  The determination of cluster centers generally depends on the scale that we
use to analyze the data to be clustered. Inappropriate scale usually leads to
unreasonable cluster centers and thus unreasonable results. In this study, we
first consider the similarity of elements in the data as the connectivity of
nodes in an undirected graph, then present the concept of a connection center
and regard it as the cluster center of the data. Based on this definition, the
determination of cluster centers and the assignment of class are very simple,
natural and effective. One more crucial finding is that the cluster centers of
different scales can be obtained easily by the different powers of a similarity
matrix and the change of power from small to large leads to the dynamic
evolution of cluster centers from local (microscopic) to global (microscopic).
Further, in this process of evolution, the number of categories changes
discontinuously, which means that the presented method can automatically skip
the unreasonable number of clusters, suggest appropriate observation scales and
provide corresponding cluster results.
