Consensus Dynamics in a Dolphin Social Network

Abdelmalik Moujahid 1

, BlanaCases 1

and FranisoJavier

Olasagasti 1

1

ComputationalIntelligene Group. Dept. of Computational

Siene and Artitial Intelligene, University of Basque Country.

Abstrat. In this work we investigate the onsensus between

individuals in a soial network of bottlenose dolphins by simulat-

ingthe OCR (OpinionChanging Rate) model reently proposed by

Pluhini et al. in ref [1℄. This model is a soial adaptation of the

Kuramotooneinwhihtheoneptofopinionhangingrate,i.e,the

natural tendeny to hange opinion, transforms the usual problem

of opinion onsensus into a lass of synhronization. We study the

emergene of synhronized groups of individuals both in terms of

natural frequeny rates and entralpositions inthe network.

Keywords: Soialnetworks,Synhronization,Communitystru-

tures

1 Introdution

The study ofomplexnetwork hasattrated alotofattentionin

the sienti ommunityinreentyears [2-5℄.Indeed,manynatural,

tehnologial, biohemial and soial systems an be onveniently

modeled asnetworks made ofa large number of highlyinteronne-

ted units. In general terms a network an be represented formally

as a graph:a set of generiallyalled nodes (verties) onneted by

links representing some relationship. Reent studies have revealed

that suh systems are all haraterized by a number of distintive

topologial properties: relatively small harateristi distanes bet-

ween any two nodes, high lustering properties, power-law degree

distribution and presene of ommunity struture.

In soialnetworks, the nodes are people, and ties between them

are friendship, politial alliane or professional ollaboration. The

struture of interation network desribing who is interating with

whom,howfrequently andwith whihintensity,reets the import-

situationsinwhihitisneessary foragrouptoreahshareddeisi-

ons. Consensus makes a position stronger, and amplies its impat

on soiety. Sothe analysis of this soial network under a partiular

topology from numerial simulation of opinion dynamis models is

animportantissue tounderstand the soialgroup dynamis.

In thiswork,wedeal withtheproblemofonsensusformationin

animal soialnetwork with known ommunity struture simulating

the OCR (Opinion Changing Rate) model proposed in ref [1℄. The

networkwe studywasonstruted fromobservationofaommunity

of 62bottlenosedolphinslivinginDoubtfulSound,New Zealand[9.

Ties between dolphin pairs were established by observation of sta-

tistially signiant frequent assoiation. The paper is organized as

follow.First,wereviewthe mainfeaturesoftheKuramotoandOCR

models. Thenwe desribea dolphinsoialnetworkinterms of their

naturaldivisionsusingbetweenness-basedalgorithmofNewmanand

Girvan [7℄. In the seond part, we disuss the results of numerial

simulations of the OCR model on the network. Also,we investigate

the inueneofpartiularindividualsinmaintainingtheohesion of

ommunities.

2 From Kuramoto model to the OCR model

Originally, the Kuramoto model was motivated by the study of

olletivesynhronization,aphenomenoninwhihalargenumberof

oupled osillatorsspontaneously loks to a ommonfrequeny, de-

spite the dierenes intheirnaturalfrequenies [6,8℄.The dynamis

of the Kuramoto model is given by:

θ ˙ _i ( t ) = ω _i + 1 N

N

X

j =1

K _ij sin ( θ _j − θ _i ) ,

⁽¹⁾

where

θ _i (t)

^{denotes the phase of the osillator}

i

^{at instant}

t

^and

ω _i

its natural frequeny. The frequenies

ω _i

^are distributed aording to some probability density

g ( ω )

^.

K _ij

^{represents the oupling fore}

mean-eld interations

K _ij = K, ∀i, j

^{. The dynamis of this mo-}

del depends only on two fators: the oupling fore

K

^{whose eet}

tends to synhronize the osillators,and the frequeny distribution

thatdrivethemtostay awayeahfromotherby runningatdierent

naturalfrqueny.When theouplingissuientlyweak,theosilla-

tors run inoherently, whereas beyond a ertainthreshold olletive

synhronization emerges spontaneously. The existene ofsuh ari-

tial threshold for synhronization is very similar to the onsensus

threshold found in the majority of the opinion formation models.

Basedon this onept, Pluhinietal.[1℄ denethe OCR model asa

set of oupled ordinary dierential equations governing the rate of

hange ofagents' opinions.The dynamisofasystem of

N

^agentsis

given by:

˙

x _i (t) = ω _i + K d _i

N

X

j =1

A _ij αsin(x j − x _i )e ^{−α|x j −x i |} ,

⁽²⁾

where

x _i (t) ∈ ] − ∞ + ∞ [

^{isarealnumberthatrepresentstheopinion}

of the ith agent at time t. The

ω _i

^'s orresponding to the natural frequenies of the osillatorsin the Kuramoto model represent here

the so-allednatural opinion hanging rates (or), i.e., the intrinsi

inlinationsoftheagentstohangetheiropinions.Thevalues

ω _i

^'sare

distributedinauniform randomway withanaverage

ω 0

^{. Aording}

tothis, wean simulateonservative individualswith values of

ω _i <

ω 0

^{, exible ones with}

ω _i ≃ ω 0

^{and more exible ones with}

ω _i > ω 0

^.

K ≥ 0

^{is the oupling fore,}

d _i

^{is the degree of eah agent and}

A _ij

^{isthe adjaeny matrix. The}exponentialfator inthe oupling termensures that,foropiniondierenehigherthanaertainthres-

hold, ontrolled by the parameter

α

^{(we typially adopted}

α = 3

^),

opinionswillnomoreinuene eah other.Thisisperhapsthe main

ontributionoftheOCRmodelwithrespettotheKuramotomodel.

Thus, tostudythe opiniondynamisofthe OCRmodelwesolve

numeriallythesystemgivenby equation(2)foragivendistribution

of the

ω

^{'s and for a given oupling fore}

K

^{. As reported in [1℄, in}

order to measure the degree of opinions oherene, we use an order

rates dened as

R(t) = 1 − q 1

N

P

i ( ˙ x _i (t) − X(t)) ˙ ²

^.Here

X(t) ˙

^isthe

averageoverallagentsof

x ˙ _i (t)

^.Valuesof

R

^{approahingunitywould}

implyahigh degreeof opinionsoherene,while lowvaluesindiate

a inoherently regime.

3 Dolphin soial network

Bottlenosedolphinsommunitieshavebeendesribedasassion-

fusionsoieties and thereforeindividuals(or agents)an make dei-

sions to join or leave a group. Two soial groups (or lusters) were

identied in this population. The ommunity struture of this net-

work, obtained using betweenness-based algorithm of Newman and

Girvan[7℄, isshown inFig. 1,and the distributionof agentsineah

group is reported in Table1.

Table1.Listofagentsineahnalgroupasresultingfromusing

theNewmanandGirvanalgorithm[7℄forapartitionintwolusters.

Cluster Agentsin eah luster

1 (21) 2,6,7,8,10,14,18,20,23,26,27,28,32,33 ,40,42 ,49,55 ,57,58 ,61

2 (41) 1,3,4,5,9,11,12,13,15,16,17,19,21,22,2 4,25, 29,30, 31,34, 35,

36,37,38,39,41,43,44,45,46,47,48,50,5 1,52,5 3,54,5 6,59, 60,62

In order to measure the quality of a partiular division of a net-

workintoommunities,wehaveusedthe measureknownasModula-

rity

Q

^{introduedinref[7℄.Given apartiularpartitionofanetwork}

into

n

^{groups (or lusters), it ispossibleto dene a}

n × n

^{size sym-}

metri matrix

e

^{whose element}

e _ij

^{is the fration of all edges inthe}

networkthatlinkvertiesingroupitovertiesingroupj.Aording

to this, the trae of this matrix

T re = P

i e _ii

^{gives the fration of}

edges in the network that onnet verties in the same group, and

therefore a good division into groups should have a high value of

thistrae.Onthe otherhand,the sumofanyrow(orolumn)ofthe

matrix

e

^{, namely}

a _i = P

j e _ij

^{, give the fration of edges onneted}

Sound,extrated using the Newmanand Girvan algorithm [7℄.The

squareand irles denotethe primary splitof the network into two

groups.

togroup

i

^{.So, theexpetednumberof}intra-groupedgesisjust

a _i a _i

^.

Finally,the modularity

Q

^{is given by:}

Q = P

i (e ii − a ² _i )

^.

Valuesof

Q

^{approahing unity,whihisthe maximum,wouldim-}

plyastrongommunitystruture. Ifwetakethewholenetworkasa

singlegroup,orifthenetworkisarandomone,

Q = 0

^{.Fortheparti-}

tion intotwo groupsreportedin Table1.,the modularity

Q = 0 , 38

^.

This network is made of

N = 62

^{vertex and}

l = 159

^{edge; eah}

vertex represents anindividualand eah edgerepresentsassoiation

betweendolphinpairsourringmoreoftenthanexpetedbyhane

[19℄.

4 Numerial results

In this setion, we integrate the system of equation 2 over the

bottlenose dolphins network. In the model variable

x ( t )

^{'s represent}

deisionsofthe

62

^{agents,andthe}

ω

^{'stheirnaturaldeisionhanging}

rate dr.Forthe numerialresultswexthe ouplingfore

K = 2 . 2

and the

ω

^{'sare randomly hosen froma uniform}distribution inthe range

[ − 0 . 5 , 0 . 5]

^{with average}

ω 0 ≃ 0

^{. We set}

x _i ( t = 0) = 0 , ∀i

^{, i.e,}

intheinitialstate alldolphinssharethesamedeisionhangingrate

values. The resultsare presented in Fig. 2. Panels (a) and (b) show

thedeisionhangingrate(

x(t) ˙

^{)and theorderparameter}

R(t)

^overs

100

^{time steps in a} logarithmialsale for the absise axe.

As itan beappreiated,assoonaswe startthe simulation,the

system enters in a short unstable transient regime in whih agents

tend to synhronize their ativities due to the oupling fore (see

Fig. 2(a)). This regime is haraterized by maximum values of the

order parameter

R(t)

^{(see Fig. 2(b)).} Immediately after, the sys- temrapidly lusterizesresultingintwonal lustersinwhihagents

shareommondrvalues.Thissituationreetommunitystruture

present in bottlenose dolphins soiety. The distribution of agents in

eah nal lustersis reported inTable 2.

Table 2. Distribution of agents shown in Fig. 2(a) as resulting

fromthesimulationofOCRmodeloverthebottlenosedolphinsom-

munity for

K = 2.2

^{.Elements in}parenthesesrepresent the numbers of agentsin eahnal lusters.

Cluster Agentsin eah luster

1(21) 2,6,7,8,10,14,18,20,23,26,27,28,31,32,33, 42,49, 55,57 ,58,61

2(41) 1,3,4,5,9,11,12,13,15,16,17,19,21,22,24,2 5,29,3 0,34,3 5,36,

37,38,39,40,41,43,44,45,46,47,48,50,51,5 2,53,5 4,56,5 9,60,6 2

As it an be veried, the distribution of agents in Table 2. is al-

most the sameas obtainedin Table1. Onlytwoagents,the verties

31(

ω = − 0.29

^{) and 40(}

ω = 0.20

^{), have exhange lusters. In the}

dolphin network this two verties fall in the boundary between the

ommunities of the network (see Fig 1). Therefore, depending on

their natural deision hanging rate, an joint or leave a partiular

group. In this ase, the modularity is

Q = 0 . 3799

^{for this split into}

strength, agents in eah nal group, tend to maintain a synhro-

nized regime despite their dierent naturaldeision hanging rates.

This isdue tothe stronginuene ofthe ommunity struture pres-

ents in the network whih aet notably the deision of agents to

join or leave a group. While a subsequent inreases in the value of

the oupling strength forea ompletelysynhronized regime.

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ²

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

decision changing rate (dcr)

(a)

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ²

0.7 0.8 0.9 1

time

Order parameter

(b)

Fig. 2.Deisiondynamis of lusters synhronization in the OCR model

onthe Dolphins Network for

k = 2.2

^and

ω ∈ [−0.5, 0.5]

^.

In theotherhand,entralitymeasure (betweenness) [10℄foreah

individualofthenetworkshowthatverties2(

ω = − 0.5

^)and37(

ω = 0 . 17

^)havehighbetweenness values.Betweenness isameasureofthe inuene of individuals in a network over the ow of information

between others. So, this two individuals represent a potentially in-

formationbrokersinthisdolphinsoiety.InFig.2,panel(a),wehave

represented,inboldmarkers,thedr variablesassoiatedtothistwo

individuals. Aording to their natural deision hanging rate, ver-

ties 2 (

ω = − 0 . 5

^{) and 37(}

ω = 0 . 17

^{) represents more} onservative

soiety in the deision-making proess, we have onsidered two si-

tuation onerning the vertex 2 (the same analysis an be realized

for the vertex 37).

The rst one is when this entral agent have natural deision

hangingratesettozerovalue(

ω(2) ≃ 0

^{)simulatingaexibleagent.}

ResultsareshowninFig.3.Asitanbeappreiated,twonewsingle

groupsformedbyverties29and48merge.Itorrespondtoindividu-

als with high absolute value of

ω

^:

ω(29) = 0.41

^and

ω(48) = − 0.47

^.

For this split into4 lusters, the modularity is

Q = 0.3822

^{. Details}

of that Distribution ofagents isreported in Table 3.

Table 3. Distribution of agents shown in Fig. 3(a) as resulting

fromthesimulationofOCRmodeloverthebottlenosedolphinsom-

munity for

K = 2.2

^and

ω(2) = 0

^.

Cluster Agentsin eah luster

1 (1) 48

2(21) 2,6,7,8,10,14,18,20,23,26,27,28,32,33,40 ,42,4 9,55,5 7,58,6 1

3(39) 1,3,4,5,9,11,12,13,15,16,17,19,21,22,24, 25,30 ,34,35 ,36,

37,38,39,40,41,43,44,45,46,47,50,51,52, 53,54, 56,59 ,60,62

4 (1) 29

The seond situation simulates a more exible individual that

tends to antiipate the others members of soiety, i.e,

ω (2) = 0 . 5

^,

whih represents a maximum value of

ω

^{. Simulation results are re-}

presented inFig. 4 over200 time steps.

Table 4. Distribution of agents shown in Fig. 4(a) as resulting

fromthesimulationofOCRmodeloverthebottlenosedolphinsom-

munity for

K = 2.2

^and

ω(2) = 0.5

^.

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ²

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

decision changing rate (dcr)

(a)

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ²

0.7 0.8 0.9 1

time

Order parameter

(b)

Fig. 3.Deisiondynamis of lusters synhronization in the OCR model

onthe Dolphins Network for

k = 2.2

^and

ω(2) = 0

^{. Central} individuals arerepresented in boldmarker.

Cluster Agentsin eah luster

1 (1) 48

2 (5) 7,10,14,33,57

3 (3) 8,20,31

4 (6) 18,23,26,27,28,32

5 (38) 1,3,4,5,9,11,12,13,15,16,17,19,21,22,24,2 5,30,3 4,35,3 6,

37,38,39,41,43,44,45,46,47,50,51,52,53,5 4,56,5 9,60,6 2

6 (1) 29

7 (7) 2,6,40,42,49,55,58

8 (1) 61

In Fig. 4(a) we an see that the system reahes atransient om-

pletely synhronized regime (

R ( t ) = 1

^{in Fig. 4(b)) in whih all}

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ²

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

opinion changing rate (ocr)

(a)

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ²

0.8 1

time

Order parameter

(b)

Fig. 4.Deisiondynamis of lusters synhronization in the OCR model

onthe Dolphins Network for

k = 2.2

^and

ω(2) = 0.5

^{. Central} individuals arerepresented in boldmarker.

the

ω

^{. After, we observe that the group ontaining initiallythe in-}

dividual 2 (see Tables 2) is divided into several subgroups. As in

the previoussituation,individualswithhigh absolutevalueof

ω

^run

alone in single groups. Details of that distribution are reported in

Table 4.Compared with the initialonguration of Table 2, we see

thatthe group2isquitestable.Apartfromvertex 40thatleavethis

grouptojoinanother one(seeTable4).Moreover, themodularity

Q

(

Q = 0.3189

^{) shows a signiant derease, indiating that the par-}

tition obtained, doesnot orrespond tothe natural partitionof the

network.

It is important to stress that hanging the distribution of the

natural deisionhangingrate, the evolution of eah individualan

hange, but qualitatively the behavior of the system is the same.

5 Conlusions

This workprovide evidene that network topology isfundamen-

tallyimportantindeision-makingdynamis allowingindividualsto

entralindividualsis determinate inlusters formationproess.

Aknowledgements The Spanish Ministerio de Eduaion y

Cieniasupports this work through grantDPI2006-15346-C03-03

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