# The Influences of Self-Introspection and Credit Evaluation on Self-Organized Flocking
## 2 Models
### Two Models Based on Local Consistency
#### 2.2.2 Two Models  
To fully explore the roles of local consistency in the above two interaction mechanisms, we propose two novel models, the self-introspection model (SIM) and credit evaluation model (CEM), which endow two interaction mechanisms between individuals. These two models both exploit local consistency as the local information for individuals to take action. (1) Self-introspection model (SIM)  
In the original [PERSON] model, the orientation of the individual is obtained by averaging the orientations of all neighbors within the interaction radius. Based on this rule, we introduce the key factor of local consistency and consider a weighted model for adjusting the orientation; that is, individuals need to make real-time judgments based on their alignment with their neighbors (i.e., local consistency) to make suitable decisions. Therefore, we propose the self-introspection model (SIM):  
\[\theta_{i}(t+1)=\theta_{i}(t)+e^{\frac{1}{2}[1-\phi_{i}^{\alpha}(t)]}\big{[} \hat{\theta}_{i}(t)-\theta_{i}(t)\big{]}+\Delta\theta_{i}, \tag{5}\]  
where \(\theta_{i}(t)\) is the orientation of the \(i\)th individual at time \(t\), \(\hat{\theta}_{i}(t)\) is the average orientation of its neighbors, and \(\Delta\theta_{i}\) is the noise randomly distributed in the interval \([-\eta/2,\eta/2]\), whose amplitude is determined by \(\eta>0\). Additionally, \(\phi_{i}(t)\) is the local consistency defined in Equation (4), and \(\alpha\geq 0\) is an adjusted index. It is worth pointing out that if \(\alpha=0\), then the self-introspection model (SIM) degenerates into the original [PERSON] model, while if \(\alpha\rightarrow+\infty\), then this model is a fixed-weight model, and the characteristics of this type of model can be found in [18]. The specific relationship among these orientations in Equation (6) is shown in Figure 1.  
One can conclude from the model that, at the initial stage of flocking where the local consistency is usually poor, individuals must make great efforts to cooperate with the group behavior. But along with the stabilization of the flocking orientation, the changes in individuals' orientations will be gradually decreased until the group moves forward steadily in the same direction. When the group is gradually stablized, smaller changes in orientations are more conducive to the formation of this steady state, while larger changes, at this time, may drive the system to deviate from the steady state. On the other hand, when an individual is a neighbor of others, his or her influences will also vary with the local consistency. In this sense, the local consistency of the individual can be regarded as his or her credibility. An example is shown in Figure 2, where \(j\) and \(k\) are two neighbors of individual \(i\). From the perspective of the degree of alignment within their own neighborhood, namely the local consistency, for individual \(i\), neighbor \(j\) is more reliable than neighbor \(k\) on the orientation. The SIM is equivalent to the traditional Vicsek model with uniform weights, while the CEM uses varying weights. Figure 1: Illustration of the self-introspection model (SIM), where \(R\) represents the interaction radius of the \(i\)th individual, \(\theta_{i}(t)\) is the orientation of the \(i\)th individual at this moment, \(\theta_{i}(t+1)\) is the orientation of the \(i\)th individual at the next moment, and \(e^{\frac{1}{2}[1-\phi_{i}^{\alpha}(t)]}\) is the weight of the difference between the average orientation \(\hat{\theta}_{i}(t)\) and the current orientation \(\theta_{i}(t)\). (2) Credit Evaluation Model (CEM)  
Considering these, we take the local consistency as credibility and integrate it into the original Vicsek model in the form of weight. If the credibility of the neighbor is better, then the weight of the impact on the individual is greater.