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 Artitial 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, ree 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 amplies 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 inuen 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℄ dene 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 inuen 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 dened 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 identied 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 dene 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 ree 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 veried, 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 inuen 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 inuen 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 inuen 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 onguration 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. 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