Consensus dynamics on random rectangular graphs

(1)

ErnestoEstrada

1,2

andMatthew Sheerin

1

DepartmentofMathematis&Statistis,UniversityofStrathlyde,26RihmondStreet,

GlasgowG11XQ,U.K.;

2

CentreforMathematialResearh(CIMAT),Guanajuato,

Mexio.

Correspondingauthor: ErnestoEstrada

Telf. +44(0)1415483657

Fax. +44(0)1415483345

(2)

A random retangular graph (RRG) is a generalization of the random

ge-ometrigraph (RGG) in whih thenodes are embedded into aretangle with

sidelengths

a

and

b

= 1

/a

, insteadof onaunit square

[0

,

1]

2 .

Twonodesare

thenonnetedifandonlyiftheyareseparatedataEulideandistanesmaller

than orequalto a ertain thresholdradius

r

. When

a

= 1

the RRG is iden-tial to theRGG.Here weapply the onsensusdynamis model to the RRG.

Our main result is a lower bound for the time of onsensus, i.e., the time at

whih the network reahes a global onsensusstate. To prove this result we

need rst to nd an upperbound for the algebraionnetivity of the RRG,

i.e.,theseondsmallesteigenvalueoftheombinatorialLaplaianofthegraph.

Thisboundisbasedonatightlowerboundfoundforthegraphdiameter. Our

resultsprovethat asthe retangle in whih the nodes areembedded beomes

moreelongated,the RRG beomes a'large-world',i.e., thediameter growsto

innity, and a poorly-onneted graph, i.e., the algebrai onnetivity deays

to zero. The main onsequeneof these ndings isthe proof that thetime of

onsensusin RRGsgrowsto innity asthe retanglebeome moreelongated.

Inlosing,onsensusdynamisinRRGsstronglydependonthegeometri

har-ateristis ofthe embedding spae, and reahing the onsensusstate beomes

morediultastheretangleismoreelongated.

Keywords

•

Consensusdynamis

•

Randomgeometrigraphs

•

Graphdiameter

•

Algebraionnetivity

(3)

Manyreal-worldnetworkedsystemsareembeddedinto geometrialspaes.

Thesespatialnetworks,astheyareknown,mayrepresentmanydierentkinds

ofsenarios[1℄. For instane, in urban streetnetworksthe nodesdesribe the

intersetionofstreets,whihare representedbytheedges ofthegraph. These

streetsandtheirintersetionsareembeddedinthetwo-dimensionalspae

repre-sentingthesurfaeoupiedbytheorrespondingity. Similarsituationsour

withinfrastruturalandtransportationsystemsrangingfromwatersupply

net-works and railroads to the internet and wireless sensor networks (WSNs). In

WSNs[2℄, thenodesrepresent thesensorswhih aredeployed onagiven

geo-graphialregionandtheirommuniationdenestheonnetivityofthenodes.

Thisis analogous to manyother ommuniationsystemsranging from mobile

phonesto radio signals. On adierentsale weanmentionthe vasularand

ellularnetworks of nodesembedded into ellsand biologialtissues [3℄;

pro-tein residue networks [3℄; the networks of hannels in fratured roks [4℄; the

networks representing the orridors and galleries in animal nests [5, 6℄; and

landsapenetworks [7℄, among others. Formodeling these spatial networks it

isneessarytohaveatheoretialmodelthatapturesboththetopologial

fea-turestypial of omplexnetworks and the spatial embedding of these spei

kindsof systems. The mostommonly usedmodel forspatial networks is the

so-alled random geometri graph (RGG) [8, 9, 10, 11℄. In RGGs eah node

israndomlyassigned geometrioordinatesand thentwonodesareonneted

ifthe(Eulidean)distane betweenthem issmaller thanorequaltoaertain

threshold

r

.

The RGG model has been widelyused in the studyof wireless sensor

net-works (WSNs) and peer-to-peer networks [12, 13, 14℄, where the problem of

onsensushasreeivedgreatattentionduetothefatthatitallowsthe

ahiev-ing of tasks with a minimum overhead of ommuniation [15, 16, 17, 18, 19℄.

Intheonsensusprotools,astheyareknownintehnologialappliations,the

problemonsists ofmakingthesalarstatesof asetof agentsonvergeto the

samevalueunderloalommuniationonstraints[20,21℄. Thus,sinethe

om-muniationrequiresonlyloalinformationthereisnoongestionduetonetwork

tra. RGGsarealsousedtomodelpopulationswhiharegeographially

on-strainedinaertainregion,likeaity. Thissenarioisimportant,forinstane,

forthe analysis of epidemi spreadingin suh populations[22,17, 23, 24℄. In

this sense Rileyet al. [25℄ haveremarked that RGGs provide a nie way of

esaping the lak of loal orrelation and lustering that are impliit

proper-tiesofthe ongurationgraphsoften usedtoexplore epidemidynamis. Ina

similar fashion, RGGs anbe used to model strutured populations in whih

opinions,insteadofviruses,arepropagated. InthisasetheRGGsalsoaptures

verywellthegeographionstraintsofthepopulationand,inomparisonwith

othermodels[26℄,theyaremorerealistifor anumber ofreasons: (i)RGG is

isotropi (onaverage)while regularlattieisnot;(ii) the average degreefor an

RGGanbesettoanarbitrarypositivenumber,insteadofasmallxednumber

(4)

IntheformulationoftheRGGmodelitisassumedthat thenodesare

uni-formlyandindependentlydistributedonaunitsquare(orahigherdimensional

hyperube in thegeneralase) [8, 9℄. This unit square representstheareaon

whih the agentsare interating to reah aonsensusstate, and itould be a

workplae,aity,oraforest,justtomentionsomeexamples. Suhasquare-like

areaistypialofmanyreal-worldsenarios.Forinstane,theityofSan

Fran-iso(USA)isknownasthe"seven-by-seven-milesquare",duetothefatthat

themainlandpartoftheityisasquareofnearly11kmby11km. However,if

weonsiderotherities, likeManhattan,thepiturelooksverydierent.

Man-hattanis13.4miles(21.6km)longand2.3miles(3.7km)wide,whihresembles

aretangularshapeinsteadofasquareone. Basedonthisneessityof

onsid-eringtheinuene of theretangular shapeonthe topologial anddynamial

properties of the random networks deployed on these areas we have reently

introdued therandomretangular graph(RRG) model [27℄. Inthis ase, the

nodesareuniformlyandindependentlydistributedonaunit retangleofgiven

sidelengths. When both sidesare of thesamelength we reovertheRGG in

suhawaythattheRRGmodelgeneralizestheRGGone.

Here, we are interested in investigating analytially and omputationally

howtheelongationoftheretangleintheRRGaetstheonsensusdynamis

takingplaeonthenodesandedgesof thenetworksonstrutedonthem. We

startbyintroduingtheoneptoftherandomretangulargraph(RRG),and

ontinuewith thedesriptionof theonsensusmodelto beonsidered. Then,

westatethemain resultofthisworkwhihprovesthatforaRRGwithaxed

numberofnodesandagivenonnetionradius,thetimeforreahingonsensus

growsto innity whentheretangle isveryelongated. Wenally support our

analytiresultswithomputationalsimulationsforRRGs.

2. Preliminaries

Here we present somedenitions, notations, and properties whih will be

usedinthis work(see[3℄). Forthebasidenitions aboutnetworksthereader

is direted to the literature (see for instane [3℄). The notation used here is

standard. Forinstane,

ki

designatethedegreeofthenode

i

. Thematrix

K

=

diag

(

ki

)

designatethe degreematrixofthegraphand thematrix

L

=

K

−

A

isthegraphLaplaian,where

A

standsfortheadjaenymatrixof thegraph. Ithasentries

L

uv

=







ki

if

u

=

v

−

1

if

(

u, v

)

∈

E

0

otherwise

∀

u, v

∈

V.

TheeigenvaluesoftheLaplaianmatrixaredenotedhereby:

0 =

µ1

≤

µ2

≤

· · · ≤

µn

. If the network is onneted the multipliity of the zero eigenvalue

(5)

thematrixoforthonormalizedeigenvetors

ψj

~

of

L

,i.e.,

U

=

h

~

ψ1

· · ·

ψn

~

i

.

Theeigenvetor

ψ2

~

assoiated withthe algebraionnetivityis known asthe Fiedlervetor[28℄. Let

Λ

bethediagonalmatrixofeigenvaluesoftheLaplaian matrix. Then,

L

=

U ΛU

T

.

2.1. Random Retangular Graphs

The RGG is dened by distributing uniformlyand independently

n

points in the unit

d

-dimensional ube

[0

,

1]

d

[8℄. Then, two points are onnetedby

anedgeiftheirEulideandistaneis atmost

r

,whihis agivenxednumber known as theonnetion radius. That is, wereate adisk of radius

r

entred ateahnode,andeverynodeinside thatdiskisonnetedtotheentralnode.

Thisdiskplaystheroleoftheareaofinueneofagivennode,suhasthearea

ofoverageofamobileorwirelesssensor.

In [27℄ wehaveonsidered aunit hyperretangleasthe Cartesianprodut

[

a1, b1

]

×

[

a2, b2

]

× · · · ×

[

ad, bd

]

where

ai, bi

∈

R

, ai

≤

bi

,and

1 ≤

i

≤

d

instead

of the unit square of the RGG. Hereafter we will restrit ourselves to the

2-dimensionalase,whihorrespondstoaretangleofunitarea. Now,theRRG

hasbeendened by distributing uniformly and independently

n

pointsin the unitretangle

[

a, b

]

andthenonnetingtwopointsbyanedgeiftheirEulidean distaneisatmost

r

. Therestoftheonstrutionproessremainsthesameas fortheRGG.This impliesthat

RRG

→

RGG

as

(

a/b

)

→

1

and onsequently theRRGisageneralizationoftheRGG.

InFig. 1weillustratetwoRRGswithdierentvaluesoftheretangleside

length

a

andthesamenumberofnodesandedges. Intherstasewhen

a

= 1

the graph orresponds to the lassial random geometri graph in whih the

(6)

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.5

1

1.5

2

0

0.2

0.4

Figure1: IllustrationofaRRGreatedwith250nodesembeddedintoaunitsquare,

a

= 1

,

(top)andaunitretanglewith

a

= 2

(bottom). Inbothasesthenodesareonnetedifthey

areataEulideandistanesmallerthanorequalto

r

= 0

.

15

.

A fewimportantstruturalparametersofRRGshavebeendetermined

an-alytially in a previous work by the urrent authors (see [27℄). They inlude

theaveragedegree,theprobabilitythat thegraphsareonneted,theirdegree

distributions,averagepathlengthandlusteringoeient.

2.2. Consensusdynamis on graphs

Letusonsiderthatthestateofthenodesofthegraphattime

t

arestored in thevetor

~u

(

t

)

. Then, thevariation ofthestate ofthenode

i

with timeis ontrolledbytheequation[21,20℄:

~

_˙

ui

(

t

) =

X

(i,j)

∈E

(

uj

~

(

t

)

−

ui

~

(

t

))

, i

= 1

,

2 , . . . , n,

(1)

whih,forthekindofgraphsweanalyzeinthis workanbewrittenas

~

_˙

ui

(

t

) =

−

n

X

j=1

aij

(

ui

~

(

t

)

−

uj

~

(

t

))

, i

= 1

,

2 , . . . , n.

(2)

(7)

Itisobviousnowthatweanwrite1byusingtheLaplaianmatrixofthegraph:

~

_˙

u

(

t

) =

−L

~u

(

t

)

,

(3)

~u

(0) =

~z.

(4)

Intheequation(3)theLaplaianmatrixisatingoverthevetor

~u

(

t

)

whih isupdatedin time. Thatis,

~up

(

t

)

is asalarwhih representsthe'opinion'of thenode

p

atthetime

t

. Thesolutionofthis equationis:

~u

(

t

) =

e

−tL

_~z.

(5)

where

0 =

µ1

< µ2

≤ · · · ≤

µn

aretheeigenvaluesand

ψj,p

~

the

p

thentryof

theorresponding

j

th eigenvetoroftheLaplaianmatrix. Then, thesolution

oftheonsensusequationonthegraphisgivenby

~

u

(

t

) =

e

−tµ

₁

_ψ1

_~

· ~z

ψ1

~

+

e

−tµ

₂

_ψ2

_~

· ~z

ψ2

~

+

· · ·

+

e

−tµ

n

~

ψn

· ~z

ψn,

~

(6)

where

~x

· ~y

representstheinnerprodutoftheorrespondingvetors. When thetimetendstoinnityeverynodetendstothestateditatedbytheaverageof

thevaluesoftheinitialondition. Thisstateisusuallyknownastheonsensus

set [21℄anditanbeformallydened astheset

A ⊆

R

n

whihisthesubspae

span

{

1 }

,

i.e.,

A

=

{

~u

∈

R

n

|

ui

~

=

uj,

~

∀

i, j

∈

V

}

.

(7)

The following is a well-known result in the theory of onsensus dynamis on

networks.

Lemma1. ([21℄p. 46) Let

G

be aonnetedgraph. Then, the onsensus dy-namisonvergestotheagreementsetwitharateofonvergenethatisditated

by

µ2

.

Proof. As

t

→ ∞

~u

(

t

)

→

ψ1

~

· ~z

ψ1

~

=

~

1 ~z

n

~

1

(8)

and hene

~ut

→ A

. As

µ2

is the smallestpositiveeigenvalueof the graph Laplaian,itditatestheslowestmodeof onvergenein theequation6.

For thesake of simulationsit is sometimes useful to onsider the

(8)

~ui

(

k

+ 1) =

~ui

(

k

) +

ǫ

n

X

j=1

aij

(

~uj

(

k

)

−

~ui

(

k

))

,

(9)

where

0 < ǫ < k

−

1 max

isthetimestepforthesimulation. Theequation9an

bewrittenin matrixform asfollows:

~u

(

k

+ 1) = (

I

−

ǫL

)

~u

(

k

)

,

(10)

where

I

istheidentitymatrix. Thematrix

(

I

−

ǫL

)

isusuallyknownasthe Perronmatrix[20℄.

3. Algebraionnetivity and diameterof RRGs

Aswehaveseenin theprevioussetiontheso-alledalgebraionnetivity

µ2

[28,29℄ditatestheslowestmodeofonvergeneintheonsensusdynamis.

Thatis,therateatwhihagivengroupofnodesonnetedinanetworkreahes

theglobalonsensusismainlydeterminedbytheseondsmallesteigenvalueof

theLaplaianmatrix. Consequently,weobtaintherstresulthere,whihisan

upperboundforthealgebraionnetivityof aRRG.

Theorem2. Let

GR

(

n, a, r

)

beaonnetedRRGwith

n

nodesembeddedin a retangle of sides with lengths

a

and

b

=

a

−1

,and onnetion radius

r

. Then, the algebrai onnetivity, i.e., the seond smallest eigenvalueof the Laplaian

matrix,isboundedas

µ2

(

GR

)

≤

8 (

n

−

1) (

ar

)

2 a

4 _{+ 1}

log

2

2 n.

(11)

InordertoproveTheorem2weneedthefollowingresult.

Lemma 3. Let

GR

(

n, a, r

)

be a onneted RRG with

n

nodes embedded in a retangleofsideswithlengths

a

and

b

=

a

−

1

,andtwonodesareonnetedifand

onlyif their at aEulidean distane smaller or equal than

r

. Let

D

=

D

(

GR

)

bethe diameterof the orresponding RRG.Then,

D

(

GR

)

≥

√

a

4 _{+ 1}

ar

.

(12)

Proof. Thenodes ofthe RRG are uniformlyand independently distributed in

theunit retangle. Then, letusassumethatthe

n

pointsareequallyspaedin theareaoftheretangleseparatedbyaEulideandistaneof

r

. Inthisasethe largestnumberofpointsareonnetedalongthemaindiagonaloftheretangle.

Ifthelengthofthemaindiagonalis

c

thereare

c

r

onnetednodesin thisline.

Thus,themaximumshortestpathdistanein theRRGis

c

r

with

c

=

√

(9)

r

,thenthediameterof

GR

willbelargerthan

c

r

,whihprovestheresult.

Now,weonsiderthefollowingboundobtainedbyAlonandMilman[30℄for

thealgebraionnetivityofanysimplegraph.

Theorem4. ([30 ℄). The seondsmallesteigenvalueofthe Laplaianmatrixof

anygraphisboundedas

µ2

(

G

)

≤

8 kmax

D

2 log

2

2 n.

(13)

Then,bysubstituting12into13wehave

µ2

(

G

)

≤

8 _dD

kmax

2 log

2

2 n

≤

8 kmax

(

ar

)

2 a

4 _{+ 1}

log

2

2 n

≤

8 (

n

−

1) (

ar

)

2 a

4 _{+ 1}

log

2

2 n,

(14)

wherethelastinequalityusesthefatthatforanysimplegraph

kmax

≤

n

−

1

, whihnally provestheTheorem2.

Remark 5. Theresultsinthissetionprovethattheelongationoftheretangle

in the RRG makes the graphs drastially less onneted for a given radius.

However,asanbe seenin theinequalities(11) and(12)the redutionof the

algebraionnetivitywith the retangleelongationanbeompensatedwith

theinreaseof theonnetionradius,whihalsodereases thediameterof the

graph. Forinstane, ifweareonsideringthedeploymentofwirelesssensorsin

veryelongatedregionitisustomarytousesensorswhihhaveoverageradius

signiantlylargerthanthosetypiallyusedforoveringmoresquaredregions.

Otherwise,there isahighriskthatthewhole network isdisonneted.

4. Consensustime

Hereweareinterestedinanalyzingtheinueneofthealgebraionnetivity

onthetimeof onsensus

tc

,i.e., thetimeforwhih

|

ui

~

−

uj

~

| ≤

δ

,where

δ

is a giventhreshold. Then,westateourmainresultofthiswork.

Theorem 6. Let

GR

(

n, a, r

)

be a onneted RRG with

n

nodes embedded in a retangle of sides with lengths

a

and

b

=

a

−

1

, and two nodes are onneted

if andonly if they are ata Eulidean distane smaller than or equal to

r

. Let

h

tc

i

be the timeofonsensusaveragedfor all the nodesinthe graph. Let

µ2

be

the algebrai onnetivityof the RRGand

ψ2

~

the orresponding Fiedler vetor. Then

h

tc

i ≥

1 nµ2

n

X

p=1

ln

~

ψ2,p

ψ2

~

· ~z

δ

(10)

Proof. First,wewritetheeq. 6foragivennode

p

as

~up

(

t

) =

n

X

q=1

~zq

n

X

j=1

~

ψj,p

ψj,q

~

e

−tµ

j

_,

(15)

whih represents the evolution of the state of the orresponding node as

timeevolves. Now,letusonsiderthat thetimetendstothetimeofonsensus

t

→

tc

, where

tc

is thetimeat whih

u

(

t

)

→

~

ψ

T

1 ~z

~

ψ1

. Letus designatethis

timeby

t

−

c

~up

t

−

c

=

1 n

n

X

q=1

~zq

+

n

X

j=2

~

ψj,pe

−t

−

c

(p)µ

j

n

X

q=1

~

ψj,q~zq

!

,

(16) here

t

−

c

(

p

)

means the time at whih the node

p

is lose to reahing the

onsensusstate. Let

h

~u0

i

=

1 n

P

n

q=1

~zq

andletuswrite16asfollows

~up

t

−

c

− h

~z

i

=

n

X

j=2

~

ψj,pe

−t

−

c

(p)µ

j

n

X

q=1

~

ψj,q~zq

!

.

(17)

Letusseletanode

p

suhthat

ψ2,p

~

hasthesamesignas

ψ2

~

· ~z

.

Sine

µ2

orrespondingto

j

= 2

isthesmallesteigenvalueinthesumonthe righthandoftheexpression,thistermstendsto0slowerthanthetermsforthe

othervaluesof

j

. Thismeansthat,ifwehooseasmallenoughvalueof

δ

, the valuesof

tc

and thus

t

−

c

will be verylarge. Thus, weanensurethat theleft

sideofthe equationissmall enoughthat

n

P

j=3

~

ψj,pe

−

t

−

c

(p)µ

j

₍

_ψj

~

_·

_~z

₎

<

0

. This

impliesthat

~up

t

−

c

− h

~z

i

< ~

ψ2,pe

−t

−

c

(p)µ

2 ~

ψ2

· ~z

.

(18)

Now,beause

|

~up

(

t

−

c

)

− h

~z

i| ≥

δ

wehave

δ

≤

~up

t

−

c

− h

~z

i

<

ψ2,pe

~

−t

−

c

(p)µ

2 _ψ2

~

_·

_~z

.

(19)

Then,thetimeatwhihtheonsensusisreahed

tc

(

p

)

isbounded by

tc

(

p

)

≥

t

−

c

(

p

)

≥

1 µ

₂

ln

~

ψ2,p

ψ2

~

· ~z

δ

.

(20)

(11)

h

tc

i ≥

_µ2n

1 n

X

p=1

ln

~

ψ2,p

ψ2

~

· ~z

δ

,

(21)

whihprovestheresult.

Ifweareusingadisrete-timeapproahliketheonegivenin 10then

h

tc

i ≥

_ǫµ2n

1 n

X

p=1

ln

~

ψ2,p

ψ2

~

· ~z

δ

.

(22)

TheimportaneoftheTheorem6isthatwhen

µ2

→

0

thetimeofonsensus grows to innity. Previously, we have already proved in Theorem 2 that the

elongationofarandomgeometrigraphwithagivennumberofnodesandaxed

onnetionradiusmeansthat thealgebraionnetivitygoesasymptotiallyto

zero. The immediate onsequene of this result is that the onsensus time

growstoinnityin RRGwhen

a

→ ∞

due totheinverserelationbetweenthe onsensustimeandthealgebraionnetivity.

5. Simulations

Inthissetionwearryoutsimulationswiththemain goalofinvestigating

howtheelongationof aretangleinuenestheonsensustime. However,due

to its importane for the urrent paper as well as in general for the further

studyof RRGwerstinvestigatetheinuene ofthe retanglesidelengthon

the diameter of the graphsand on their algebraionnetivity. Here we will

onsiderRRGsonstrutedbyplaing

n

= 500

nodesinaunitretangleofside lengths

a

and

b

=

a

−

1

. Theonnetionradiuswill bexedto

r

= 0

.

15

andwe systematiallyvary the sidelengthfrom

a

= 1

to

a

= 12

. Foreah valueof

a

we take100 random realizations of the RRGand report the averagevalue of

theorresponding property. InFig. 2(a)weillustrate the plotof the average

valuesofthediameter

h

D

i

versusthevaluesof

a

(bluesquares). Asanbeseen the diameter inreases linearly with the elongationof the retangle. In fat,

h

D

i ≈

7 .

927 a

−

0 .

237

. Thisresultagreeswithouranalytialones(seeTheorem 3)whihindiatesthatas

a

→ ∞

thediameteroftheRRGalsogrowstoinnity. Thelowerbound (12)isalso plottedin Fig. 2(a)(redirles)where itanbe

seen that it follows idential trend as the observed value of the diameter for

RRG. Indeed,theobserved diameter linearlyorrelateswiththe oneobtained

byeq. 12withaorrelationoeientof0.999.

Thediameter ismainly usedhereto ndanupperbound forthealgebrai

onnetivity

µ2

ofthegraph,i.e.,theseondsmallesteigenvalueoftheLaplaian matrix. Wehavealulatedtheaveragevaluesofthealgebraionnetivityfor

[image:11.595.239.378.135.175.2] [image:11.595.233.389.240.280.2]

(12)

lengthof theretangle:

µ2

≈

0 .

7487

a

−

1.968 ₋

₀

_.

₀₀₆₆₁

withPearsonorrelation

oeientequalto 0.9999. This onrmsour analytialresultthat as

a

→ ∞

the algebrai onnetivity deays to zero. We also inlude in this gure the

plot orresponding to the values of the upper bound found for the algebrai

onnetivity (red irles). As an be seen, although the observed values are

signiantly smaller than the ones provided by the upper bound, they both

deayfollowingsimilarpower-lawswiththehangeoftheretanglesidelength.

Theobserved valuesof thealgebraionnetivity donothange linearlywith

thoseprovided by theupperbound. Instead, theyare relatedby apower-law

relation of the type:

h

µ2

i ≈

4476ˆ

µ

0.61

2 −

84 .

65

, where

µ2

ˆ

is the upper bound

obtainedbytheeq. (11).

Themainonlusionofthispartoftheworkisthat whentheretangle

be-omesmoreelongated,theRRGbeomesalargerworld(thediameterinreases)

and alsoit displaysless onnetivity (derease of

µ2

), whih makesthe graph morevulnerable to be split into isolated omponents by removing just a few

nodes/edges. Theonsequenesofthis resultfor theanalysisof theonsensus

dynamisareanalyzedbelow.

0

2

4

6

8

10

12

0

10

20

30

40

50

60

70

80

90

100 Rectangle side length, a

Network diameter

0

2

4

6

8

10

12

10 −3

10 −2

10 −1

10

0

10

1

10

2

10

3

10

4 Rectangle side length, a

Algebraic connectivity

Figure2: Changeof thediameter(a)and thealgebrai onnetivity(b)of RRGswiththe

variationinthesidelength oftheretangle,

a.

Allthegraphshave

n

= 500

nodesand the

onnetionradiusis

r

= 0

.

15

.Thesquaresorrespondtotheaveragevaluesobservedforthe

RRGafter100randomrealizations,andtheirlesrepresentstheboundsobtainedbyeq.12

andeq. 11,respetively.Notiethesemilogplotonthey-axisfortheplot(b).

Wenowstudy theinuene of theretangle elongationovertheonsensus

dynamisonRRGs. Wetakearewiththeelongationproesssothatthegraphs

donot beome disonneted. First weompare the disrete-timeevolutionof

(13)

shorterthanthat fortheelongatedRRG(seefurther). Beause theseplotsare

the results of only one random realization we perform a systemativariation

of the retangle side length and report the average of the time of onsensus

after 100 random realizations for eah valueof

a

with astopping riterion of

δ

= 10

−

4

. Theresultsareillustratedin Fig. 5(), where weplotthevaluesof

theaveragetimeforonsensusversustheretanglesidelengths(bluesquares).

As an be seen the time for onsensus inreases with the elongation of the

retangle. The best t for this orrelation is a 4th order polynomial:

h

tc

i ≈

0 .

1885

a

4 ₋

₁

_.

₆₅₁

_a

3 _{+ 19}

_.

₅₉

_a

2 ₋

₃₇

_.

₀₆

_a

2 _{+ 30}

_.

₅₉

;thethasaPearsonorrelation

oeientof0.9997. Usingthismodelweanobtainamorepreiseestimation

ofthe averagetime for onsensusofthe random realizationillustratedin Fig.

5(a) and(b). Foravalueof

δ

= 10

−

4

theonsensusisreahedfor

a

= 1

at a timeof11.66,whilefor

a

= 10

at atimeof1853. Wewill gobaktothis kind ofanalysislateroninthispaper.

Theestimatedtimesforonsensusobtainedfromtheequation(21)arealso

plottedinFig. 5()(redirles),whereitanbeseenthattheyfollowthesame

trendastheobservedvalues. Indeed,theplotoftheobservedvaluesversusthose

expetedfromtheeq. (21)(seeFig. 5(d))indiatesaperfetlinearorrelation

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0

100

200

300

400

500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 time

state

0 1000

2000

3000

4000

5000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 time

state

0

2

4

6

8

10

12

0

500 1000

1500

2000

2500

3000

3500

4000

Rectangle side length

Time for consensus

0

500 1000

1500

2000

2500

3000

0

500 1000

1500

2000

2500

3000

3500

Observed time for consensus

Estimated time for consensus

Figure3:IllustrationoftheonsensusdynamisforaRRGwith

a

= 1

(a)andfor

a

= 5

(b).

Thesimulationswerearriedoutusingadisretetimeonsensusmodel(see10)witharandom

alloationof initialstatesforthe nodes. Bothnetworkshave500nodesandthe onnetion

radiusis

r

= 0

.

15 .

Notiethatthesaleforthetimeaxishashangedbyafatorof10from

oneplottotheother. ()Dependeneofthetimeforonsensuswiththelengthofthesideof

theretangle. Herethesquaresrepresenttheaveragevaluesofthe100simulationsand the

irlesarethevaluesobtainedbytheequation21. Thesolidlinesrepresentthebesttwhih

wereobtainedusing4th orderpolynomials. (d)Linear plotof the observedand estimated

(usingequation21)forthetimeofonsensusoftheRRGswith500nodesand

r

= 0

.

15 .

Finally, we plot in Fig. 4 the dependene of the time of onsensus with

respet to both theonnetion radius andthe retangle side length. Theline

that divides the region of relatively fast onsensus (deep blue region in the

ontourplot) fromthatofrelativelyslowoneisgivenby

a

=

κ

· r

−

1 .

5

,where

κ

= 28

fortheanalytialand

κ

= 26

fortheobservedresults. Thus,aondition

forfastonsensusinRRGwith

n

= 500

anbesimplyapproximatedby

a

+ 1

.

5

[image:14.595.135.477.125.459.2]

(15)

haveonthestudyofonsensusinreal-worldsituations. Aswehaveseenin the

IntrodutionaitylikeManhattanhasdimensionswhih resemblearetangle

morethanasquare. Thatis,Manhattanis21.6kmlongand3.7kmwide. This

anbe represented as a unit retangle of dimensions

a

≈

2 .

42

and

b

≈

0 .

41

. Usingourttedmodel,andonsideringthatweembed500nodes,e.g.,wireless

sensorsto monitortheity,weobtaintheexpetedtime foronsensusonthis

RRG,whihis38.3. Thistimeis3.3timeslongerthantheoneexpetedifthe

network is onsideredto beembedded into aunit square, i.e.,

a

= 1

. That is, wewould beunderestimatingthetimefor onsensusof thesensors byafator

ofthree. Also, aordingto (23)weanestimatethat afastonsensusanbe

reahedin thisnetwork onlyif

r >

0 .

157

.

Connection radius

Rectangle side length

0.1

0.2

0.3

0.4

0.5

0.6

2

4

6

8

10

12

14

50

100

150

200

250

300

350

400 Connection radius

Rectangle side length

0.1

0.2

0.3

0.4

0.5

0.6

2

4

6

8

10

12

14

50

100

150

200

250

300

350

400

450

500

550

Figure4: Contourplotshowingthedependeneofthetimeofonsensuswiththeonnetion

radius and the retangle side length in RRGs with 500 nodes. a) Analytial results. b)

Observedresultsfromthesimulations. Thediagonalwhitelineorrespondstotheequations

a

=

κ

· r

−

1 .

5

,where

κ

= 28

fortheanalytialand

κ

= 26

fortheobservedresults.

6. Conlusions

Inthisworkwehavefoundsomeinterestingstruturalanddynamial

prop-ertiesofgraphsembeddedintoretangularareas. Thereentlydenedrandom

retangulargraphs(RRGs)aountforthespatialdistributionofnodesallowing

thevariationoftheshapeoftheunit retangleommonlyused inrandom

geo-metrigraphs(RGGs). Inpartiular, wehavefound anexellentlowerbound

for the diameter of RRGs. The diameter is an important parameter per se

as well as for its inlusion in many inequalities for other network strutural

parameters. For instane, we have used this bound to nd an upper bound

forthealgebraionnetivityof RRGs. Thealgebrai onnetivity,theseond

smallesteigenvalueofthegraphLaplaian,isoneofthemostimportant

param-etersrelatingnetworkstrutureanddynamialproessestakingplaeonthem,

e.g., onsensus/diusion dynamis, synhronization. Finally, we have studied

(16)

polynomially with theside lengthof the retangle. The simulationresults

al-lowed us to onrm these results and to nd empirial relations between the

topologialanddynamialparameterswiththeretanglesidelength.

The results obtained in this work have important pratial onsequenes

formodeling real-worldsenarios. First,modeling areal-worldsenariowhih

is not geometrially similar to a square using the `lassial' RGG, produes

a signiant error in estimating important strutural and dynamial network

parameters. Moreimportantly,theRRGprovidesamodelingsenarioinwhih

we an simulate the inuene of the shape of a geographial region on the

topologialanddynamialpropertiesofthegraphsembeddedonthem.

TherearemanynewresearhavenuesthatthestudyofRRGsopenfor the

study ofspatially embedded graphs. One ofthem is the analysisof other

dy-namialproesses,suh assynhronization,and epidemi spreadingonRRGs.

AnotherareaofdevelopmentistheextensionofRRGstohigherdimensions,

spe-iallytothree-dimensional(3D)ones. 3D-RRGswillallowtheeetivemodeling

of many real-worldsenariosin whih thenodes are embedded into elongated

ubiregionsofthe3Dspae. Finally,athirdareaofinterestingdevelopmentis

theonsiderationofotherproximitygraphs,suhasGabrielgraphsandrandom

neighborhoodgraphs,embeddedintoretangularregionsinsteadofunitsquared

ones. Wehopethese developmentswill ontributeto bebetterunderstanding

ofnetworksembeddedintogeometrialspaes.

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