’(A)satisfies1 withboundaryconditions

E(Ao)

E(A,)=0andF(A) 0. Thus

F(A)

isadiscreteharmonicfunctionand must be uniquely given by

F(A)

0 for 1,2,

n.Therefore

E(A)

E’(A),

and any functionsatisfying2 while vanishingatthe endpointsis

Since E(Ai)

n2ala2

satisfies 2andthe convex coordinatesofV1,

V2

are

(1,0), (0,1),

respectively,wehave theexpectedlength. That is, theexpectedlengthofarandomwalk

starting from

A,

ontheonedimensionalorlineargraphofn

+

points andnsegmentsis_E(A,)

n2ac

wherec,1,

c

aretheconvexcoordinatesof

A,.

A

RANDOM

WALK

ON A TRIANGLE. Consider theclosed triangular regionnamed

byitsvertices as

V1V2V

3. Forconvenience,wetake the triangletobe equilateral. Weconsider

eachsideto beacopy oftheonedimensionalgraphofn

+

1equallyspacedpointsandn connecting segmentsdescribed above. Ifaline is drawn within

VV2V

paralleltoside

ViVa

then each point of thelinehas thesameconvexcoordinatea,,k i,j

Suppose,

then,that threelinesaredrawnsothat each isparalleltoadifferent side ofthe triangleandsothatthe threelinesareconcurrent. Eachisalineof constantconvexcoordinate withrespect to a different vertex. Thustheconvexcoordinatesfor the pointPcommontothe threelinescanbe read from the pointsat whichthelines intersectthesideof the triangleasindicated inFigure2.

FIGURE2. TheConvexCoordinates ofP:

(a,b,c).

Nowletus"drawthree setsofn- lineseachparallel tooneofthe three sidessothat the lines dividethesides inton congruent segments.

In

Figure 3,weshowthe three setsoflines

for thecasein whichn 6. Theinterior intersectionpoint

P

hassixadjacent points.

FIGURE3. The Triangular Graphforn 6.

Thearrayof points andsegmentsconstitutesatwodimensional, triangular graphupon

which wewillconsiderarandom walk. Each stepis amovebetween adjacent points of the

graph alongthe segment connecting them. Ifwedenote the points adjacentto interiorpoint

P

by

A,

for 1,2,3,4,5,6,wecanstatethe followingrulesforthe random walk.

1 1. The probability ofamovefrom

P

to

A

isgiven byp(P

A)

.

2. Oncethewalk reachesaboundary point, the walkmustremainontheboundary. 3. Theverticesofthetriangleserveastraps. The walk mustterminateonceitreaches

The probabilityofa movefromany boundarypoint otherthanavertex toeitherof

its twoneighboring boundary pointsis

1/2.

That is,oncethe walkhasreachedaside

of thetriangle, the walk becomesaonedimensional walkasdescribedabove.

Letusnowsuppose that Figure3 represents thegeneralcaseforthetriangulargraph

with n > 3. Then,iftheconvexcoordinates of interiorpoint

P

are

a(P)

(al,a2,

as),

the convexcoordinatesof its nearestneighboringpointsare

(A,)

(,-

,

+

,

),

()

(,-

,

,

+

),

()

(,,,-

,

+

),

(,)=

(,

+

,

,-

,

),

By

ect

computation,westhat

H

P

is udypoint shown

()

((,)

+

(n,)).

(3)

FIGURE4. Point

P

onSide

ViVj

Letp(P)denote theprobabilitythatarandomwalkstarting frompointPofthe

triangulargraphwillreachvertexV_1. Thefunctionp(P)isagainadiscrete harmonic function.

Itsatisfiesthecondition

6 6

for eachinterior pointP. Ontheboundaryp(P)

1/2

(p(A1)+ P(A2))withthenotationtkea fromFigttre4.

Equation 3 and the results for theone dimensionalwalk imply that p(P) al. That is, the probability ofreaching

V1

starting from point

P

is al. Likewise, the probabilities of reaching

V

andVsarea and as, respectively.

The walkcanbeextended tographsontetrahedron

V1V2V3V

4. Each facecontainsa

triangular graphand eachedgecontainsalineargraph. The probability ofarandom walk’s

reachingvertex

V,

1,2,3,4 from starting point

P

isa, theconvex coordinate of

P

with

respectto vertex

V,.

The walkcanbeextendedinductivelytographson the k-simplex, each face of whichisa(k-1)-simplex.

ELECTRICAL APPLICATION

1.

For

eachrandom walk thereexists anelectrical potentialproblemhaving thesame solution withtheonlydifferencebetween theproblemsbeing thephysicalmeaning of the variables.

[4]

Imaginethatthe triangular gridofFigure5is anelectricalnetworkof identicalresistors in whicheach segment connecting adjacent points hasresistance

R.

Letthepotentialat

V1

be

maintained at 1voltwhileall pointsof

V2Vz

areheldatzeropotential. The potentialsonsides

VIV

and

V1V

3 decrease linearly from to0 volts. Thus the boundary conditionson

potential match those oftherandomwalk.

orv(P)= v(A,). Thus theaveragevalueproperty given by Equation3 issatisfied. We

can

conclude’’ht

theelectrical potential atpoint

P

is

a.

Figure5. AnElectricalNetwork.

THE

EXPECTED LENGTH OF

THE

TRIANGULAR RANDOM WALK. Letus supposethatarandom walkstartsfrom point

P

of the triangulargraphand terminates atone ofthevertices. Letusdenote theexpectedlengthofthewalk byE(P).

If

P

isaninterior pointhavingnearestneighbors

A,,

1,2,3,4,5,6,itfollows that

E(P) (I+E(Ai))

+

E(A,).

=1 =1

()

If

P

isabounda.,Tpointwith

bounda

neighbors

A1, A

on

V,Vj,

wealready know that

E(P)

n2aiaj

satisfies

E(P)=

1+1/2

(E(A1) +(E(A2)).

Furthermore,

E(V)

E(Vi)

0.

SinceE(P)

n2(la

+

aa

3

+

aas)

satisfiesEquation4and also reducestothe

boundarysolution,itfollows thatwehavediscoveredthe expectedvalueof thelengthof the randomwalk.

That is, the expectedlengthofawalk starting from point

P

of the triangulaz graphis

E(P)

n2(ala2

q-_ala q-

a2a3)

wherea, a,a_zeconvexcrdinatesof with

rt

to

V,

V,

Vz

inthat order.

AN EXAMPLE.

Letn 12. Thecentroidof the trigle hconvexcrdinates

(,

,

)

,disapoint of theaph.

The probability thatawMk stating

om

thecentroid

will__reach

a

ptic..x

vtexis_]

steps.

COMMENTS ON BOUNDARY CONDITIONS. A nst

problemconcernsthe cotructionofdiscretehocfunctionsontheinteMorpoints of thetrigul aphwhen the functionserequiredtomt bitrlysetboundyconditions. Weconsider the

problemwitha

sci

ce.

Letuschgethe fourth ofourroles for rdomwksontMglesby supsing thatthe entireboundyof

V]VVz

serves trap.

In

otherwords, the rdom wk mustcome to aht wheneveritrh

bound

point. Letusfindorindicatehowtofinddiscrete

honicnctionsintesof], a,d

z

inclosed fowhichwill give theprobabityof

denote the desiredprobability by p(P).

Since thevertices

V1,V,V

3cannot be reached directly fromany interiorpoint of the

array, wecanexclude themfromourcalculations.

In

Figure 6,weindicatethe triangulargraph

withV

1,V2,V

3 removed.

v!

Figure6.

AV1V2V

3

{VI,V,V3}.

The boundaryconditions forp(P)arep(P) 0ifPe

V1V2

U

VV

{V,V,V3}

and

p(P) ifPe

VV

{V2,

V3}.

Oncewefindafunction p(P} satisfyingboththeseboundary

conditionsand the average value property given by 3, that probability function will beunique.

However,

the forms of thefunctions will notbeunique. Forexample, p(P) ot

+

a

is

harmonic,yet ithasthedifferent, but equivalent,form p(P) 1 _03. Furthermore,sincethe

functionsneedbe evaluated onlyat a finitenumber of points,variouscombinationsof

continuousfunctionsmaytakeon thesamevalues.

A

bitof calculationindicatesthatp(P) cosnra _{cosnrct co3rtrct3}satisfiesthe average value property, but the value ofp(P}is at allinteriorpoints of thegraphif nis evenand

ifnisodd.

Thisexample involvingthe trigonometric functionssuggeststhatweinvestigateproducts of exponentialfunctions inoursearchfor other,moreuseful, harmonicfunctions. Therefore,

suppose that

f(P)

analbna2c

ha3_{for nonmegative real numbers a,b,c. Wecan rewritethe} na nl a n2

function asf(P)=

ana’bnC’2c

n’nc’1"na=c

()

()

implyingthatweneed only considerproductsof theformp(P) ana,b

na=.

If p(P)

analb

ha2_is

to satisfy the average value property,werequire that

Simplifyingthisequation,wefindthat

a’Ib+a-+b

"l+ab

"+a+b=6

or

(a+l)b

2+(a

6a+l)b+(a

2+a)=0.

In

order thatbtakeonrealvalues,it isnecessarythat the discriminant

D

be nonnegative. The discriminant takes the simplified form

D

(a- 1) (a

-

14a

+

1)

implyingthatashouldbe chosen from

(-cx,

?-4x/r]

U

[7

+

4V/,

cx). In

addition,neithera norb may bezero.

Sinceweintend tousesumsoffunctionsof the form

(constant)-a’mlb

ha2tosatisfy the

boundaryconditions of ourproblem, letuschooseaand btoyieldassimple calculationsas

possible. Tothat end,welistall of the admissible values ofathat give discriminants whichare

-6 14 15 20

5929

169

3136

43681

S or 15 21 10

-

or

"T

-6 or

-12

-35-

or

Table 1. Values of a,D,b.

Since therolesofaand b may beinterchanged,wecanconstructfifteen different functions ina anda whichhaveoneintegral basewhilethe other baseis rational.

In

addition,we can usethepairs of convex coordinatescq,c,3orc,2,aaasindependentvariables. Of

course,aand b need not be rational, butourimmediate purposeis toproduce simple examples.

In

Figure 7,weshowthe simplegraphforn 3. The probability that awalkstartingat

P,

the onlyinteriorpoint,willreachapointof

VV

_aisclearly

1/2.

However,

we usethisgraph as atestcasetodisplaythe processfor constructingadiscrete harmonic functiongiving the

desiredprobability.

Figure 7. TheCaseforn 3.

Theproblemis symmetric withrespectto

a

and

aa.

Sincesumsof harmonic functions areharmonic, letusfind constantsA,B,C, and

D

sothat

p(P)

[z,(-)

+

*(-)

+

Bo,

+

C,

+

D

(S)

vishesif

(a,a,a)

0, 0, d h thevMue if

(a,

a3)

0,

,

(0,

,

)}.

Notethatsyetryimphes thatifp(P) visheson

VV,

it will

Mso

vhon

VV.

Substitution of appropriate values ofa,a,aintoEquation 5 yields thefollongline

equations:

-3375A+2B

+

D=0, 1665A+ B +2D=0, 90A / C+2D=3,

90A

+2C+

D=3.

Solvingthese equations,weobtain

A

1/2205, B

187/147,

C

D

149/147.

Checkingourwork byevaluationof the probabilityfunction withthesecoefficients at

Figure 8. TheCaseforn 4.

probabilities ofreaching

V2V

3 from

,

thatprobabilitybyx. The averagevalueproperty impliesthat

0+0+l+l+z+y 0+0+0+0+x+z

6 =x and 6 =Y"

Thus

Nowletusconstructp(P)andthencompareourfunction values with x and y. Recalling that theform of p(P)is notunique,wesimply chooseourformforreasonsofconvenience. Let

p(P)

A[154a’(_6)

’In,

+

154a’(-6)

4%

+

Oa

+ E% +

Fa.

Using theconvexcrdinat of points from

VV

vV

{V,V,V}

d noting that

E

F,

we cwritethe equations

-16874A 8780.8B 5062.5C

+

.75D

+

.25F 0 8325A

+

3653.44B

+1631.25C

+

.5D

+

.5F 0

-3225A -1023.232B- 219.375C

+

.25D

+

.75F 0

22A -78.288B 18.125C

+

F

1

72A

+

35.28B

+

12.5C

+

F

A

computer programtoimplement aGaussianeliminationprocedureyields

A

1.425 x 10

"4, B

1.029 x 10-4 _{C -8.189 x 10}

"4, D

_{-1.449, and}

E

F

_{0.996. Evaluating}

easilyexplainedbyround-offerrorsincomputation.

Weconclude thatlinearcombinations offunctionsofthe formahal b

ha2,

ahal b

n%,

an% b

n%,

andc1,c2,o3willproducediscrete harmonic functionssatisfying

arbitrarilyset boundaryconditionsfor triangular random walks. Unfortunately, the computations do notseemanesthetically pleasing, but they dogeneralizethe subject.

ELECTRICAL APPLICATION

2.

Let

usconsider thenetworkbelow whichisderived

VIV

{Vl, V}

and

VlV

{Vl,

V}

aremaintained atapotential of+1volt.

Figure9. AnotherElectricalNetwork.

1 3

Itfollows fromourlastcomputationsthat the potentialsatpoints

P,

and

,

respectively.

THE

EXPECTED LENGTH OF

THE

RANDOM

WALK WHEN THE BOUNDARIES

ARE

TRAPS.

In

thecasethat all points of the boundary oftriangular graph

V1V2V

3serveas

traps, the expectedlengthofarandom walk frominteriorpointP:

(ch,c,,a3)

isgiven by

E(P)

3nala2a3

Thisresult follows from the fact that

E(P)=

Again,theaverageistakenoveradjacentpoints

A,,

1,2,3,4,5,6. Theclaimcanbe verified

quite easily bydirect computationoverthesix pointsadjacent to

P

in thegraphhavingn

+

pointsoneachside. Itisalso clear thatE(P)vanishesontheboundaryof thetrianglesince at

leastone convex coordinate mustbezeroat anyboundarypoint. The resultisunique by argumentssimilar tothose givenin theonedimensionalcase.

In

the simplecasewithn 3, E(P) 1forthe walkstarting fromthe centroid. Ifn

n 12and

P

isthecentroid, then

E(P)

16.

FURTHER COMMENTS

ON EXPECTED LENGTHS. Again,it isthe result of

straightforwardcomputationthat, forinterior point P:

(a,a,c,),

each of thefollowing

functions

n2aa,

n22a3, n2al3,

and

n2l2a3

satisfies theaverage valueproperty

f(A)+1/2

f(P).

Itfollows thatE(P)

n(aaa

+

bac,

+

caa3

+

da=a3)

satisfiesEquation4 and

becomesacandidateforanexpectedlengthifthereal coefficients,a,b,c,d,sumto 3.

For

example, ifa b c 1, d 0,wehave theexpectedlengthfor the walk in which the verticesaretheonly trapsand for whichoriginal boundarycondition 4holds.

Ifa b c 0, d 3,wehavethe expectedlengthif allboundary pointsserve as traps.

Suppose

thatallpoints of

VV=

U

V1V

serveastrapsbut that boundarycondition 4

holdstrueon

VV3.

ThenE(P)isgiven bya 0,b 1. c 0, d 2sincesuchafunction

uniquelysatisfiestheboundaryconditions.

REFERENCES

GRAY,

J. "MSbius’ GeometricalMechanics," inMSbius

.and

His Band

(J.

Fauvel,

R. Flood, andR. Wilson,Editors),OxfordUniversity

Press,

New York,pp.

78-103, 1993.

HANSNER,

M. "TheCenterofMassand AffineGeometry," Th._.eAmerican MathematicalMonthly69

(October 1962),

724-737.

BOYD, J.N.,

and

RAYCHOWDHURY,

P.N. "CountingPartitions ofthe

Integers

with

ConvexCoordinates," Bulletinof Number TheoryXV

(1991),

28-37.

DOYLE,

P.G.and

SNELL, J.L.,

"Random WalksandElectrical Networks"