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In this study, we employ the Sachs graph theory to formulate the conduction properties of a single-molecular junction consisting of a molecule in which one carbon atom of an alternant hydrocarbon is replaced with a heteroatom. The derived formula includes odd and even powers of the adjacency matrix, unlike the graph of the parental structure. These powers correspond to odd- and even-length walks. Furthermore, because the heteroatom is represented as a self-loop of unit length in the graph, an odd number of passes of the self-loop will change the parity of the length of the walk. To confirm the aforementioned effects of heteroatoms on conduction in an actual sample, the conduction behavior of meta-connected molecular junctions consisting of a heterocyclic six-membered ring, whose conductive properties have already been experimentally determined, was analyzed based on the enumerated number of walks. [ABSTRACT FROM AUTHOR]

We study flat bands of periodic graphs in a Euclidean space. These are infinitely degenerate eigenvalues of the corresponding adjacency matrix, with eigenvectors of compact support. We provide some optimal recipes to generate desired bands and some sufficient conditions for a graph to have flat bands, we characterize the set of flat bands whose eigenvectors occupy a single cell, and we compute the list of such bands for small cells. We next discuss the stability and rarity of flat bands in special cases. Additional folklore results are proved, and many questions are still open. [ABSTRACT FROM AUTHOR]

The brain is a complex network, and diseases can alter its structures and connections between regions. Therefore, we can try to formalize the action of diseases by using operators acting on the brain network. Here, we propose a conceptual model of the brain, seen as a multilayer network, whose intra-lobe interactions are formalized as the diagonal blocks of an adjacency matrix. We propose a general and abstract definition of disease as an operator altering the weights of the connections between neural agglomerates, that is, the elements of the brain matrix. As models, we consider examples from three neurological disorders: epilepsy, Alzheimer–Perusini's disease, and schizophrenia. The alteration of neural connections can be seen as alterations of communication pathways, and thus, they can be described with a new channel model. [ABSTRACT FROM AUTHOR]

Reservoir computing is a machine learning framework that has been shown to be able to replicate the chaotic attractor, including the fractal dimension and the entire Lyapunov spectrum, of the dynamical system on which it is trained. We quantitatively relate the generalized synchronization dynamics of a driven reservoir during the training stage to the performance of the trained reservoir computer at the attractor reconstruction task. We show that, in order to obtain successful attractor reconstruction and Lyapunov spectrum estimation, the maximal conditional Lyapunov exponent of the driven reservoir must be significantly more negative than the most negative Lyapunov exponent of the target system. We also find that the maximal conditional Lyapunov exponent of the reservoir depends strongly on the spectral radius of the reservoir adjacency matrix; therefore, for attractor reconstruction and Lyapunov spectrum estimation, small spectral radius reservoir computers perform better in general. Our arguments are supported by numerical examples on well-known chaotic systems. [ABSTRACT FROM AUTHOR]

Energy is an important network indicator defined by the eigenvalues of an adjacency matrix that includes the neighbor information for each node. This article expands the definition of network energy to include higher-order information between nodes. We use resistance distances to characterize the distances between nodes and order complexes to extract higher-order information. Topological energy (T E), defined by the resistance distance and order complex, reveals the characteristics of the network structure from multiple scales. In particular, calculations show that the topological energy can be used to distinguish graphs with the same spectrum well. In addition, topological energy is robust, and small random perturbations of edges do not significantly affect the T E values. Finally, we find that the energy curve of the real network is significantly different from that of the random graph, thus showing that T E can be used to distinguish the network structure well. This study shows that T E is an indicator that distinguishes the structure of a network and has some potential applications for real-world problems. [ABSTRACT FROM AUTHOR]

Networked dynamical systems are common throughout science in engineering; e.g., biological networks, reaction networks, power systems, and the like. For many such systems, nonlinearity drives populations of identical (or near-identical) units to exhibit a wide range of nontrivial behaviors, such as the emergence of coherent structures (e.g., waves and patterns) or otherwise notable dynamics (e.g., synchrony and chaos). In this work, we seek to infer (i) the intrinsic physics of a base unit of a population, (ii) the underlying graphical structure shared between units, and (iii) the coupling physics of a given networked dynamical system given observations of nodal states. These tasks are formulated around the notion of the Universal Differential Equation, whereby unknown dynamical systems can be approximated with neural networks, mathematical terms known a priori (albeit with unknown parameterizations), or combinations of the two. We demonstrate the value of these inference tasks by investigating not only future state predictions but also the inference of system behavior on varied network topologies. The effectiveness and utility of these methods are shown with their application to canonical networked nonlinear coupled oscillators. [ABSTRACT FROM AUTHOR]

Complex network approaches have attracted a growing interest in the analysis of nonlinear time series. Among other reconstruction methods, it has been shown that the recurrence plot can be used as the adjacency matrix for recurrence networks, expanding the applications of the already successful recurrence analysis. We study here the potential benefits of a directed formulation of recurrence networks through a simple modification of the recurrence plot. As it is directly related to the recurrence analysis field, this approach takes advantage of the progresses regarding the creation and treatment of the recurrence plot. It appears that directed recurrence networks provide more robust results than their undirected counterpart for transitions detection as well as temporal patterns discovery and clustering. New applications for network cleaning and data modeling are also demonstrated. [ABSTRACT FROM AUTHOR]

The flywheel energy storage (FES) array system plays an important role in smoothing the power output of wind farms. Therefore, how to allocate the total charging and discharging power of wind farms to individual FES system (FESS) units has long been a research hotspot. We propose a hierarchical coordinated control strategy applying the concept of the state of energy (SOE) of FESS units. Energy allocation is achieved by investigating the consensus algorithm of the SOE for both an undirected graph and unbalanced directed graph. Then, we propose another approach to convert the adjacency matrix into a column stochastic matrix using the idea of the PageRank algorithm for an unbalanced directed graph and achieve an SOE consensus of FESS units using the power iteration method. Simulation results demonstrate that the proposed algorithm is effective. [ABSTRACT FROM AUTHOR]

We perform a Koopman spectral analysis of elementary cellular automata (ECA). By lifting the system dynamics using a one-hot representation of the system state, we derive a matrix representation of the Koopman operator as the transpose of the adjacency matrix of the state-transition network. The Koopman eigenvalues are either zero or on the unit circle in the complex plane, and the associated Koopman eigenfunctions can be explicitly constructed. From the Koopman eigenvalues, we can judge the reversibility, determine the number of connected components in the state-transition network, evaluate the period of asymptotic orbits, and derive the conserved quantities for each system. We numerically calculate the Koopman eigenvalues of all rules of ECA on a one-dimensional lattice of 13 cells with periodic boundary conditions. It is shown that the spectral properties of the Koopman operator reflect Wolfram's classification of ECA. [ABSTRACT FROM AUTHOR]

This article, titled "Modeling a neurological disorder as the result of an operator acting on the brain: A first sketch based on network channel modeling," was published in the journal Chaos. The article acknowledges errors in the first two lines of the final section of Equation (19) and in the designators of Fig. 4. These errors have been corrected in the online versions of the article as of May 28, 2024. The article presents a simple brain network and discusses the modification of this network due to the K-operator, which changes the thickness of the edges in the lobes and modifies the weights of the corresponding elements inside the brain-adjacency matrix. The authors of the article are Maria Mannone, Peppino Fazio, and Norbert Marwan. [Extracted from the article]

Molecular dynamics simulation produces three-dimensional data on molecular structures. The classification of molecular structure is an important task. Conventionally, various order parameters are used to classify different structures of liquid and crystal. Recently, machine learning (ML) methods have been proposed based on order parameters to find optimal choices or use them as input features of neural networks. Conventional ML methods still require manual operation, such as calculating the conventional order parameters and manipulating data to impose rotational/translational invariance. Conversely, deep learning models that satisfy invariance are useful because they can automatically learn and classify three-dimensional structural features. However, in addition to the difficulty of making the learned features explainable, deep learning models require information on large structures for highly accurate classification, making it difficult to use the obtained parameters for structural analysis. In this work, we apply two types of graph neural network models, the graph convolutional network (GCN) and the tensor embedded atom network (TeaNet), to classify the structures of Lennard-Jones (LJ) systems and water systems. Both models satisfy invariance, while GCN uses only length information between nodes. TeaNet uses length and orientation information between nodes and edges, allowing it to recognize molecular geometry efficiently. TeaNet achieved a highly accurate classification with an extremely small molecular structure, i.e., when the number of input molecules is 17 for the LJ system and 9 for the water system, the accuracy is 98.9% and 99.8%, respectively. This is an advantage of our method over conventional order parameters and ML methods such as GCN, which require a large molecular structure or the information of wider area neighbors. Furthermore, we verified that TeaNet could build novel order parameters without manual operation. Because TeaNet can recognize extremely small local structures with high accuracy, all structures can be mapped to a low-dimensional parameter space that can explain structural features. TeaNet offers an alternative to conventional order parameters because of its novelty. [ABSTRACT FROM AUTHOR]

The scattering functions of nine generations of ideal tri-functional comb and dendrimer polymers are computed by novel graph techniques. The properties of polymers having from 9 to 3069 branches are explored. The g-ratios and the scattering functions indicate that as the number of branches increases, comb polymers behave more and more like linear polymers with half the number of branches, whereas dendrimers become more like spherical objects. [ABSTRACT FROM AUTHOR]

Contemporary graph learning algorithms are not well-suited for large molecules since they do not consider the hierarchical interactions among the atoms, which are essential to determining the molecular properties of macromolecules. In this work, we propose Multiresolution Graph Transformers (MGT), the first graph transformer architecture that can learn to represent large molecules at multiple scales. MGT can learn to produce representations for the atoms and group them into meaningful functional groups or repeating units. We also introduce Wavelet Positional Encoding (WavePE), a new positional encoding method that can guarantee localization in both spectral and spatial domains. Our proposed model achieves competitive results on three macromolecule datasets consisting of polymers, peptides, and protein-ligand complexes, along with one drug-like molecule dataset. Significantly, our model outperforms other state-of-the-art methods and achieves chemical accuracy in estimating molecular properties (e.g., highest occupied molecular orbital, lowest unoccupied molecular orbital, and their gap) calculated by Density Functional Theory in the polymers dataset. Furthermore, the visualizations, including clustering results on macromolecules and low-dimensional spaces of their representations, demonstrate the capability of our methodology in learning to represent long-range and hierarchical structures. Our PyTorch implementation is publicly available at https://github.com/HySonLab/Multires-Graph-Transformer. [ABSTRACT FROM AUTHOR]

Identifying a reduced set of collective variables is critical for understanding atomistic simulations and accelerating them through enhanced sampling techniques. Recently, several methods have been proposed to learn these variables directly from atomistic data. Depending on the type of data available, the learning process can be framed as dimensionality reduction, classification of metastable states, or identification of slow modes. Here, we present mlcolvar, a Python library that simplifies the construction of these variables and their use in the context of enhanced sampling through a contributed interface to the PLUMED software. The library is organized modularly to facilitate the extension and cross-contamination of these methodologies. In this spirit, we developed a general multi-task learning framework in which multiple objective functions and data from different simulations can be combined to improve the collective variables. The library's versatility is demonstrated through simple examples that are prototypical of realistic scenarios. [ABSTRACT FROM AUTHOR]

The Atlantic Meridional Overturning Circulation (AMOC) is a significant component of the global ocean system, which has so far ensured a relatively warm climate for the North Atlantic and mild conditions in regions, such as Western Europe. The AMOC is also critical for the global climate. The complexity of the dynamical system underlying the AMOC is so vast that a long-term assessment of the potential risk of AMOC collapse is extremely challenging. However, short-term prediction can lead to accurate estimates of the dynamical state of the AMOC and possibly to early warning signals for guiding policy making and control strategies toward preventing AMOC collapse in the long term. We develop a model-free, machine-learning framework to predict the AMOC dynamical state in the short term by employing five datasets: MOVE and RAPID (observational), AMOC fingerprint (proxy records), and AMOC simulated fingerprint and CESM AMOC (synthetic). We demonstrate the power of our framework in predicting the variability of the AMOC within the maximum prediction horizon of 12 or 24 months. A number of issues affecting the prediction performance are investigated. [ABSTRACT FROM AUTHOR]

A multi-stage planning method for independent energy storage (IES) based on dynamically updating key transmission sections (KTS) is proposed to address issues such as uneven power flow distribution and transmission congestion resulting from the high penetration of renewable energy sources and load growth. First, an IES planning model considering KTS constraints is established by searching and updating the KTS and associated constraints. This model takes into account IES investment and operation costs, maintenance costs, revenue from spot price differences, and penalties for system congestion. Then, a multi-stage planning method for energy storage is proposed based on the dynamic updating of KTS and the annual planning results. To verify the effectiveness and feasibility of the proposed method, a case study simulation is conducted using the improved IEEE 39-bus New England standard system. The results demonstrate the ability of the method to achieve multi-stage rolling optimization of IES, providing a reference for future efforts toward multi-stage planning of IES. [ABSTRACT FROM AUTHOR]

External flows of energy, entropy, and matter can cause sudden transitions in the stability of biological and industrial systems, fundamentally altering their dynamical function. How might we control and design these transitions in chemical reaction networks? Here, we analyze transitions giving rise to complex behavior in random reaction networks subject to external driving forces. In the absence of driving, we characterize the uniqueness of the steady state and identify the percolation of a giant connected component in these networks as the number of reactions increases. When subject to chemical driving (influx and outflux of chemical species), the steady state can undergo bifurcations, leading to multistability or oscillatory dynamics. By quantifying the prevalence of these bifurcations, we show how chemical driving and network sparsity tend to promote the emergence of these complex dynamics and increased rates of entropy production. We show that catalysis also plays an important role in the emergence of complexity, strongly correlating with the prevalence of bifurcations. Our results suggest that coupling a minimal number of chemical signatures with external driving can lead to features present in biochemical processes and abiogenesis. [ABSTRACT FROM AUTHOR]

Using quantum computers for computational chemistry and materials science will enable us to tackle problems that are intractable on classical computers. In this paper, we show how the relative energy of defective graphene structures can be calculated by using a quantum annealer. This simple system is used to guide the reader through the steps needed to translate a chemical structure (a set of atoms) and energy model to a representation that can be implemented on quantum annealers (a set of qubits). We discuss in detail how different energy contributions can be included in the model and what their effect is on the final result. The code used to run the simulation on D-Wave quantum annealers is made available as a Jupyter Notebook. This Tutorial was designed to be a quick-start guide for the computational chemists interested in running their first quantum annealing simulations. The methodology outlined in this paper represents the foundation for simulating more complex systems, such as solid solutions and disordered systems. [ABSTRACT FROM AUTHOR]

Many biological processes are regulated by allosteric mechanisms that communicate with distant sites in the protein responsible for functionality. The binding of a small molecule at an allosteric site typically induces conformational changes that propagate through the protein along allosteric pathways regulating enzymatic activity. Elucidating those communication pathways from allosteric sites to orthosteric sites is, therefore, essential to gain insights into biochemical processes. Targeting the allosteric pathways by mutagenesis can allow the engineering of proteins with desired functions. Furthermore, binding small molecule modulators along the allosteric pathways is a viable approach to target reactions using allosteric inhibitors/activators with temporal and spatial selectivity. Methods based on network theory can elucidate protein communication networks through the analysis of pairwise correlations observed in molecular dynamics (MD) simulations using molecular descriptors that serve as proxies for allosteric information. Typically, single atomic descriptors such as α-carbon displacements are used as proxies for allosteric information. Therefore, allosteric networks are based on correlations revealed by that descriptor. Here, we introduce a Python software package that provides a comprehensive toolkit for studying allostery from MD simulations of biochemical systems. MDiGest offers the ability to describe protein dynamics by combining different approaches, such as correlations of atomic displacements or dihedral angles, as well as a novel approach based on the correlation of Kabsch–Sander electrostatic couplings. MDiGest allows for comparisons of networks and community structures that capture physical information relevant to allostery. Multiple complementary tools for studying essential dynamics include principal component analysis, root mean square fluctuation, as well as secondary structure-based analyses. [ABSTRACT FROM AUTHOR]

We introduce a method for solving the "inverse" phase equilibria problem: How should the interactions among a collection of molecular species be designed in order to achieve a target phase diagram? Using techniques from convex optimization theory, we show how to solve this problem for phase diagrams containing a large number of components and many coexisting phases with prescribed compositions. We apply our approach to commonly used mean-field models of multicomponent fluids and then use molecular simulations to verify that the designed interactions result in the target phase diagrams. Our approach enables the rational design of "programmable" fluids, such as biopolymer and colloidal mixtures, with complex phase behavior. [ABSTRACT FROM AUTHOR]

Allostery is a constitutive, albeit often elusive, feature of biomolecular systems, which heavily determines their functioning. Its mechanical, entropic, long-range, ligand, and environment-dependent nature creates far from trivial interplays between residues and, in general, the secondary structure of proteins. This intricate scenario is mirrored in computational terms as different notions of "correlation" among residues and pockets can lead to different conclusions and outcomes. In this article, we put on a common ground and challenge three computational approaches for the correlation estimation task and apply them to three diverse targets of pharmaceutical interest: the androgen A2A receptor, the androgen receptor, and the EGFR kinase domain. Results show that partial results consensus can be attained, yet different notions lead to pointing the attention to different pockets and communications. [ABSTRACT FROM AUTHOR]

Efficiently identifying the most important communities and key transition nodes in weighted and unweighted networks is a prevalent problem in a wide range of disciplines. Here, we focus on the optimal clustering using variational kinetic parameters, linked to Markov processes defined on the underlying networks, namely, the slowest relaxation time and the Kemeny constant. We derive novel relations in terms of mean first passage times for optimizing clustering via the Kemeny constant and show that the optimal clustering boundaries have equal round-trip times to the clusters they separate. We also propose an efficient method that first projects the network nodes onto a 1D reaction coordinate and subsequently performs a variational boundary search using a parallel tempering algorithm, where the variational kinetic parameters act as an energy function to be extremized. We find that maximization of the Kemeny constant is effective in detecting communities, while the slowest relaxation time allows for detection of transition nodes. We demonstrate the validity of our method on several test systems, including synthetic networks generated from the stochastic block model and real world networks (Santa Fe Institute collaboration network, a network of co-purchased political books, and a street network of multiple cities in Luxembourg). Our approach is compared with existing clustering algorithms based on modularity and the robust Perron cluster analysis, and the identified transition nodes are compared with different notions of node centrality. [ABSTRACT FROM AUTHOR]

Allosteric regulation of proteins continues to be an engaging research topic for the scientific community. Models describing allosteric communication have evolved from focusing on conformation-based descriptors of protein structural changes to appreciating the role of internal protein dynamics as a mediator of allostery. Here, we explain a "violin model" for allostery as a contemporary method for approaching the Cooper–Dryden model based on redistribution of protein thermal fluctuations. Based on graph theory, the violin model makes use of community network analysis to functionally cluster correlated protein motions obtained from molecular dynamics simulations. This Review provides the theory and workflow of the methodology and explains the application of violin model to unravel the workings of protein kinase A. [ABSTRACT FROM AUTHOR]

The existence of a first-order phase transition between a low-density liquid (LDL) and a high-density liquid (HDL) form of supercooled water has been a central and highly debated issue of physics and chemistry for the last three decades. We present a computational study that allows us to determine the free-energy landscapes of supercooled water over a wide range of pressure and temperature conditions using the TIP4P/2005 force field. Our approach combines topology-based structural transformation coordinates, state-of-the-art free-energy calculation methods, and extensive unbiased molecular dynamics. All our diverse simulations cannot detect any barrier within the investigated timescales and system size, for a discontinuous transition between the LDL and HDL forms throughout the so-called "no man's land," until the onset of the solid, non-diffusive amorphous forms. [ABSTRACT FROM AUTHOR]

A reservoir computer is a way of using a high dimensional dynamical system for computation. One way to construct a reservoir computer is by connecting a set of nonlinear nodes into a network. Because the network creates feedback between nodes, the reservoir computer has memory. If the reservoir computer is to respond to an input signal in a consistent way (a necessary condition for computation), the memory must be fading; that is, the influence of the initial conditions fades over time. How long this memory lasts is important for determining how well the reservoir computer can solve a particular problem. In this paper, I describe ways to vary the length of the fading memory in reservoir computers. Tuning the memory can be important to achieve optimal results in some problems; too much or too little memory degrades the accuracy of the computation. [ABSTRACT FROM AUTHOR]