## Appendix A Algorithm Description of AP Clustering and Its Flow Chart  
AP clustering is a clustering algorithm based on the information propagation between data points. Unlike other clustering algorithms, AP clustering does not require a prespecified number of clusters, and the chosen cluster centers are actual points within the data set. The algorithm operates by using an input similarity matrix \(S\) derived from the data, where \(S(i,v)\) represents the suitability of data point \(v\) as a potential cluster center for data point \(i\). During the iterative process, the algorithm propagates two types of information, responsibility and availability, as depicted in Figure 11. In Figure 11, each \(\delta_{v}\) is represented by a  
sibility \(r(i,v)\) is conveyed from data point \(i\) to exemplar \(v\), reflecting the degree to which \(v\) is a credible cluster center for \(i\), while taking into account other exemplars. Availability \(a(i,v)\) is conveyed from exemplar \(v\) to data point \(i\), representing the likelihood that \(v\) will be selected as the cluster center for \(i\), considering the support from other data points for \(v\)[28]. The AP algorithm iteratively updates the responsibility and availability for each data point, with these values collectively determining whether a sample can become an exemplar, thereby leading to the final clustering results. Assume that \(\tilde{E}=\{\tilde{E}_{1},\cdots,\tilde{E}_{Z}\}\) is a data set, \(\tilde{E}_{i}\in R^{n}(i=1,\cdots,Z)\). The procedure of AP clustering is described as given below, and its process diagram is displayed in Figure A3. Figure A3: Flow chart of the AP clustering algorithm. **Step 1**: Calculate the similarity matrix \(S\) of the data pair. The similarity between any two sample points \(\hat{E}_{i}\) and \(\hat{E}_{v}\) is measured by the negative Euclidean distance: \(s(i,\
u)=-\big{\|}\hat{E}_{i}-\hat{E}_{v}\big{\|}^{2}\). Then, the similarity matrix \(S\) is normalized as in (A1)  
\[S=\frac{S-\min(\min(S))}{\max(\max(S)-\min(\min(S)))}.\] (A1)  
**Step 2**: Set the maximum number of iteration \(T\), damping factor \(\lambda\), and reference degree \(p(\
u)=s(\
u,\
u)=median(s(i,\
u))\). **Step 3**: Update the responsibility matrix, and the update rule is given as follows:  
\[r(i,\
u)\gets s(i,\
u)-\max_{\
u^{\prime}\
eq\
u}(a(i,\
u^{\prime}))+s(i,\
u^{\prime}).\] (A2)  
\[r(\
u,\
u)=p(\
u)-\max_{j\
eq\
u}\{a(\
u,j)+s(\
u,j)\}.\] (A3)  
Using damping factor \(\lambda\), the responsibility matrix \(V^{(\
u)}\) at the \(k\)th iteration is updated as in (A4):  
\[V^{(k)}=(1-\lambda)V^{(k)}+\lambda V^{(k-1)}.\] (A4)  
**Step 4**: Update the availability matrix, and the update rules are given as follows:  
\[a(i,\
u)\leftarrow\min\{0,r(\
u,\
u)+\sum_{i^{\prime}\
otin\{i,\
u\}}\max(0,r (i^{\prime},\
u))\}.\] (A5)  
\[a(\
u,\
u)\leftarrow\sum_{i^{\prime}\
eq\
u}\max(0,r(i^{\prime},\
u)).\] (A6)  
Using damping factor \(\lambda\), the availability matrix \(A^{(j)}\) at the \(j\)th iteration is updated as in (A7):  
\[A^{(j)}=(1-\lambda)A^{(j)}+\lambda A^{(j-1)}.\] (A7)  
**Step 5**: The decision matrix is obtained by summing the availability matrix and the responsibility matrix. Find the point in the decision matrix where the diagonal is greater than 0 as the cluster representative center. The sample points are classified according to the distance between the sample points and the cluster representative points. **Step 6**: Determine the number of iterations and the iteration stop condition. The current iteration number has reached the set maximum iteration number T, or the current iteration number is less than T, but the change in its class representative center after several iterations is less than a small value, the AP clustering algorithm ends the iteration, and we assign the data points to the corresponding cluster centers. Although the AP clustering algorithm does not require specifying the number of clusters explicitly, there are some key clustering parameters, namely, the reference value \(P\) and the damping factor \(\lambda\)[29].