Random Network Theory Research Articles

Determining structural sites of the IGPD protein family from patterns in property based correlation matrices

Abstract We use the statistical approach of random matrix and network theory to tackle the problem of identifying the important motifs responsible for the crucial functioning of the IGPD protein family. This addresses directly the question of patterns of interaction between amino acid residues (based on properties) in proteins that contribute to protein function. We use the mathematical tools of inverse participation ratio and Shannon entropy to determine the locations of the important groups of correlated amino acid positions which gives us the structural sites of the IGPD protein. These tools isolate the smallest eigenvalues/outliers corresponding to eigenmodes as the most localized which give the crucial sites for the family. We also create the threshold network of the IGPD protein and find that at certain threshold, similar sites emerge from the network analysis which in addition gives us the strongest connected sites. This strengthens our method of finding the structural and functional sites. As a bonus we find these important sites also match with experiments.

On the mechanism of helium permeation through silica glass

<abstract> <p>Although the densities of crystal quartz and vitreous silica differ only by about 17% (2.65 and 2.20 g/cm<sup>3</sup>, respectively), the helium permeability of silica glass is six orders more elevated than that of crystal quartz. This vast difference has puzzled researchers for decades considering that silica glass and quartz crystal have the same chemical composition. This work discusses the mechanism of high helium permeation through silica glass. It briefly reviews the experimental data and its contradictions with the continuous random network theory. A recently proposed nanoflake model for silica glass structure is utilized to explain the origin of glass permeation to helium. According to the nanoflake model, the formation of nanoflakes not only brings a one-dimensional medium-range ordering structure into silica glass but simultaneously creates regions where van der Waals bonds replace the oxygen-silicon covalent bonds. It is the weakness of van der Waals bonds that causes the helium mobility in these areas to increase.</p> </abstract>

Study of Conditional Glass Formers (II) on Basics of Optical Electronegativity

Concept of Electronegativity is used by so many researchers to calculate Refractive index Band energy, Dielectric constant, Metallization criterion, Basicity, Oxide ion polarizability, Bond ionicity, Third order non-linear optical susceptibility to study the so many glass systems. Here the concept of optical electronegativity is used to study these parameters for conditional glass formers refer as (II) of group V(A,B) and VI(A,B). They are Ta2O5, Nb2O5, V2O5, Cr2O3, Bi2O3, TeO2, MoO3, As2O3, WO3, Se2O3. It is observed that on decreasing the values of optical electronegativity, optical band energy, metallization criterion, and bond iconicity decreases; and optical dielectric constant, optical basicity, oxide ion polarizability refractive index, third order non linear optical susceptibility increases. All conditional glass forming oxides (II) are according to Pauling's packing rule, Goldschmidt radius ratio rule, Zachariasen's random network theory and rules for formation of glass and Sun’s single bond strength theory. For Conditional glass formers (II) radius ratio rc/ra = 0.250-0.492, [except Bi2O3 (0.75) & TeO2 (0.87)] and single bond strength greater than 90 Kcal/mol. According to Sun’s, high bond strength oxides are not good glass formers because they themselves do not forms glass; but when they forms small ring in the melt of these materials which would result in easy crystallization. Key Words: Refractive index Band energy, Dielectric constant, Metallization criterion, Basicity, Oxide ion polarizability, Bond ionicity, Third order non-liner optical susceptibility.

STUDY OF GLASS FORMERS ON BASICS OF OPTICAL ELECTRONEGATIVITY

Concept of electronegativity is used by so many researchers to study Refractive index, Band energy, Dielectric constant, Metallization criterion, Basicity, Oxide ion polarizability, Bond ionicity, Third order non-linear optical susceptibility to study so many glass systems. Here the concept of optical electronegativity is used to study these parameters in glass formers. It is observed that on decreasing the values of optical electronegativity; optical band energy, metallization criterion, and bond ionicity decreases; where as optical dielectric constant, optical basicity, oxide ion polarizability refractive index and third order non linear optical susceptibility increases. All glass forming oxides are according to Pauling's packing rule, Goldschmidt radius ratio rule, Zachariasen's random network theory and rules for formation of glass and Sun’s single bond strength theory. For glass formers radius ratio rc/ra = 0.14-0.40, [except P2O5] and single bond strength greater than 90 Kcal/mol. Key Words: Band energy, Basicity, Optical dielectric constant, Oxide ion polarizability, Refractive index.

Study of Conditional Glass Formers (I) on Basics of Optical Electronegativity

Concept of Electronegativity is used by so many researchers to calculate Refractive index, Band energy, Dielectric constant, Metallization criterion, Basicity, Oxide ion polarizability, Bond ionicity, Third order non-linear optical susceptibility to study the so many glass systems. Here the concept of optical electronegativity is used to study these parameters for conditional glass formers referred as (1) taken from group III (A,B) and IV (A,B); they are Y2O3, HfO2, ZrO2, Sc2O3, TiO2, Al2O3, In2O3, Ga2O3, and SnO2. It is observed that optical band energy, metallization criterion, and bond iconicity decreases; and optical dielectric constant, optical basicity, oxide ion polarizability refractive index and third order non linear optical susceptibility increases. All conditional glass formers (I) oxides are according to Pauling's packing rule, Goldschmidt radius ratio rule, Zachariasen's random network theory, rules for formation of glass and Sun’s single bond strength theory. For conditional glass formers (I) radius ratio rc/ra = 0.40-0.75 and single bond strength greater than 90 Kcal/mol. According to Sun’s, high bond strength oxides are not good glass formers because they themselves do not forms glass; but when they forms small ring in the melt of these materials which result in easy crystallization. Key Words: Refractive index Band energy, Dielectric constant, Metallization criterion, Basicity, Oxide ion polarizability, Bond ionicity, Third order non-liner optical susceptibility.

Localized eigenvectors on metric graphs

We analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. In accordance with random network theory, localization of an eigenvector is rare and the network should be tuned to observe exactly localized eigenvectors. We derive the resonance conditions to obtain localized eigenvectors for various geometric configurations and their combinations to form more complicated resonant structures. These localized eigenvectors suggest new indicators based on the energy density; in contrast to standard criteria, ours provide the number of active edges. We also suggest practical ways to design resonating systems based on metric graphs. Finally, numerical simulations of the time-dependent wave equation on the metric graph show that localized eigenvectors can be excited by a broadband initial condition, even with leaky boundary conditions.

A Random Matrix Network Model for the Network Teaching System of College Music Education Courses

How to improve the teaching management model has always been an important part of the research and exploration of university music teaching management. Based on the random matrix network theory, this paper builds a random matrix network model and uses the random matrix network of Internet/Intranet to realize the electronic and networked multimedia information management, which makes the data query more flexible and convenient. In the random matrix network environment, the security of the model network is greatly improved, which solves the expansion problem of the random matrix network. During the simulation process, the model integrates the relevant image, audio, video, animation, and other multimedia processing technologies and storage technologies. The system adopts object-oriented analysis and design ideas to analyze the requirements, adopts the random matrix network architecture, and uses tomcat as the server, and the SQL Server database launched by Microsoft is used as the back-end data support, and the Struts architecture is designed by using the MVC development mode to ensure the system has good maintainability and enhanced data processing capabilities. The experimental results show that the response time of the system login verification is less than 2 seconds, the response time of adding users is less than 2 seconds, the response time of downloading assignments is less than 2 seconds, the response time of uploading courseware is less than 3 seconds, and the response time of checking the results is less than 2 seconds. The separation of technology effectively improves the scalability and maintainability of the system.

Revisiting Lebedev’s one-century old experiment

One hundred years ago, world-famous scientist A. A. Lebedev performed a set of classical measurements on annealed optic crown glasses. He found that these glasses exhibited characteristic endothermic effects in a particular temperature range. To explain these phenomena, Lebedev proposed a hypothesis that the glasses contain tiny quartz crystals. This initial hypothesis was quickly disapproved, and the origin of the endothermic effect of glasses remains an unsolved puzzle. This work uses recently proposed nanoflake model of silica glass structure to explain the endothermic effect of various glasses. The new model differs from the popular continuous random network theory in that it emphasizes the medium-range ordering structure of glasses. According to the nanoflake based theory, the endothermic effect of glasses is caused by the transition from ordered one-dimensional structures into disordered structure in glasses. The new theory also predicts that the temperature range of the endothermic effect is dependent on both glass composition and cooling rates during glass formation.

Topological Ordering of Memory Glass on Extended Length Scales.

Identifying ordering in non-crystalline solids has been a focus of natural science since the publication of Zachariasen's random network theory in 1932, but it still remains as a great challenge of the century. Literature shows that the hierarchical structures, from the short-range order of first-shell polyhedra to the long-range order of translational periodicity, may survive after amorphization. Here, in a piece of AlPO4, or berlinite, we combine X-ray diffraction and stochastic free-energy surface simulations to study its phase transition and structural ordering under pressure. From reversible single crystals to amorphous transitions, we now present an unambiguous view of the topological ordering in the amorphous phase, consisting of a swarm of Carpenter low-symmetry phases with the same topological linkage, trapped in a metastable intermediate stage. We propose that the remaining topological ordering is the origin of the switchable "memory glass" effect. Such topological ordering may hide in many amorphous materials through disordered short atomic displacements.

Effective medium theory of random regular networks

Disordered spring networks can exhibit rigidity transitions, due to either the removal of material in over-constrained networks or the application of strain in under-constrained ones. While an effective medium theory (EMT) exists for the former, there is none for the latter. We, therefore, formulate an EMT for random regular, under-constrained spring networks with purely geometrical disorder to predict their stiffness via the distribution of tensions. We find a linear dependence of stiffness on strain in the rigid phase and a nontrivial dependence on both the mean and standard deviation of the tension distribution. While EMT does not yield highly accurate predictions of shear modulus due to spatial heterogeneities, it requires only the distribution of tensions for an intact system, therefore making it an ideal starting point for experimentalists quantifying the mechanics of such networks.

Transient Chaotic Dimensionality Expansion by Recurrent Networks

Neurons in the brain communicate with spikes, which are discrete events in time and value. Functional network models often employ rate units that are continuously coupled by analog signals. Is there a qualitative difference implied by these two forms of signaling? We develop a unified mean-field theory for large random networks to show that first- and second-order statistics in rate and binary networks are in fact identical if rate neurons receive the right amount of noise. Their response to presented stimuli, however, can be radically different. We quantify these differences by studying how nearby state trajectories evolve over time, asking to what extent the dynamics is chaotic. Chaos in the two models is found to be qualitatively different. In binary networks, we find a network-size-dependent transition to chaos and a chaotic submanifold whose dimensionality expands stereotypically with time, while rate networks with matched statistics are nonchaotic. Dimensionality expansion in chaotic binary networks aids classification in reservoir computing and optimal performance is reached within about a single activation per neuron; a fast mechanism for computation that we demonstrate also in spiking networks. A generalization of this mechanism extends to rate networks in their respective chaotic regimes.7 MoreReceived 7 May 2020Revised 18 March 2021Accepted 23 April 2021DOI:https://doi.org/10.1103/PhysRevX.11.021064Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasFluctuations & noiseNeural encodingNeuronal networksStatistical field theoryTechniquesChaos & nonlinear dynamicsDynamical mean field theoryIsing modelLangevin equationStatistical Physics

Beyond R0: heterogeneity in secondary infections and probabilistic epidemic forecasting.

The basic reproductive number, R0, is one of the most common and most commonly misapplied numbers in public health. Often used to compare outbreaks and forecast pandemic risk, this single number belies the complexity that different epidemics can exhibit, even when they have the same R0. Here, we reformulate and extend a classic result from random network theory to forecast the size of an epidemic using estimates of the distribution of secondary infections, leveraging both its average R0 and the underlying heterogeneity. Importantly, epidemics with lower R0 can be larger if they spread more homogeneously (and are therefore more robust to stochastic fluctuations). We illustrate the potential of this approach using different real epidemics with known estimates for R0, heterogeneity and epidemic size in the absence of significant intervention. Further, we discuss the different ways in which this framework can be implemented in the data-scarce reality of emerging pathogens. Lastly, we demonstrate that without data on the heterogeneity in secondary infections for emerging infectious diseases like COVID-19 the uncertainty in outbreak size ranges dramatically. Taken together, our work highlights the critical need for contact tracing during emerging infectious disease outbreaks and the need to look beyond R0.

Dynamic Networks that Drive the Process of Irreversible Step-Growth Polymerization

Many research fields, reaching from social networks and epidemiology to biology and physics, have experienced great advance from recent developments in random graphs and network theory. In this paper we propose a generic model of step-growth polymerisation as a promising application of the percolation on a directed random graph. This polymerisation process is used to manufacture a broad range of polymeric materials, including: polyesters, polyurethanes, polyamides, and many others. We link features of step-growth polymerisation to the properties of the directed configuration model. In this way, we obtain new analytical expressions describing the polymeric microstructure and compare them to data from experiments and computer simulations. The molecular weight distribution is related to the sizes of connected components, gelation to the emergence of the giant component, and the molecular gyration radii to the Wiener index of these components. A model on this level of generality is instrumental in accelerating the design of new materials and optimizing their properties, as well as it provides a vital link between network science and experimentally observable physics of polymers.

ON CONNECTED COMPONENT DISTRIBUTION OF RANDOM BLOCK-HIERARCHICAL NETWORKS USING THE AUTOMATED SYSTEM $ \textit{XRANDNET} $

In the paper the result of a research done by using the automated system xRandNet is presented, which is designed and implemented for generating and analyzing the main topological properties of some hierarchical models of random networks. The research is related to the connected component distribution of random block-hierarchical networks, which are quite new objects in the random network theory.

Multi-objective active distribution networks expansion planning by scenario-based stochastic programming considering uncertain and random weight of network

This paper presents a novel multi-objective model of active distribution network planning based on stochastic programming and uncertain random network (URN) theory. The planning model is proposed to find the final scheme with optimal alternative, location, size and operational strategy for the candidate distribution lines, transformer substations (TSs), distribution generations (DGs), static var compensators (SVCs) and on-load tap changers (OLTCs). Firstly, a scenario-based approach is developed to analyse the uncertainties in network system, such as the demand and intermittency of renewable sources. Since the impact of multiple uncertain factors on network cannot be ignored, a network frame is then modelled by uncertain and random weights of spanning tree (ST) instead of fixed value. In order to achieve the minimization of total cost, and further the selection of a minimum spanning tree (MST) with the uncertain and random weight, a 3-dimensional uncertain space is constructed based on the combination of the previous two targets. In addition, a second-order cone programming (SOCP) is applied to cope with the multi-objective, mixed-integer nonlinear nature of the proposed planning model. Simulation is performed on a modified Pacific Gas and Electric Company (PG&E) 69-bus distribution system, and the results demonstrate the effectiveness of the proposed model.

Dependence of glass transition on the structure in Bi[sbnd]B[sbnd]Zn oxide glass

In this work, the properties and structures of a series BiBZn oxide glass with a high Bi2O3 content from 30 to 60 mol% were studied. It was found that when molar ratio of Bi to B exceeded 4:1, crystal phase formed in the glass after quenching. The densities were observed to increase with increase of the Bi2O3 content. When molar ratio of Bi to B is between 40% and 51%, the glasses had a higher rate of densities growth increase. The glass transition temperature (Tg) and crystallization temperature (Tx) values decrease with increase of the Bi2O3 content. The infrared spectra and the Raman spectra both presented [BiO6] increased with increase of Bi2O3 content. The network structure transformation from [BO3] to [BO4] was observed when molar ratio of Bi to B was below 4:1. With further increase of the Bi2O3 content, a structure transformation from [BiO3] to [BiO6] unit was observed. The changes in the structure resulted in the decrease of glass characteristic temperatures. A theory based on the random network theory was proposed to explain the transition of the structure and the property of the glasses.

Interdependence of Resistance and Optical Transmission in Conductive Nanowire Networks

AbstractMetallic nanowire networks show great potential for use in transparent conductive displays. The coupled relationship between the principal materials properties of interest for such applications, optical transmittance and sheet resistance, is known to be affected by both nanowire dimensions and area coverage. Existing theory to describe the observed interdependence requires consideration of both bulk and surface phases. Here, an analysis of structural variables in these materials is presented confirming that bulk considerations do not typically apply. Accordingly, a model using statistical theory for random line networks is developed that requires consideration of a surface phase only. The resultant simple expression relates sheet resistance and optical transmission of heterogeneous nanowire assemblies in terms of nanowire dimensions. Comparison of our model with experimental and simulation data from the literature shows excellent agreement. In closing, theoretical insights into the observed influence on sheet resistance of polydisperse nanowire lengths and structural variability are provided.

Modelling the atomic arrangement of amorphous 2D silica: a network analysis.

The recent experimental discovery of a semi two-dimensional silica glass has offered a realistic description of the random network theory of a silica glass structure, initially discussed by Zachariasen. To study the structure formation of silica in two dimensions, we introduce a two-body force field, based on a soft core Yukawa potential. The different configurations, sampled via Molecular dynamics simulations, can be directly compared with the experimental structures, which have been provided in the literature. The parameters of the force field are obtained from comparison of the nearest-neighbor distances between experiment and simulation. Further key properties such as angle distributions, distribution of ring sizes and triplets of rings are analyzed and compared with the experiment. Of particular interest is the spatial correlation of ring sizes. In general, we observe a very good agreement between experiment and simulation. Additional insight from the simulations is provided about the temporal and spatial stability of the rings in dependence of their size.

Population density equations for stochastic processes with memory kernels.

We present a method for solving population density equations (PDEs)--a mean-field technique describing homogeneous populations of uncoupled neurons-where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation-a recent result from random network theory-describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.

A framework for analyzing contagion in assortative banking networks.

We introduce a probabilistic framework that represents stylized banking networks with the aim of predicting the size of contagion events. Most previous work on random financial networks assumes independent connections between banks, whereas our framework explicitly allows for (dis)assortative edge probabilities (i.e., a tendency for small banks to link to large banks). We analyze default cascades triggered by shocking the network and find that the cascade can be understood as an explicit iterated mapping on a set of edge probabilities that converges to a fixed point. We derive a cascade condition, analogous to the basic reproduction number R0 in epidemic modelling, that characterizes whether or not a single initially defaulted bank can trigger a cascade that extends to a finite fraction of the infinite network. This cascade condition is an easily computed measure of the systemic risk inherent in a given banking network topology. We use percolation theory for random networks to derive a formula for the frequency of global cascades. These analytical results are shown to provide limited quantitative agreement with Monte Carlo simulation studies of finite-sized networks. We show that edge-assortativity, the propensity of nodes to connect to similar nodes, can have a strong effect on the level of systemic risk as measured by the cascade condition. However, the effect of assortativity on systemic risk is subtle, and we propose a simple graph theoretic quantity, which we call the graph-assortativity coefficient, that can be used to assess systemic risk.