# Common Nearest Neighbor Clustering--A Benchmark
## 4 Results
### Core-Set Models of Molecular Dynamics  
MD simulations are not only used to sample and analyze the Boltzmann distribution, but also to extract and analyze the slow dynamic processes of the system. The slow dynamic processes of pseudo-dynamics (i.e., generated by a MCMC algorithm) on the PES in Figure 1 are shown in Figure 8d. The first process \(l_{1}\) is the stationary process and is equal to the Boltzmann distribution. Formally, it is associated to a time scale of infinity. The slowest dynamic process with a finite time scale is \(l_{2}\) and represents the dynamic exchange between the minimum at the bottom and the two minima at the top, that is, across the largest barrier. Its time scale is \(\approx\)70 time steps, where each iteration of the MCMC algorithm is interpreted as a time step. The next slowest process, \(l_{3}\), represents the dynamic exchange between the two minima at the top and is associated to a time scale of \(\approx\)24 time steps. There are several methods known that can extract information from such a trajectory, such as sequence kernels and support vector machines [45; 46; 47; 48; 49]. However, most of these are currently not applied to MD data, as these data are described in terms of non-linear internal coordinates in a high-dimensional conformational space. The most commonly used method to extract the slow dynamic processes from a trajectory are Markov state models [50; 51; 52; 53; 54; 55] (MSM). They represent a discretization of the dynamical propagator [50; 55] and are based on a discretization of the molecular state space into crisp non-overlapping states. The dynamics are approximated as a jump process (Markov process) between pairs of states within a time lag \(\tau\) (Figure 8a), which yields a matrix of pairwise transition probabilities, the transition matrix \(\mathbf{T}(\tau)\). The transition probabilities can be estimated from a trajectory. The transition matrix is however only a model of the true dynamics, and the approximation error depends sensitively on the discretization [56]. Recently, methods to discretize the dynamical propagator with respect to arbitrary ansatz functions have emerged [57; 58; 59; 60; 29]. In core-set models [24; 25; 26; 28], the state space is discretized into core sets, which do not fully partition the state space. Associated to each core set \(\tilde{C}_{i}\) is a committor function \(q_{i}:\Omega\rightarrow[0,1]\), which represents the probability that the dynamic process will reach \(\tilde{C}_{i}\) before it reaches any of the other core sets. Thus, \(q_{i}(\mathbf{x})=1\) if \(\mathbf{x}\in\tilde{C}_{i}\), \(q_{i}(\mathbf{x})=0\) if \(\mathbf{x}\in\tilde{C}_{j\
eq i}\), and \(0<q_{i}(\mathbf{x})<1\) otherwise. If the core sets represent long-lived conformations (which usually coincide with maxima in the Boltzmann distribution), discretizing the dynamical propagator with respect to the associated committor functions yields models with a very small approximation error. The approximation error of a model is usually judged by the drift of the implied time scale \(t_{k}\) as a function of the time lag \(\tau\) (Figure 8b,_c_):  
\[t_{k}(\tau)=-\frac{\tau}{\ln\lvert\lambda_{k}(\tau)\rvert} \tag{4}\]  
where \(\tau\) is the time lag of the model and \(\lambda_{k}(\tau)\) is the eigenvalue associated to the \(k\)th eigenvector of the discretized propagator [51]. If the dynamics were fully Markovian within the chosen discretization, the implied time scale would be constant, that is, independent of \(\tau\). However, even for models with a discretization error, the implied time scales typically reach a plateau region for large \(\tau\). However, large values of \(\tau\) also imply a low time resolution of the model. In general, the smaller the value of \(\tau_{\text{Markov}}\) for which the drift of the implied time scale becomes negligible, the smaller the approximation error. At time lags shorter than this \(\tau_{\text{Markov}}\), the implied time scales are typically underestimated.