=Paper=
{{Paper
|id=Vol-442/paper-2
|storemode=property
|title=Consensus Dynamics in a Dolphin Social Network
|pdfUrl=https://ceur-ws.org/Vol-442/p2_MoujahidCasesOlasagasti.pdf
|volume=Vol-442
}}
==Consensus Dynamics in a Dolphin Social Network==
https://ceur-ws.org/Vol-442/p2_MoujahidCasesOlasagasti.pdf
Consensus dynami s in a dolphin so ial network
Abdelmalik Moujahid1, Blan a Cases1 and Fran is o Javier
Olasagasti1
1
Computational Intelligen e Group. Dept. of Computational
S ien e and Arti tial Intelligen e, University of Basque Country.
Abstra t. In this work we investigate the onsensus between
individuals in a so ial network of bottlenose dolphins by simulat-
ing the OCR (Opinion Changing Rate) model re ently proposed by
Plu hini et al. in ref [1℄. This model is a so ial adaptation of the
Kuramoto one in whi h the on ept of opinion hanging rate, i.e, the
natural tenden y to hange opinion, transforms the usual problem
of opinion onsensus into a lass of syn hronization. We study the
emergen e of syn hronized groups of individuals both in terms of
natural frequen y rates and entral positions in the network.
Key words: So ial networks, Syn hronization, Community stru -
tures
1 Introdu tion
The study of omplex network has attra ted a lot of attention in
the s ienti ommunity in re ent years [2-5℄. Indeed, many natural,
te hnologi al, bio hemi al and so ial systems an be onveniently
modeled as networks made of a large number of highly inter onne -
ted units. In general terms a network an be represented formally
as a graph: a set of generi ally alled nodes (verti es) onne ted by
links representing some relationship. Re ent studies have revealed
that su h systems are all hara terized by a number of distin tive
topologi al properties: relatively small hara teristi distan es bet-
ween any two nodes, high lustering properties, power-law degree
distribution and presen e of ommunity stru ture.
In so ial networks, the nodes are people, and ties between them
are friendship, politi al allian e or professional ollaboration. The
stru ture of intera tion network des ribing who is intera ting with
whom, how frequently and with whi h intensity, re e ts the import-
an e of topology in so ial dynami s. On the other hand, onsensus is
a key aspe t of so ial group dynami s. Everyday life presents many
situations in whi h it is ne essary for a group to rea h shared de isi-
ons. Consensus makes a position stronger, and ampli es its impa t
on so iety. So the analysis of this so ial network under a parti ular
topology from numeri al simulation of opinion dynami s models is
an important issue to understand the so ial group dynami s.
In this work, we deal with the problem of onsensus formation in
animal so ial network with known ommunity stru ture simulating
the OCR (Opinion Changing Rate) model proposed in ref [1℄. The
network we study was onstru ted from observation of a ommunity
of 62 bottlenose dolphins living in Doubtful Sound, New Zealand [9.
Ties between dolphin pairs were established by observation of sta-
tisti ally signi ant frequent asso iation. The paper is organized as
follow. First, we review the main features of the Kuramoto and OCR
models. Then we des ribe a dolphin so ial network in terms of their
natural divisions using betweenness-based algorithm of Newman and
Girvan [7℄. In the se ond part, we dis uss the results of numeri al
simulations of the OCR model on the network. Also, we investigate
the in uen e of parti ular individuals in maintaining the ohesion of
ommunities.
2 From Kuramoto model to the OCR model
Originally, the Kuramoto model was motivated by the study of
olle tive syn hronization, a phenomenon in whi h a large number of
oupled os illators spontaneously lo ks to a ommon frequen y, de-
spite the dieren es in their natural frequen ies [6,8℄. The dynami s
of the Kuramoto model is given by:
N
1 X
θ̇i (t) = ωi + Kij sin(θj − θi ), (1)
N j=1
where θi (t) denotes the phase of the os illator i at instant t and ωi
its natural frequen y. The frequen ies ωi are distributed a ording
to some probability density g(ω). Kij represents the oupling for e
between units. The original model studied by Kuramoto assumed
mean- eld intera tions Kij = K, ∀i, j . The dynami s of this mo-
del depends only on two fa tors: the oupling for e K whose ee t
tends to syn hronize the os illators, and the frequen y distribution
that drive them to stay away ea h from other by running at dierent
natural frquen y. When the oupling is su iently weak, the os illa-
tors run in oherently, whereas beyond a ertain threshold olle tive
syn hronization emerges spontaneously. The existen e of su h a ri-
ti al threshold for syn hronization is very similar to the onsensus
threshold found in the majority of the opinion formation models.
Based on this on ept, Plu hini et al.[1℄ de ne the OCR model as a
set of oupled ordinary dierential equations governing the rate of
hange of agents' opinions. The dynami s of a system of N agents is
given by:
N
KX
ẋi (t) = ωi + Aij αsin(xj − xi )e−α|xj −xi | , (2)
di j=1
where xi (t) ∈] − ∞ + ∞[ is a real number that represents the opinion
of the ith agent at time t. The ωi 's orresponding to the natural
frequen ies of the os illators in the Kuramoto model represent here
the so- alled natural opinion hanging rates (o r), i.e., the intrinsi
in linations of the agents to hange their opinions. The values ωi 's are
distributed in a uniform random way with an average ω0 . A ording
to this, we an simulate onservative individuals with values of ωi <
ω0 , exible ones with ωi ≃ ω0 and more exible ones with ωi > ω0 .
K ≥ 0 is the oupling for e, di is the degree of ea h agent and
Aij is the adja en y matrix . The exponential fa tor in the oupling
term ensures that, for opinion dieren e higher than a ertain thres-
hold, ontrolled by the parameter α (we typi ally adopted α = 3),
opinions will no more in uen e ea h other. This is perhaps the main
ontribution of the OCR model with respe t to the Kuramoto model.
Thus, to study the opinion dynami s of the OCR model we solve
numeri ally the system given by equation (2) for a given distribution
of the ω 's and for a given oupling for e K . As reported in [1℄, in
order to measure the degree of opinions oheren e, we use an order
parameter related to the standard deviation of the opinion hanging
q
rates de ned as R(t) = 1 − N1 i (ẋi (t) − Ẋ(t))2 . Here Ẋ(t) is the
P
average over all agents of ẋi (t). Values of R approa hing unity would
imply a high degree of opinions oheren e, while low values indi ate
a in oherently regime.
3 Dolphin so ial network
Bottlenose dolphins ommunities have been des ribed as a ssion-
fusion so ieties and therefore individuals (or agents) an make de i-
sions to join or leave a group. Two so ial groups (or lusters) were
identi ed in this population. The ommunity stru ture of this net-
work, obtained using betweenness-based algorithm of Newman and
Girvan [7℄, is shown in Fig. 1, and the distribution of agents in ea h
group is reported in Table1.
Table 1. List of agents in ea h nal group as resulting from using
the Newman and Girvan algorithm [7℄ for a partition in two lusters.
Cluster Agents in ea h 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) 1,3,4,5,9,11,12,13,15,16,17,19,21,22,24,25,29,30,31,34,35,
36,37,38,39,41,43,44,45,46,47,48,50,51,52,53,54,56,59,60,62
In order to measure the quality of a parti ular division of a net-
work into ommunities, we have used the measure known as Modula-
rity Q introdu ed in ref [7℄. Given a parti ular partition of a network
into n groups (or lusters), it is possible to de ne a n × n size sym-
metri matrix e whose element eij is the fra tion of all edges in the
network that link verti es in group i to verti
P es in group j. A ording
to this, the tra e of this matrix T re = i eii gives the fra tion of
edges in the network that onne t verti es in the same group, and
therefore a good division into groups should have a high value of
this tra e. On the other P
hand, the sum of any row (or olumn) of the
matrix e, namely ai = j eij , give the fra tion of edges onne ted
Fig. 1. Community stru ture in the bottlenose dolphins of Doubtful
Sound, extra ted using the Newman and Girvan algorithm [7℄. The
square and ir les denote the primary split of the network into two
groups.
to group i. So, the expe ted number of intra-group edges is just ai ai .
Finally, the modularity Q is given by:Q = i (eii − a2i ).
P
Values of Q approa hing unity,whi h is the maximum, would im-
ply a strong ommunity stru ture. If we take the whole network as a
single group, or if the network is a random one, Q = 0. For the parti-
tion into two groups reported in Table 1., the modularity Q = 0, 38.
This network is made of N = 62 vertex and l = 159 edge; ea h
vertex represents an individual and ea h edge represents asso iation
between dolphin pairs o urring more often than expe ted by han e
[19℄.
4 Numeri al results
In this se tion, we integrate the system of equation 2 over the
bottlenose dolphins network. In the model variable x(t)'s represent
de isions of the 62 agents, and the ω 's their natural de ision hanging
rate d r. For the numeri al results we x the oupling for e K = 2.2
and the ω 's are randomly hosen from a uniform distribution in the
range [−0.5, 0.5] with average ω0 ≃ 0. We set xi (t = 0) = 0, ∀i, i.e,
in the initial state all dolphins share the same de ision hanging rate
values. The results are presented in Fig. 2. Panels (a) and (b) show
the de ision hanging rate (ẋ(t)) and the order parameter R(t) overs
100 time steps in a logarithmi al s ale for the abs ise axe.
As it an be appre iated, as soon as we start the simulation, the
system enters in a short unstable transient regime in whi h agents
tend to syn hronize their a tivities due to the oupling for e (see
Fig. 2(a)). This regime is hara terized by maximum values of the
order parameter R(t) (see Fig. 2(b)). Immediately after, the sys-
tem rapidly lusterizes resulting in two nal lusters in whi h agents
share ommon d r values. This situation re e t ommunity stru ture
present in bottlenose dolphins so iety. The distribution of agents in
ea h nal lusters is reported in Table 2.
Table 2. Distribution of agents shown in Fig. 2(a) as resulting
from the simulation of OCR model over the bottlenose dolphins om-
munity for K = 2.2. Elements in parentheses represent the numbers
of agents in ea h nal lusters.
Cluster Agents in ea h 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) 1,3,4,5,9,11,12,13,15,16,17,19,21,22,24,25,29,30,34,35,36,
37,38,39,40,41,43,44,45,46,47,48,50,51,52,53,54,56,59,60,62
As it an be veri ed, the distribution of agents in Table 2. is al-
most the same as obtained in Table 1. Only two agents, the verti es
31(ω = −0.29) and 40(ω = 0.20), have ex hange lusters. In the
dolphin network this two verti es fall in the boundary between the
ommunities of the network (see Fig 1). Therefore, depending on
their natural de ision hanging rate, an joint or leave a parti ular
group. In this ase, the modularity is Q = 0.3799 for this split into
2 lusters.
It is important to stress that, in this region of the oupling
strength, agents in ea h nal group, tend to maintain a syn hro-
nized regime despite their dierent natural de ision hanging rates.
This is due to the strong in uen e of the ommunity stru ture pres-
ents in the network whi h ae t notably the de ision of agents to
join or leave a group. While a subsequent in reases in the value of
the oupling strength for e a ompletely syn hronized regime.
0.5
(a)
0.4
0.3
decision changing rate (dcr)
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
1 −2 −1 0 1 2
10 10 (b) 10 10 10
Order parameter
0.9
0.8
0.7
−2 −1 0 1 2
10 10 10 10 10
time
Fig. 2.De ision dynami s of lusters syn hronization in the OCR model
on the Dolphins Network for k = 2.2 and ω ∈ [−0.5, 0.5].
In the other hand, entrality measure (betweenness) [10℄ for ea h
individual of the network show that verti es 2(ω = −0.5) and 37(ω =
0.17) have high betweenness values. Betweenness is a measure of the
in uen e of individuals in a network over the ow of information
between others. So, this two individuals represent a potentially in-
formation brokers in this dolphin so iety. In Fig. 2, panel (a), we have
represented, in bold markers, the d r variables asso iated to this two
individuals. A ording to their natural de ision hanging rate, ver-
ti es 2 (ω = −0.5) and 37(ω = 0.17) represents more onservative
and exible individuals respe tively.
To test how entral individuals in uen e the other members of
so iety in the de ision-making pro ess, we have onsidered two si-
tuation on erning the vertex 2 (the same analysis an be realized
for the vertex 37).
The rst one is when this entral agent have natural de ision
hanging rate set to zero value (ω(2) ≃ 0) simulating a exible agent.
Results are shown in Fig. 3. As it an be appre iated, two new single
groups formed by verti es 29 and 48 merge. It orrespond to individu-
als with high absolute value of ω : ω(29) = 0.41 and ω(48) = −0.47.
For this split into 4 lusters, the modularity is Q = 0.3822. Details
of that Distribution of agents is reported in Table 3.
Table 3. Distribution of agents shown in Fig. 3(a) as resulting
from the simulation of OCR model over the bottlenose dolphins om-
munity for K = 2.2 and ω(2) = 0.
Cluster Agents in ea h luster
1 (1) 48
2 (21) 2,6,7,8,10,14,18,20,23,26,27,28,32,33,40,42,49,55,57,58,61
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 se ond situation simulates a more exible individual that
tends to anti ipate the others members of so iety, i.e, ω(2) = 0.5,
whi h represents a maximum value of ω . Simulation results are re-
presented in Fig. 4 over 200 time steps.
Table 4. Distribution of agents shown in Fig. 4(a) as resulting
from the simulation of OCR model over the bottlenose dolphins om-
munity for K = 2.2 and ω(2) = 0.5.
0.5
(a)
0.4
0.3
decision changing rate (dcr)
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5 −2 −1 0 1 2
1 10 10 10 10 10
Order parameter
(b)
0.9
0.8
0.7 −2 −1 0 1 2
10 10 10 10 10
time
Fig. 3.De ision dynami s of lusters syn hronization in the OCR model
on the Dolphins Network for k = 2.2 and ω(2) = 0. Central individuals
are represented in bold marker.
Cluster Agents in ea h 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) 1,3,4,5,9,11,12,13,15,16,17,19,21,22,24,25,30,34,35,36,
37,38,39,41,43,44,45,46,47,50,51,52,53,54,56,59,60,62
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 rea hes a transient om-
pletely syn hronized regime (R(t) = 1 in Fig. 4(b)) in whi h all
individuals run a the same d r value, generally the average of all
0.5
(a)
0.4
0.3
opinion changing rate (ocr)
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5 −2 −1 0 1 2
Order parameter
1 10 10 (b) 10 10 10
0.8
−2 −1 0 1 2
10 10 10 10 10
time
Fig. 4.De ision dynami s of lusters syn hronization in the OCR model
on the Dolphins Network for k = 2.2 and ω(2) = 0.5. Central individuals
are represented in bold marker.
the ω . After, we observe that the group ontaining initially the in-
dividual 2 (see Tables 2) is divided into several subgroups. As in
the previous situation, individuals with high absolute value of ω run
alone in single groups. Details of that distribution are reported in
Table 4. Compared with the initial on guration of Table 2, we see
that the group 2 is quite stable. Apart from vertex 40 that leave this
group to join another one (see Table 4). Moreover, the modularity Q
(Q = 0.3189) shows a signi ant de rease, indi ating that the par-
tition obtained, does not orrespond to the natural partition of the
network.
It is important to stress that hanging the distribution of the
natural de ision hanging rate, the evolution of ea h individual an
hange, but qualitatively the behavior of the system is the same.
5 Con lusions
This work provide eviden e that network topology is fundamen-
tally important in de ision-making dynami s allowing individuals to
join or to leave parti ular groups depending on their positions in the
network. On the other hand, the natural de ision hanging rate of
entral individuals is determinate in lusters formation pro ess.
A knowledgements The Spanish Ministerio de Edu a ion y
Cien ia supports this work through grant DPI2006-15346-C03-03
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