Clustering conditions and the cluster formation process in a dynamical model of multidimensional attracting agents

We consider a multiagent clustering model where each agent belongs to a multidimensional space. We investigate its long term behavior, and we prove emergence of clustering behavior in the sense that the velocities of the agents approach asymptotic values, independently of the initial conditions; agents with equal asymptotic velocities are said to belong to the same cluster. We propose a set of relations governing these asymptotic velocities. These results are compared with results obtained earlier for the model with agents belonging to a one-dimensional space and are then explored for the case of an infinite number of agents. For the particular case of a spherically symmetric configuration of an infinite number of agents a rigorous analysis of the relations governing the asymptotic velocities is possible, assuming that a continuity property established for the finite case remains true for the infinite case. This leads to a characterization of the onset of cluster formation in terms of the evolution of the cluster size with varying coupling strength. A remarkable point is that the cluster formation process depends critically on the dimension of the agent state space; considering the cluster size as an order parameter, the cluster formation in the one-dimensional case may be seen as a second-order phase transition, while the multidimensional case is associated with a first-order phase transition. We provide bounds for the critical coupling strength at the onset of the cluster formation, and we illustrate the results with two examples in three dimensions

Finding long and similar parts of trajectories

A natural time-dependent similarity measure for two trajectories is their average distance at corresponding times. We give algorithms for computing the most similar subtrajectories under this measure, assuming the two trajectories are given as two polygonal, possibly self-intersecting lines. When a minimum duration is specified for the subtrajectories, and they must start at exactly corresponding times in the input trajectories, we give a linear-time algorithm for computing the starting time and duration of the most similar subtrajectories. The algorithm is based on a result of independent interest: We present a linear-time algorithm to find, for a piece-wise monotone function, an interval of at least a given length that has minimum average value. When the two subtrajectories can start at different times in the two input trajectories, it appears difficult to give an exact algorithm for the most similar subtrajectories problem, even if the duration of the desired two subtrajectories is fixed to some length. We show that the problem can be solved approximately, and with a performance guarantee. More precisely, we present (1 + e)-approximation algorithms for computing the most similar subtrajectories of two input trajectories for the case where the duration is specified, and also for the case where only a minimum on the duration is specified

Unsupervised landmark analysis for jump detection in molecular dynamics simulations

Molecular dynamics is a versatile and powerful method to study diffusion in solid-state ionic conductors, requiring minimal prior knowledge of equilibrium or transition states of the system's free energy surface. However, the analysis of trajectories for relevant but rare events, such as a jump of the diffusing mobile ion, is still rather cumbersome, requiring prior knowledge of the diffusive process in order to get meaningful results. In this work, we present a novel approach to detect the relevant events in a diffusive system without assuming prior information regarding the underlying process. We start from a projection of the atomic coordinates into a landmark basis to identify the dominant features in a mobile ion's environment. Subsequent clustering in landmark space enables a discretization of any trajectory into a sequence of distinct states. As a final step, the use of the smooth overlap of atomic positions descriptor allows distinguishing between different environments in a straightforward way. We apply this algorithm to ten Li-ionic systems and conduct in-depth analyses of cubic Li$_{7}$La$_{3}$Zr$_{2}$O$_{12}$, tetragonal Li$_{10}$GeP$_{2}$S$_{12}$, and the $\beta$-eucryptite LiAlSiO$_{4}$. We compare our results to existing methods, underscoring strong points, weaknesses, and insights into the diffusive behavior of the ionic conduction in the materials investigated

Fast conformational clustering of extensive molecular dynamics simulation data

We present an unsupervised data processing workflow that is specifically designed to obtain a fast conformational clustering of long molecular dynamics simulation trajectories. In this approach we combine two dimensionality reduction algorithms (cc\_analysis and encodermap) with a density-based spatial clustering algorithm (HDBSCAN). The proposed scheme benefits from the strengths of the three algorithms while avoiding most of the drawbacks of the individual methods. Here the cc\_analysis algorithm is for the first time applied to molecular simulation data. Encodermap complements cc\_analysis by providing an efficient way to process and assign large amounts of data to clusters. The main goal of the procedure is to maximize the number of assigned frames of a given trajectory, while keeping a clear conformational identity of the clusters that are found. In practice we achieve this by using an iterative clustering approach and a tunable root-mean-square-deviation-based criterion in the final cluster assignment. This allows to find clusters of different densities as well as different degrees of structural identity. With the help of four test systems we illustrate the capability and performance of this clustering workflow: wild-type and thermostable mutant of the Trp-cage protein (TC5b and TC10b), NTL9 and Protein B. Each of these systems poses individual challenges to the scheme, which in total give a nice overview of the advantages, as well as potential difficulties that can arise when using the proposed method