The density peaks clustering (DPC) is a clustering method proposed by Rodriguez and Laio (Science, 2014), which sets up a decision graph to identify the cluster centers of data points. Because the improper selection of its parameter cut-off distance will lead to the wrong selection of initial cluster centers with no corrective actions in the subsequent assignment process, DPC may not identify cluster centers with different densities accurately. Especially, all cluster centers are settled as soon as they are detected, after which the DPC simply assigns each point to the same cluster as its nearest neighbor of higher density. This tends to cause the erroneous assignments of data and thus degrade the efficiency of clustering. In this paper, we propose a robust clustering method which establishes a symmetric neighborhood graph over all data points, based on the k -nearest neighbors and reverse k-nearest neighbors of each point. In order to distinguish the density peaks from all data points, local densities of each point are calculated using the reverse k-nearest neighbors. After that, initial centers for clusters are estimated over the peaks and similar clusters are aggregated on the symmetric neighborhood graph, which ends up with every point being successfully assigned to a cluster. To testify the efficiency of the new clustering method, numerical experiments and comparison works have been done on a variety of artificial and real data sets for clustering.

Classification of asynchronous elementary cellular automata (AECAs) was explored in the first place by Fates et al. (Complex Systems, 2004) who employed the asymptotic density of cells as a key metric to measure their robustness to stochastic transitions. Unfortunately, the asymptotic density seems unable to distinguish the robustnesses of all AECAs. In this paper, we put forward a method that goes one step further via adopting a metric entropy (Martin, Complex Systems, 2000), with the aim of measuring the asymptotic mean entropy of local pattern distribution in the cell space of any AECA. Numerical experiments demonstrate that such an entropy-based measure can actually facilitate a complete classification of the robustnesses of all AECA models, even when all local patterns are restricted to length 1. To gain more insights into the complexity concerning the forward evolution of all AECAs, we consider another entropy defined in the form of Kolmogorov–Sinai entropy and conduct preliminary experiments on classifying their uncertainties measured in terms of the proposed entropy. The results reveal that AECAs with low uncertainty tend to converge remarkably faster than models with high uncertainty.