DOI: 10.7906/indecs.11.3.1 Regular article

In a social network, individual opinions and interpersonal relationships always interact and coevolve. This continuously leads to self-organization of opinion clusters in the whole network. In this article we study how the coevolution on the two kinds of complex networks and the self-organization of opinion clusters are differently affected by the dynamic parameters, the structural parameters and the propagating parameters. It is found that the two dynamic parameters are homogeneous bringing about the strong and weak relations, while the two structural parameters are heterogeneous having equivalent relations. Moreover, the impact of the propagating parameter has been found only above its threshold.


INTRODUCTION
Opinions which individuals hold and relationships among them always interplay and interact, which can lead to self-organization of opinion clusters.The definition of opinion clusters has two: one is the sets in which individuals have the same opinion and the association among them [1,2]; the other is the sets in which individuals also have the same opinion, but with or without association among them is no limited [3,4].The latter is employed in this article.
In social systems, the formation of opinion clusters is affected by many factors, such as diffusional dynamic parameters, network structural parameters and different network characteristics.As it is very difficult to study the social systems for the complexity of them, many researchers recently investigated the social system by networks [5,6].Many real systems, such as social systems, ecological systems, and cellular systems, can be represented as networks, in which nodes denote the objects of interest and edges that connect nodes describe the relationships between them [7,8].However, current researches for opinion clustering detection focus more on finding algorithms that can identify opinion clusters in all contexts [6][7][8], than on the effects of different factors on opinion clustering in the same context.
In this study, we simulate information propagation in different conditions on networks to result in self-organization of opinion clusters, in order to ascertain the effects of dynamic parameters, structural parameters and networks characteristics on it.

THE MODEL
Many discoveries [9][10][11] show that a number of large-scale complex networks, including the electric power grid for Southern California, the network of movie-actor collaborations, and the neuronal network of the worm Caenorhabditis elegans, are scale-free and small world.The Watts-Strogatz (WS) small world model exhibits a high degree of clustering as in the regular network and a small average distance between vertices as in the random network.The Barabási-Albert (BA) model suggests that the two main ingredients of self-organization of a network in a scale-free structure are growth and preferential attachment [10].
Let us consider opinion synchronous diffusion 1 on WS and BA networks.

OPINION MODEL
Each of N vertices denotes an individual and each of M links denotes a relationship between two individuals in the network.We consider O i possible opinions of which every individual must hold one, and two relationships (called +1 and -1) denote positive sentiment (friends) and negative sentiment (enmities), according to balance theory.Opinion model in this study is the majority-friends-rule model extending the majority-rule model [12,13].It assumes that individuals preferentially follow the friends instead of following the crowd (the majority-rule 2 ) in their opinion update.In each step, every vertex has the same opportunity to update its opinion or relations by the following rules: 1) majority friends' preference (MFP): with probability P, the focal individual accepts the specific opinion held by a majority of its friends (i.e., the opinion is the one or one of that has the largest supporter among the friends).If the specific opinion is more than one, random one of them is chosen.We call this process P action, 2) cognitive consistence (CC) [14]: with probability 1 -P, the focal individual keeps his opinion unchanged and updates (keeps or flips) its relationship.It will keep the sign of edge if the focal pair is cognitive consistence: holding the same opinion with positive (+1) relation or different opinions with the negative (-1) relation.On the country, an edge will flip the sign if the focal pair is cognitive inconsistence: the same opinion hold by focal pair individuals with negative (-1) relation or vice versa.When flipping is activated, with likelihood Q, the focal individual will flip all of links to neighbors, or with likelihood 1 -Q, it will flip random one of links to neighbors.We call these processes Q action and 1 -Q action, 3) repeating previous two steps, the system will converge to consensus state.The consensus state has two sub-states: one is opinion consensus sub-state; another is relation consensus sub-state.They respectively represents that all opinions and all relations currently hold by all individuals do not change over time.After reaching the two sub-states, the system can only reach consensus state.It also claims that the coevolution between opinions and relations has been completed.The particular algorithm is shown in Figure 1.

THE NUMBER OF OPINION CLUSTERS
It is similar on the self-organization of opinion clusters of BA (Figs 2a) and 2c)) and WS network (Figs 2b) and 2d)), because scale-free networks are also small-world networks [14], because (i) they have clustering coefficients much larger than random networks [11] and (ii) their diameter increases logarithmically with the number of vertices N [9].The two dynamical parameters are homogenous to the process (Figs 2a) and 2b)).Both the P and the Q promote to self-organization and deduce the number of opinion clusters with the increasing of them.While the effect of P is larger than that of Q for the Q action occurs on the condition probability of 1 -P.As is shown in equation However, the two structural parameters are heterogeneous in that the increasing of average degree k hinders the self-organization of opinion clusters.It is in contrast with the fact that increasing P ne accelerates it.That indicates that the increase of edge density is advantageous to the density of the opinion clusters if the increases are of opposite signs.If P ne = 1, opinion clustering cannot proceed, the number of opinion clusters will not cannot be inferred using the rule of MFP as it is almost invalidated, thus an individual in such a network has few friends.

THE RATIO OF SURVIVAL OPINION CLUSTER
The ratio the survival opinion cluster is denoted as R S , R S = O f /O i where O f is the number of survival opinion cluster and O i the number of initial opinion cluster.As shown in Figure 3, the difference of effects between BA and WS network on R S is quite obvious.Under the same If O i  0,1N then R S will reach 1 (thus all the opinion clusters will survive) whatever other parameters are.

THE SCALE OF OPINION CLUSTERS
In this section we analyze the effects of three types of factors on the self-organization of the opinion clusters, measured by the top 10 of opinion clusters' (abbr.top 10) sizes after coevolution.The average size of an opinion cluster is inversely proportional to the number of opinion clusters and is analysed further in the text.Figure 4 shows that top 10 sizes in BA network is always larger than that in WS network in most cases at the same condition.This is maybe due to that degree distribution of BA network is power-law and tendency to form bigger community (community is always opinion cluster under the rule of MFP, though not vice versa) than that of WS network which has average degree distribution.
The effect of structural parameters is that the sizes of top 10 are proportional to P and Q for constant values of other parameters.While the effect degree of Q is smaller than that of P, as shown in Figure 4 (three curves in Fig. 4c) and Fig. 4d) are mutually closer than the corresponding curves in Fig. 4a) and Fig. 4b)).The effect indicates that the bigger the P value, the greater the probability that each individual supports the popular opinion (the opinion which most of friends holding).It is a benefit that advantageous opinion clusters (top 10) enlarge advantages (i.e.top 10 have more supporters).The effect of Q is similar to that of P.
If Q increases, the probability of advantageous individuals getting more friends also increases.It also results in the larger sizes of top 10.Whatever the way the Q action occurs in the condition probability 1 -P, the effect of Q is usually smaller than that of P (except if P is relativelly very small and close to zero).
It is clear that structural parameters effect on the sizes of top 10: the sizes are proportional to k value and inversely proportional to P ne .As k value rises, the clustering coefficients in BA and WS network both increase.It benefits advantageous opinion to increase supporter microscopically, thus larger sizes of opinion cluster form macroscopically.
On the contrary, with the increase of P ne value, each individual will decrease its friends in microscopical scales, thus it also leads to advantageous opinion clusters decrease sizes in macroscopical scales.

Figure 2 .
Figure 2. The effects of dynamical parameters, structural parameters and network types on the self-organization opinion clusters during evolution, with N = 1000.In a) and b) k = 4 and P ne = 0,5, while in c) and d) P = 0,5 and Q = 0,5.

Figure 3 .
Figure 3.The effects of the types of networks and the number of initial opinions on the ratio of survival opinion cluster.condition, R S in WS network is larger than that in BA network.As O i decreases, O f always increases both in WS and BA networks.

Figure 4 .
Figure 4.The effects of dynamical parameters on top 10 sizes.The rank of top 10 sizes versus the fraction of top 10 with a) and b) P equal to 0,1, 0,5 and 0,9 for constant Q = 0,5, c) and d) Q equal to 0, 0,5 and 1 for constant P = 0,5.In all graphs N = 1000, k = 4 and P ne = 0,5.

Figure 5 .
Figure 5.The effects of dynamical parameters on top 10 sizes.The rank of top 10 sizes versus the fraction of top 10 for a) and b) P ne = 0,5 and k values equal to 4, 6, 8 and 10, c) and d) k = 4 for P ne values equal to 0, 0,5 and 1.In all graphs N = 1000, P = 0,5 and Q = 0,5.
The two networks (BA and WS) which have same parameters and scopes of parameters.The total number (N) of vertices is fixed (N = 1000, O i = 100).There are three types of variables: structural parameters, dynamical parameters and diffusive parameter.Structural parameters include average degree k  4, 6, 8, 10, as well as a proportion of negative edge P ne  0, 1.Dynamical parameters include the probability of opinions propagation P  (0, 1), as well as the likelihood of relations evolution, Q  0, 1.There is one diffusive parameter, the initial number of opinion clusters, O i  100, 200, 290, 366, 433.The final number of opinion clusters after evolution is O f .
Figure 1.Main algorithm of opinions diffusion and relations evolution.