This paper defines two different ways of the process of the opinion formation and focuses on the effect of the proportion of the two strategies of process and the structure of the network. A multiopinion model is proposed in this paper, which includes two strategies of opinion formation. At the first part, the change of the structured network and the change of the single node are used as two strategies of the process of the opinion formation. We focus on how the proportion of the two strategies can affect the outcome of the process. At the second part, as the proportion of the two strategies is fixed, the edges are dense in the community and sparse outside. Thus we can construct a bifurcation diagram to be verified through experimental study. The phase transition is studied in the network which contains more than four opinions and two strategies of process. Our results show that the size of the group and the dense of edges are the two important features for the process of opinion formation.
In the field of social networks analysis, the study of opinions dynamics has attracted a growing amount of work [
One of the most important properties for the social networks is the community structure [
In order to understand the effect of the community structure and the different strategies of opinion process in the opinions dynamics, for simplicity, a multiopinion model is proposed in this paper, including two strategies of opinion formation. It does not aim at an exact description of reality. However, it focuses on discovering some essential and fundamental features of an otherwise very complex and multiple phenomena by doing some crude approximations. Therefore, we use Monte Carlo method to simulate the reality and determine the sensitivity of the model in the different situation.
In this paper, to understand the mechanism of the community structure in the opinions dynamics, we focus on the following four questions.
How can the different proportion of the process of the opinion formation affect the dynamic of the model? At which proportion the model is most sensitive to the change of the proportion?
How can the size of the community contribute to the outcome of the model when the proportion of the process is fixed?
Could the community structure affect the evolution of the network?
How can different kinds of the noise contribute to the evolution of the network?
In the first part of this paper, we investigate the outcome of the model with different ways of the process of the opinion formation. The first way of transition of the opinion is the change of the structure of the network and the second way is the intersecting opinion of the different points in the network. The different proportion of the ways of processing will lead to the different outcome of the dynamic model.
If the model only uses the first way, the opinion of every node will not change, while the structure of the network will change. Hence, at the final phase of the processing, the entire network will evolute into several small communities and there should be no intersects between two different networks.
If the model only uses the second way, the opinion of different nodes will change while the structure will remain the same. In this situation, the model will develop into single network where the structure remains the same.
First of all, we define the term “win,” that is, if one of the several opinions have occupied more than the 95% of nodes in the network in less than 1000 steps. Using Monte Carlo method, we simulate the situation for 5000 times each for its first 1000 steps. We count the probability of win for every 5% change of the proportion of the two strategies.
In the second part of this paper, we focus on the last three questions. In the same way, we use Monte Carlo method to simulate the different situation. At first, we define a small community in the network of four different opinions. Then we change the size of the community and find out the relation of the size and the probability for the community to occupy the entire network. Secondly, we fix the size of the community and change the structure of the community. And we discover the relation of the structure of the community and the outcome.
At last, we add two different noises into the model. The first kind of noise is the white noise and the second kind of noise is the linearly decreasing noise. Changing the strength of the noise, we do the same study listed above. Therefore, we can find the role that the noises play in the evolution.
Table
Symbol table.
Symbol  Description 


The number of nodes in the entire network 

The number of opinions 

The number of nodes of the small community 

The average number of edges for every node 

The number of edges for the nodes in the small community 

The proportion of the first way of the process 

Time (or the number of the steps) 

The opinion of node 

The different opinion 

The number of nodes that have the opinion 

The number of nodes which connect to the node 

The probability for small community to win 
In the first way of the process, the opinion of every point will not change, while the structure of the network will change in every step. Hence, just like Figure
Things of one kind come together.
The second way of process is the opinions intersected with the different points in the network. We can see from Figure
One takes the behavior of one’s company.
Determine the probability of the opinion of node
At the second part, we consider a network with
Then we fix the proportion of two ways of the process and change the size of the small community.
In the following model, we consider
First, we consider the distant noise.
First, we determine
Hence, the
Consider the probability of the opinion of node
Second, we consider the linear increasing noise.
We determine
Hence, although the exception of the noise is still the same, the variance is depending on
Then, we consider
In order to determine how the proportion of two ways can affect the model, we first stimulate it without noise. We do experiment every 5% from 0% to 100%.
We consider the situation where the
According to Figure
The
We do some preprocessing to choose the suitable noise for the model. Since
From Figure
The
We do the experiments when
We consider
In the linear increasing noise model, we add the noise that increases with the time. It is reasonable because it is our common sense that as the time goes by, it is more difficult for people to have more information about the entire situation and hence they cannot make reasonable decision.
According to Figure
The
In order to compare with the community with a different structure, we first study the small community in the random network. First, we put 1000 nodes and randomly add 5000 nodes into the network so that
As we can see from Figure
The
Compared with the formal experiment (the model that only contains two opinions), the probability does not fall dramatically. When the rate of nodes of small community to other community is near 90%, the probability for win is near 12%. However, in the two opinions model, when the rate of size of the small community to others is 90%, the probability for win is 0. This phenomenon reflects that, in the model that contains more opinions, the influence of
Then we add noise into the model. When the noise is strong, the probability for every opinion to win is very low. Therefore, in this experiment, we fix the noise with 5%. The probability to win decreases with the coming of noise. As Figure
The
We determine
In the last experiment, we find that when the
The
In conclusion, when there is no noise, the cohesion also can contribute to the probability for win.
In Figure
The
Compared with another formal study, the model in this paper focused on the effect of the proportion of two strategies of the opinion formation. Furthermore, we add five different opinions in the model which made the model more complex and more close to the reality. In the experiments, we found the following.
The point of phase transition is near 70% in the model without noise. Near the proportion of 70%, the probability is strongly sensitive to the change of the proportion of two strategies.
When there are more kinds of opinions in the network, the probability for small community to win is less sensitive to the change of its size compared with the model with only two opinions.
The density of edges in the small community can contribute to the probability for it to win. However, it also contributes less compared with the model with only two kinds of opinions.
From the last two remarks, we find that the network with more kinds of opinions is less sensitive to the change of the structure. Hence, it is more stable.