Graph Theoretical Tools Research Articles

Puberty interacts with sleep and brain network organization to predict mental health.

Along with pubertal development, the transition to adolescence brings about increased risk for sleep disturbances and mental health problems. Functional connectivity of overlapping large-scale brain networks, such as increased connectivity between the default mode and dorsal attention networks, has been reported to relate to both sleep and mental health problems. Clarifying whether pubertal development interacts with sleep disturbances and functional brain networks to predict mental health may provide information to improve the timing and design of interventions targeting sleep disturbances in adolescents. To examine how pubertal status and tempo relate to sleep disturbances and shape the relationship between sleep disturbances and mental health problems, we harnessed a large sample of children aged 10-14 years from the Adolescent Brain and Cognitive Development (ABCD) Study (N ~ 3,000-10,000). We used graph theoretical tools to probe how pubertal development concurrently interacts with sleep disturbances and brain network organization to predict mental health problems. We found that advanced pubertal status, but not pubertal tempo, predicted sleep disturbances; however, both pubertal status and tempo interact with sleep disturbances to predict mental health problems and engage in three-way interactions with sleep and brain network organization to predict mental health problems. Overall, this work suggests that less advanced pubertal status and slower tempo are risk factors for the strongest links between sleep disturbances, brain organization, and mental health problems. Further, our findings speak to the importance of accounting for interactions in the constellation of factors that surround complex behavioral and clinical syndromes, here internalizing and externalizing disorders, and provide new context to consider for targeted interventions.

Subcortical influences on the topology of cortical networks align with functional processing hierarchies

fMRI of the human brain reveals spatiotemporal patterns of functional connectivity (FC), forming distinct cortical networks. Lately, subcortical contributions to these configurations are receiving renewed interest, but investigations rarely focus explicitly on their effects on cortico-cortical FC. Here, we employ a straightforward multivariable approach and graph-theoretic tools to assess subcortical impact on topological features of cortical networks. Given recent evidence showing that structures like the thalamus and basal ganglia integrate input from multiple networks, we expect increased segregation between cortical networks after removal of subcortical effects on their FC patterns. We analyze resting state data of young and healthy participants (male and female; N = 100) from the human connectome project. We find that overall, the cortical network architecture becomes less segregated, and more integrated, when subcortical influences are accounted for. Underlying these global effects are the following trends: ‘Transmodal’ systems become more integrated with the rest of the network, while ‘unimodal’ networks show the opposite effect. For single nodes this hierarchical organization is reflected by a close correspondence with the spatial layout of the principal gradient of FC (Margulies et al., 2016). Lastly, we show that the limbic system is significantly less coherent with subcortical influences removed. The findings are validated in a (split-sample) replication dataset. Our results provide new insight regarding the interplay between subcortex and cortical networks, by putting the integrative impact of subcortex in the context of macroscale patterns of cortical organization.

Computational Analysis of Some More Rectangular Tessellations of Kekulenes and Their Molecular Characterizations.

Cycloarene molecules are benzene-ring-based polycyclic aromatic hydrocarbons that have been fused in a circular manner and are surrounded by carbon-hydrogen bonds that point inward. Due to their magnetic, geometric, and electronic characteristics and superaromaticity, these polycyclic aromatics have received attention in a number of studies. The kekulene molecule is a cyclically organized benzene ring in the shape of a doughnut and is the very first example of such a conjugated macrocyclic compound. Due to its structural characteristics and molecular characterizations, it serves as a great model for theoretical research involving the investigation of π electron conjugation circuits. Therefore, in order to unravel their novel electrical and molecular characteristics and foresee potential applications, the characterization of such components is crucial. In our current research, we describe two unique series of enormous polycyclic molecules made from the extensively studied base kekulene molecule, utilizing the essential graph-theoretical tools to identify their structural characterization via topological quantities. Rectangular kekulene Type-I and rectangular kekulene Type-II structures were obtained from base kekulene molecules arranged in a rectangular fashion. We also employ two subcases for each Type and, for all of these, we derived ten topological indices. We can investigate the physiochemical characteristics of rectangular kekulenes using these topological indices.

Graph Theoretical Analysis of local ultraluminous infrared galaxies and quasars

We present a methodological framework for studying galaxy evolution by utilizing Graph Theory and network analysis tools. We study the evolutionary processes of local ultraluminous infrared galaxies (ULIRGs) and quasars and the underlying physical processes, such as star formation and active galactic nucleus (AGN) activity, through the application of Graph Theoretical analysis tools. We extract, process and analyze mid-infrared spectra of local (z ¡ 0.4) ULIRGs and quasars between 5-38μm through internally developed Python routines, in order to generate similarity graphs, with the nodes representing ULIRGs being grouped together based on the similarity of their spectra. Additionally, we extract and compare physical features from the mid-IR spectra, such as the polycyclic aromatic hydrocarbons (PAHs) emission and silicate depth absorption features, as indicators of the presence of star-forming regions and obscuring dust, in order to understand the underlying physical mechanisms of each evolutionary stage of ULIRGs. Our analysis identifies five groups of local ULIRGs based on their mid-IR spectra, which is quite consistent with the well established fork classification diagram by providing a higher level classification. We demonstrate how graph clustering algorithms and network analysis tools can be utilized as unsupervised learning techniques for revealing direct or indirect relations between various galaxy properties and evolutionary stages, which provides an alternative methodology to previous works for classification in galaxy evolution. Additionally, our methodology compares the output of several graph clustering algorithms in order to demonstrate the best-performing Graph Theoretical tools for studying galaxy evolution.

Chemical space of the singlet C4H8O2 species. A systematic theoretical analysis on their structural and thermochemical properties

The goal of this study was to explore and define the thermodynamic properties of one of the subspaces of ‘chemical space’ using a mixture of graph theory and theoretical chemistry tools. Therefore, all possible molecular structures with C4H8O2stoichiometry were generated, considering constitutional isomers and molecular complexes. The thermodynamic properties of the obtained isomers have been obtained by G3MP2B3 protocol. The classification of the obtained isomers was simplified by using thermodynamic maps, which is an effective method for the comparison of thermodynamic stability for entities of complex molecular systems.Modern computational methods can be used to understand larger systems, which has made it possible to characterize a chemical subspace not only by selecting individual entities, but also as a whole. With this procedure one can catch a glimpse into the diversity of a molecular system and predict further uses of newly discovered molecules or design molecules with predefined properties.

Enabling in-depth analysis in heat exchanger network synthesis via graph-theoretic tool: Experiences in Swinburne University of Technology Sarawak Campus

The ability and capability to analyze and benchmark alternative designs on top of the optimal network are deemed valuable competencies for current and future chemical engineers. In this context, a process graph (P-graph)-inspired tool – P-HENS is introduced to an integrated plant design unit in an undergraduate chemical engineering degree program at Swinburne University of Technology Sarawak Campus in Malaysia. The energy recovery aspect is one of the key design elements in the integrated plant design unit. The introduction of P-HENS, which is capable of mathematically determining multiple optimal and sub-optimal solutions is considered useful for the students to (i) identify plausible heat exchanger networks (HENs) structures that may be overlooked using conventional approaches and (ii) enable a more in-depth analysis to justify the selected design. Overall, the implementation of P-HENS shows positive outcomes, where this free-of-charge software complements the learning of conventional manual approaches used in HENs synthesis. Furthermore, recommendations suggested by the users (students) are collected and compiled for potential future software development. This work serves as an essential reference for other chemical engineering educators who are teaching pinch analysis or heat integration-related courses.

The learnability of Pauli noise

Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today’s quantum devices. A fundamental issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question what information is learnable and what is not, which is unclear even for a single CNOT gate. Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates using graph theoretical tools. Our results reveal the optimality of cycle benchmarking in the sense that it can extract all learnable information about Pauli noise. We experimentally demonstrate noise characterization of IBM’s CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we show that an attempt to extract unlearnable information by ignoring state preparation noise yields unphysical estimates, which is used to lower bound the state preparation noise.

On Some Ev-Degree and Ve-Degree Dependent Indices of Benes Network and營ts Derived Classes

One of the most recent developments in the field of graph theory is the analysis of networks such as Butterfly networks, Benes networks, Interconnection networks, and David-derived networks using graph theoretic parameters. The topological indices (TIs) have been widely used as graph invariants among various graph theoretic tools. Quantitative structure activity relationships (QSAR) and quantitative structure property relationships (QSPR) need the use of TIs. Different structure-based parameters, such as the degree and distance of vertices in graphs, contribute to the determination of the values of TIs. Among other recently introduced novelties, the classes of ev-degree and ve-degree dependent TIs have been extensively explored for various graph families. The current research focuses on the development of formulae for different ev-degree and ve-degree dependent TIs for dimensional Benes network and certain networks derived from it. In the end, a comparison between the values of the TIs for these networks has been presented through graphical tools.

A Spectral Clustering Approach for the Evolution of the COVID-19 Pandemic in the State of Rio Grande do Sul, Brazil

The aim of this paper is to analyse the evolution of the COVID-19 pandemic in Rio Grande do Sul by applying graph-theoretical tools, particularly spectral clustering techniques, on weighted graphs defined on the set of 167 municipalities in the state with population 10,000 or more, which are based on data provided by government agencies and other sources. To respond to this outbreak, the state has adopted a system by which pre-determined regions are assigned flags on a weekly basis, and different measures go into effect according to the flag assigned. Our results suggest that considering a flexible approach to the regions themselves might be a useful additional tool to give more leeway to cities with lower incidence rates, while keeping the focus on public safety. Moreover, simulations show that the combination of pendulum migration and isolation data used in this paper leads to a coherent qualitative description of the evolution of the pandemic in Rio Grande do Sul. These simulations also confirm the dampening effect of isolation on the dissemination of the disease.

ReNeGate: A Reaction Network Graph-Theoretical Tool for Automated Mechanistic Studies in Computational Homogeneous Catalysis.

Exploration of the chemical reaction space of chemical transformations in multicomponent mixtures is one of the main challenges in contemporary computational chemistry. To remove expert bias from mechanistic studies and to discover new chemistries, an automated graph-theoretical methodology is proposed, which puts forward a network formalism of homogeneous catalysis reactions and utilizes a network analysis tool for mechanistic studies. The method can be used for analyzing trajectories with single and multiple catalytic species and can provide unique conformers of catalysts including multinuclear catalyst clusters along with other catalytic mixture components. The presented three-step approach has the integrated ability to handle multicomponent catalytic systems of arbitrary complexity (mixtures of reactants, catalyst precursors, ligands, additives, and solvents). It is not limited to predefined chemical rules, does not require prealignment of reaction mixture components consistent with a reaction coordinate, and is not agnostic to the chemical nature of transformations. Conformer exploration, reactive event identification, and reaction network analysis are the main steps taken for identifying the pathways in catalytic systems given the starting precatalytic reaction mixture as the input. Such a methodology allows us to efficiently explore catalytic systems in realistic conditions for either previously observed or completely unknown reactive events in the context of a network representing different intermediates. Our workflow for the catalytic reaction space exploration exclusively focuses on the identification of thermodynamically feasible conversion channels, representative of the (secondary) catalyst deactivation or inhibition paths, which are usually most difficult to anticipate based solely on expert chemical knowledge. Thus, the expert bias is sought to be removed at all steps, and the chemical intuition is limited to the choice of the thermodynamic constraint imposed by the applicable experimental conditions in terms of threshold energy values for allowed transformations. The capabilities of the proposed methodology have been tested by exploring the reactivity of Mn complexes relevant for catalytic hydrogenation chemistry to verify previously postulated activation mechanisms and unravel unexpected reaction channels relevant to rare deactivation events.

Quantitative Structure-Property Relationship Analysis on Priority PAHs Using Certain Closed Neighbourhood Topological Indices

Molecular structures and their physicochemical properties are related harmoniously. Having the potential uses in our everyday life, graph theory has become one of the most iconic and discussed areas of applied mathematics. Chemical graph theory (CGT) is a branch of graph theory that incorporates chemical aspects. The topological index (TI) is a well-defined graph-theoretical tool that offers a mathematical relationship with molecular structures and also characterizes their topology. TI has real-time applications such as isomer discrimination, drug design, QSPR, and QSAR studies in various domains of chemistry, including nanotechnology and biochemistry. Herein, seven closed neighborhood TIs are investigated. Using QSPR regression analysis, the chemical signature of the indices in predicting the physicochemical properties of priority polycyclic aromatic hydrocarbons (PAHs) is explored. These Tis exhibited a strong correlation with certain properties of priority PAHs. Certain statistical aspects of these indices are discussed, and the significant results have also been represented graphically.

The expanding horizons of network neuroscience: From description to prediction and control

The field of network neuroscience has emerged as a natural framework for the study of the brain and has been increasingly applied across divergent problems in neuroscience. From a disciplinary perspective, network neuroscience originally emerged as a formal integration of graph theory (from mathematics) and neuroscience (from biology). This early integration afforded marked utility in describing the interconnected nature of neural units, both structurally and functionally, and underscored the relevance of that interconnection for cognition and behavior. But since its inception, the field has not remained static in its methodological composition. Instead, it has grown to use increasingly advanced graph-theoretic tools and to bring in several other disciplinary perspectives—including machine learning and systems engineering—that have proven complementary. In doing so, the problem space amenable to the discipline has expanded markedly. In this review, we discuss three distinct flavors of investigation in state-of-the-art network neuroscience: (i) descriptive network neuroscience, (ii) predictive network neuroscience, and (iii) a perturbative network neuroscience that draws on recent advances in network control theory. In considering each area, we provide a brief summary of the approaches, discuss the nature of the insights obtained, and highlight future directions.

The Price of Connectivity in Fair Division

We study the allocation of indivisible goods that form an undirected graph and quantify the loss of fairness when we impose a constraint that each agent must receive a connected subgraph. Our focus is on well-studied fairness notions including envy-freeness and maximin share fairness. We introduce the price of connectivity to capture the largest gap between the graph-specific and the unconstrained maximin share, and derive bounds on this quantity which are tight for large classes of graphs in the case of two agents and for paths and stars in the general case. For instance, with two agents we show that for biconnected graphs it is possible to obtain at least $3/4$ of the maximin share with connected allocations, while for the remaining graphs the guarantee is at most $1/2$. In addition, we determine the optimal relaxation of envy-freeness that can be obtained with each graph for two agents, and characterize the set of trees and complete bipartite graphs that always admit an allocation satisfying envy-freeness up to one good (EF1) for three agents. Our work demonstrates several applications of graph-theoretic tools and concepts to fair division problems.

Retrosplenial Cortex Effects Contextual Fear Formation Relying on Dysgranular Constituent in Rats.

Animal contextual fear conditioning (CFC) models are the most-studied forms used to explore the neural substances of posttraumatic stress disorder (PTSD). In addition to the well-recognized hippocampal–amygdalar system, the retrosplenial cortex (RSC) is getting more and more attention due to substantial involvement in CFC, but with a poor understanding of the specific roles of its two major constituents—dysgranular (RSCd) and granular (RSCg). The current study sought to identify their roles and underlying brain network mechanisms during the encoding processing of the rat CFC model. Rats with pharmacologically inactivated RSCd, RSCg, and corresponding controls underwent contextual fear conditioning. [18F]-fluorodeoxyglucose positron emission tomography/computed tomography (18F-FDG PET/CT) scanning was performed for each animal. The 5-h and 24-h retrieval were followed to test the formation of contextual memory. Graph theoretic tools were used to identify the brain metabolic network involved in encoding phase, and changes of nodal (brain region) properties linked, respectively, to disturbed RSCd and RSCg were analyzed. Impaired retrieval occurred in disturbed RSCd animals, not in RSCg ones. The RSC, hippocampus (Hip), amygdala (Amy), piriform cortex (Pir), and visual cortex (VC) are hub nodes of the brain-wide network for contextual fear memory encoding in rats. Nodal degree and efficiency of hippocampus and its connectivity with amygdala, Pir, and VC were decreased in rats with disturbed RSCd, while not in those with suppressed RSCg. The RSC plays its role in contextual fear memory encoding mainly relying on its RSCd part, whose condition would influence the activity of the hippocampal–amygdalar system.

Using Graph-Theoretic Machine Learning to Predict Human Driver Behavior

Studies have shown that autonomous vehicles (AVs) behave conservatively in a traffic environment composed of human drivers and do not adapt to local conditions and socio-cultural norms. It is known that socially aware AVs can be designed if there exists a mechanism to understand the behaviors of human drivers. We present an approach that leverages machine learning to predict, the behaviors of human drivers. This is similar to how humans implicitly interpret the behaviors of drivers on the road, by only observing the trajectories of their vehicles. We use graph-theoretic tools to extract driver behavior features from the trajectories and machine learning to obtain a computational mapping between the extracted trajectory of a vehicle in traffic and the driver behaviors. Compared to prior approaches in this domain, we prove that our method is robust, general, and extendable to broad-ranging applications such as autonomous navigation. We evaluate our approach on real-world traffic datasets captured in the U.S., India, China, and Singapore, as well as in simulation.

Computing some Novel Closed Neighborhood Degree-Based Topological Indices of Graphene Structures

Chemical graph theory (CGT) is a crucial part of mathematical chemistry, which characterizes the chemical compounds as molecular graphs using graph-theoretical tools. By IUPAC terminology, a numerical parameter that represents the topology of a molecular graph is known as a topological index (TI). In this scenario, our research interest is to employ newly defined seven TIs, namely closed neighborhood first Zagreb index (CM1(G)), modified closed neighborhood first Zagreb index (CM1*(G)), closed neighborhood second Zagreb index (CM2(G)), closed neighborhood Forgotten index (CF(G)), modified closed neighborhood Forgotten index (CF*(G)), closed neighborhood first hyper Zagreb index (CHM1(G)) and closed neighborhood second hyper Zagreb index (CHM2(G)) as well as explore them mathematically. Without applying quantum mechanics, these TIs could obtain the maximum information behind the graphene structures.

Network nonlocality via rigidity of token counting and color matching

Network Nonlocality is the study of the Network Nonlocal correlations created by several independent entangled states shared in a network. In this paper, we provide the first two generic strategies to produce nonlocal correlations in large classes of networks without input. In the first one, called Token-Counting (TC), each source distributes a fixed number of tokens and each party counts the number of received tokens. In the second one, called Color-Matching (CM), each source takes a color and a party checks if the color of neighboring sources match. Using graph theoretic tools and Finner's inequality, we show that TC and CM distributions are rigid in wide classes of networks, meaning that there is essentially a unique classical strategy to simulate such correlations. Using this rigidity property, we show that certain quantum TC and CM strategies produce correlations that cannot be produced classicality. This leads us to several examples of Network Nonlocality without input. These examples involve creation of coherence throughout the whole network, which we claim to be a fingerprint of genuine forms of Network Nonlocality. This work extends a more compact parallel work [Nonlocality for Generic Networks, arXiv:2011.02769] on the same subject and provides all the required technical proofs.

Modified Zagreb Connection Indices for Benes Network and Related Classes

The study of networks such as Butterfly networks, Benes networks, interconnection networks, David‐derived networks through graph theoretical parameters is among the modern trends in the area of graph theory. Among these graph theoretical tools, the topological Indices (TIs) have been frequently used as graph invariants. TIs are also the essential tools for quantitative structure activity relationship (QSAR) as well as quantity structure property relationships (QSPR). TIs depend on different parameters, such as degree and distance of vertices in graphs. The current work is devoted to the derivation of 2‐distance based TIs, known as, modified first Zagreb connection index and first Zagreb connection index (ZC1) for r− dimensional Benes network and some classes generated from Benes network. The horizontal cylindrical Benes network (HCB(r)), vertical cylindrical Benes network (VCB(r)), and toroidal Benes network (TB(r)) are the three classes generated by identifying the vertices of the first row with the last row, the first column with the last column of the Benes network. The obtained results are also analyzed through graphical tools.

Space: The Re-Visioning Frontier of Biological Image Analysis with Graph Theory, Computational Geometry, and Spatial Statistics

Every biological image contains quantitative data that can be used to test hypotheses about how patterns were formed, what entities are associated with one another, and whether standard mathematical methods inform our understanding of biological phenomena. In particular, spatial point distributions and polygonal tessellations are particularly amendable to analysis with a variety of graph theoretic, computational geometric, and spatial statistical tools such as: Voronoi polygons; Delaunay triangulations; perpendicular bisectors; circumcenters; convex hulls; minimal spanning trees; Ulam trees; Pitteway violations; circularity; Clark-Evans spatial statistics; variance to mean ratios; Gabriel graphs; and, minimal spanning trees. Furthermore, biologists have developed a number of empirically related correlations for polygonal tessellations such as: Lewis’s law (the number of edges of convex polygons are positively correlated with the areas of these polygons): Desch’s Law (the number of edges of convex polygons are positively correlated with the perimeters of these polygons); and Errara’s Law (daughter cell areas should be roughly half that of their parent cells’ areas). We introduce a new Pitteway Law that the number of sides of the convex polygons in a Voronoi tessellation of biological epithelia is proportional to the minimal interior angle of the convex polygons as angles less than 90 degrees result in Pitteway violations of the Delaunay dual of the Voronoi tessellation.

Structure–activity relations for antiepileptic drugs through omega polynomials and topological indices

Epilepsy has impacted more than 50 million people worldwide, and it continues to pose challenges for drug discovery. There is a dearth of reliable experimental data on antiepileptic drugs (AED) and the mechanisms of drug activity are not well understood. In the current study we develop novel mathematical tools for antiepileptic molecules as an integral part of machine learning and artificial intelligence through omega polynomials and topological indices to derive robust structure–activity relations for the correlation and prediction of anticonvulsant activities as measured by their ED50 values. We have applied the developed mathematical and computational tools to 31 potential AEDs for the development of reliable structure–activity relations. We show using both statistical and topological tools that the developed structure–activity relations on the basis of graph theoretical tools provide a robust 3D-QSAR equation for the ED50 prediction of AEDs with an R value of 0.975, and one of the key factors that determines the potency of an AED is its ability to penetrate the blood–brain–barrier (BBB).