Discover, Sample and Refine: Exploring Chemistry with Enhanced Sampling Techniques

Over the last few decades enhanced sampling methods have made great strides. Here, we exploit this progress and propose a modular workflow for blind reaction discovery and characterization of reaction paths. Central to our strategy is the use of the recently developed explore variant of the on-the-fly probability enhanced sampling method. Like metadynamics, this method is based on the identification of appropriate collective variables. Our first step is the discovery of new chemical reactions and it is performed biasing a one dimensional collective variable derived from spectral graph theory. Once new reaction pathways are detected, we construct ad-hoc tailored neural-network based collective variables to improve sampling of specific reactions and finally we refine the results using free energy perturbation theory. Our workflow has been successfully applied to both intramolecular and intermolecular reactions. Without any chemical hypothesis, we discovered several possible products, computed the free energy surface at semiempirical level, and finally refined it with a more accurate Hamiltonian. Our workflow requires minimal user input, and thanks to its modularity and flexibility, can extend the scope of ab initio molecular dynamics for the exploration and characterization of reaction space.

Geary’s c and Spectral Graph Theory

Spatial autocorrelation, of which Geary’s c has traditionally been a popular measure, is fundamental to spatial science. This paper provides a new perspective on Geary’s c. We discuss this using concepts from spectral graph theory/linear algebraic graph theory. More precisely, we provide three types of representations for it: (a) graph Laplacian representation, (b) graph Fourier transform representation, and (c) Pearson’s correlation coefficient representation. Subsequently, we illustrate that the spatial autocorrelation measured by Geary’s c is positive (resp. negative) if spatially smoother (resp. less smooth) graph Laplacian eigenvectors are dominant. Finally, based on our analysis, we provide a recommendation for applied studies.

Spectral graph theory of brain oscillations -- revisited and improved

Mathematical modeling of the relationship between the functional activity and the structural wiring of the brain has largely been undertaken using non-linear and biophysically detailed mathematical models with regionally varying parameters. While this approach provides us a rich repertoire of multistable dynamics that can be displayed by the brain, it is computationally demanding. Moreover, although neuronal dynamics at the microscopic level are nonlinear and chaotic, it is unclear if such detailed nonlinear models are required to capture the emergent meso- (regional population ensemble) and macro-scale (whole brain) behavior, which is largely deterministic and reproducible across individuals. Indeed, recent modeling effort based on spectral graph theory has shown that an analytical model without regionally varying parameters can capture the empirical magnetoencephalography frequency spectra and the spatial patterns of the alpha and beta frequency bands accurately. In this work, we demonstrate an improved hierarchical, linearized, and analytic spectral graph theory-based model that can capture the frequency spectra obtained from magnetoencephalography recordings of resting healthy subjects. We reformulated the spectral graph theory model in line with classical neural mass models, therefore providing more biologically interpretable parameters, especially at the local scale. We demonstrated that this model performs better than the original model when comparing the spectral correlation of modeled frequency spectra and that obtained from the magnetoencephalography recordings. This model also performs equally well in predicting the spatial patterns of the empirical alpha and beta frequency bands.

The Network Topology Metrics Contributing to Local-Area Frequency Stability in Power System Networks

The power system network topology influences the system frequency response to power imbalance disturbances. Here, the objective is to find the network metric(s) contributing to frequency transient stability. The graph Laplacians of six 4-node network topologies are analysed using Spectral Graph Theory. For homogeneous network connections, we show that the node degree measure indicates node robustness. Based on these analytical results, the investigation expands to a 10-node network topology consisting of two clusters, which provide further insight into the spectral results. The research then involves a simulation of a power imbalance disturbance on three 20-node networks with different topologies based on node degree, where we link the node degree measure to imbalance disturbance propagation through Wave Theory. The results provide an intuitive understanding of the impact of network topology on power system frequency stability. The analytical and simulation results indicate that a node’s sensitivity to disturbances is partially due to its node degree, reactance from disturbance location, and the link it has to other higher degree nodes (hierarchical position in network topology). Testing of the analytical and simulation results takes place on the nonhomogeneous IEEE-14 bus and IEEE-39 bus networks. These results provide insights into optimal inertia placement to improve the frequency robustness of low-inertia power systems. The network topology, considering node degrees, influences the speed at which the disturbance impact propagates from the disturbance location and how fast-standing waves form. The topology thus contributes to how fast the energy in a disturbance dissipates to zero.

Hyper-Wiener index for fuzzy graph and its application in share market

Topological indices have an important role in molecular chemistry, network theory, spectral graph theory and several physical worlds. Most of the topological indices are defined in a crisp graph. As fuzzy graphs are more generalization of crisp graphs, those indices have more application in fuzzy graphs also. In this article, we introduced the fuzzy hyper-Wiener index (FHWI) and studied this index for various fuzzy graphs like path, cycle, star, etc and provided some interesting bounds of FHWI for that fuzzy graph. A lower bound of FHWI is established for n-vertex connected fuzzy graph depending on strength of a strong edges. A relation between FHWI of a tree and its maximum spanning tree is established and this index is calculated for the saturated cycle. Also, at the end of the article, an application in the share market of this index is presented.