Topological Characterization Research Articles

Topological Characterization of Consensus in Distributed Systems

We provide a complete characterization of both uniform and non-uniform deterministic consensus solvability in distributed systems with benign process and communication faults using point-set topology. More specifically, we non-trivially extend the approach introduced by Alpern and Schneider in 1985, by introducing novel fault-aware topologies on the space of infinite executions: the process-view topology, induced by a distance function that relies on the local view of a given process in an execution, and the minimum topology, which is induced by a distance function that focuses on the local view of the process that is the last to distinguish two executions. Consensus is solvable in a given model if and only if the sets of admissible executions leading to different decision values is disconnected in these topologies. By applying our approach to a wide range of different applications, we provide a topological explanation of a number of existing algorithms and impossibility results and develop several new ones, including a general equivalence of the strong and weak validity conditions.

Distance based topological characterization, graph energy prediction, and NMR patterns of benzene ring embedded in P-type surface in 2D network

Nanostructures are tiny objects at the molecular and microscopic scale, with carbon nanotubes being the most notable among them. The elements possess exceptional microelectronic properties and other unique characteristics. Researchers have recently focused on the mathematical features of these materials. Molecular descriptors are crucial in mathematical chemistry, particularly in QSAR and QSPR modeling. Topological indices hold a significant position among them. This study presents the precise formulation of the ten most crucial topological indices for a benzene ring positioned on a P-type surface within the highly symmetric 2D lattice BCZ48\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {BCZ}_{48}$$\\end{document}. We have incorporated the computed indices to develop a predictive model for the graph energy of the 2D lattice and, in addition, provided the NMR patterns and the HOMO-LUMO gap.

Topological Characterization of Coronoid Structures via Valency Based Topological Indices.

The number of atoms producing a bond with an atom a in a molecular structure (graph) is regarded as the valency of a. A number of molecular structures have been topologically characterized via a large number of valency-based topological indices, which correlate certain physio-chemical properties like boiling point, stability, strain energy, and many more chemical compounds. Our intention is to perform the topological characterization of primitive coronoid structures initially on the base of seven different valency-based topological indices. Fundamentally, our purpose is to provide the universal formula by using the topological characterization of any primitive coronoid structures with any valency-based topological index. The technique of atom-bonds partitions according to the valency of atoms besides counting principals are used to find the results. Exact values of valency-based topological index, namely the Randic index, sum connectivity index, harmonic index, atom bond connectivity index, geometric arithmetic index, forgotten index, and symmetric division index are calculated coronoid structures initially constructed with at most three benzene lyres. All these indices are valency-dependent structural invariants that have been used to elaborate several chemical properties of compounds of the structure. Moreover, we provide the general formula to compute any valency-based topological index for any type of coronoid structure constructed with any number of benzene lyres.

Computational Characterization of Lymphocyte Topology in FSGS/Minimal Change Disease

Computational Characterization of Lymphocyte Topology in FSGS/Minimal Change Disease

Synthetic, Electrochemical, DFT, and Synchrotron X-ray Charge-Density Studies on Oxo-centered Triruthenium Clusters Supported by Electron-Withdrawing Carboxylates.

We herein report the synthesis, characterizations, and synchrotron X-ray charge-density studies of oxo-centered triruthenium(II,III,III) clusters [Ru3O(CHCl2COO)6(py)3] (1) and [Ru3O(CHCl2COO)6(CO)(py)2] (2) (py = pyridine). Dichloroacetate was chosen for its large scattering factor of the Cl atom, and its electron-withdrawing nature results in significant stabilization of the targeted lower-valent Ru3II,III,III state in the cluster framework. Multipole analysis revealed that the difference in electron populations between two crystallographically independent Ru centers is small for 1 (Δ = 0.30 e) but large for 2 (Δ = 1.46 e). Remarkable differences between 1 and 2 are also found in their static deformation density maps; substantial local charge depletion was found around the central μ3O atom for 1, which is less pronounced for 2. According to the topological characterization of Ru-μ3O bonds associated with the bond critical point, bcp, the electron density, ρbcp, is in the range of 0.79-0.89 e Å-3, and the total energy density, Hbcp, is in the range of -0.21 to -0.05 hartree Å-3. These findings represent the first charge-density distribution analysis of mixed-valence multinuclear Ru complexes including comparison between 3d and 4d transition-metal systems.

Modified reverse degree descriptors for combined topological and entropy characterizations of 2D metal organic frameworks: applications in graph energy prediction.

Topological descriptors are widely utilized as graph theoretical measures for evaluating the physicochemical properties of organic frameworks by examining their molecular structures. Our current research validates the usage of topological descriptors in studying frameworks such as metal-butylated hydroxytoluene, NH-substituted coronene transition metal, transition metal-phthalocyanine, and conductive metal-octa amino phthalocyanine. These metal organic frameworks are crucial in nanoscale research for their porosity, adaptability, and conductivity, making them essential for advanced materials and modern technology. In this study, we provide the topological and entropy characterizations of these frameworks by employing robust reverse degree based descriptors, which offer insightful information on structural complexities. This structural information is applied to predict the graph energy of the considered metal organic frameworks using statistical regression models.

Topology and functional characterization of major outer membrane proteins of Treponema maltophilum and Treponema lecithinolyticum.

Numerous Treponema species are prevalent in the dysbiotic subgingival microbial community during periodontitis. The major outer sheath protein is a highly expressed virulence factor of the well-characterized species Treponema denticola. Msp forms an oligomeric membrane protein complex with adhesin and porin properties and contributes to host-microbial interaction. Treponema maltophilum and Treponema lecithinolyticum species are also prominent during periodontitis but are relatively understudied. Msp-like membrane surface proteins exist in T. maltophilum (MspA) and T. lecithinolyticum (MspTL), but limited information exists regarding their structural features or functionality. Protein profiling reveals numerous differences between these species, but minimal differences between strains of the same species. Using protein modeling tools, we predict MspA and MspTL monomeric forms to be large β-barrel structures composed of 20 all-next-neighbor antiparallel β strands which most likely adopt a homotrimer formation. Using cell fractionation, Triton X-114 phase partitioning, heat modifiability, and chemical and detergent release assays, we found evidence of amphiphilic integral membrane-associated oligomerization for both native MspA and MspTL in intact spirochetes. Proteinase K accessibility and immunofluorescence assays demonstrate surface exposure of MspA and MspTL. Functionally, purified recombinant MspA or MspTL monomer proteins can impair neutrophil chemotaxis. Expressions of MspA or MspTL with a PelB leader sequence in Escherichia coli also demonstrate surface exposure and can impair neutrophil chemotaxis in an in vivo air pouch model of inflammation. Collectively, our data demonstrate that MspA and MspTL membrane proteins can contribute to pathogenesis of these understudied oral spirochete species.

Protein Complex Heterogeneity and Topology Revealed by Electron Capture Charge Reduction and Surface Induced Dissociation.

We illustrate the utility of native mass spectrometry (nMS) combined with a fast, tunable gas-phase charge reduction, electron capture charge reduction (ECCR), for the characterization of protein complex topology and glycoprotein heterogeneity. ECCR efficiently reduces the charge states of tetradecameric GroEL, illustrating Orbitrap m/z measurements to greater than 100,000 m/z. For pentameric C-reactive protein and tetradecameric GroEL, our novel device combining ECCR with surface induced dissociation (SID) reduces the charge states and yields more topologically informative fragmentation. This is the first demonstration that ECCR yields more native-like SID fragmentation. ECCR also significantly improved mass and glycan heterogeneity measurements of heavily glycosylated SARS-CoV-2 spike protein trimer and thyroglobulin dimer. Protein glycosylation is important for structural and functional properties and plays essential roles in many biological processes. The immense heterogeneity in glycosylation sites and glycan structure poses significant analytical challenges that hinder a mechanistic understanding of the biological role of glycosylation. Without ECCR, average mass determination of glycoprotein complexes is available only through charge detection mass spectrometry or mass photometry. With narrow m/z selection windows followed by ECCR, multiple glycoform m/z values are apparent, providing quick global glycoform profiling and providing a future path for glycan localization on individual intact glycoforms.

Computational insights into zinc silicate MOF structures: topological modeling, structural characterization and chemical predictions

Metal-organic frameworks (MOFs) play a pivotal role in modern material science, offering unique properties such as flexibility, substantial pore space, distinctive structure, and large surface area. Recently, zinc-based MOFs have attracted significant attention, particularly in the biomedical arena, owing to their versatile applications in drug delivery, biosensing, and cancer imaging. However, there remains a crucial need to explore and understand the structural properties of zinc silicate-based MOFs to fully exploit their potential in various applications. The objective of this study is to address this need by employing topological modeling techniques to characterize zinc silicate networks. Utilizing connection number concept of chemical graph theory and novel AL molecular descriptors, we aim to investigate the structural intricacies of these MOFs. More precisely, zinc silicate-based MOF networks are topologically modeled via novel AL topological indices, and derived mathematical closed form formulae for them. By comparing experimental and calculated values and constructing linear regression models, the predictive capabilities of the proposed descriptors are evaluated. Specifically, the performance of derived topological indices against the physico-chemical properties of octane isomers is assessed, which provide valuable insights into their predictive potential. The findings of this study demonstrated the potential of novel AL indices in predicting a wide range of important physico-chemical properties, further enhancing their practicality in materials science and beyond.

Interacting crystalline topological insulators in two-dimensions with time-reversal symmetry

Topology is routinely used to understand the physics of electronic insulators. However, for strongly interacting electronic matter, such as Mott insulators, a comprehensive topological characterization is still lacking. When their ground state only contains short-range entanglement and does not break symmetries spontaneously, they generically realize crystalline fermionic symmetry-protected topological phases (cFSPTs), supporting gapless modes at the boundaries or at the lattice defects. Here, we provide an exhaustive classification of cFSPTs in two dimensions with U(1) charge-conservation and spinful time-reversal symmetries, namely, those generically present in spin-orbit coupled insulators, for any of the 17 wallpaper groups. It has been shown that the classification of cFSPTs can be understood from appropriate real-space decorations of lower-dimensional subspaces, and we expose how these relate to the Wyckoff positions of the lattice. We find that all nontrivial one-dimensional decorations require electronic interactions. Furthermore, we provide model Hamiltonians for various decorations, and discuss the signatures of cFSPTs. This classification paves the way to further explore topological interacting insulators, providing the backbone information in generic model systems and ultimately in experiments. Published by the American Physical Society 2024

Clustering of negative topological charges precedes plastic failure in 3D glasses.

The deformation mechanism in amorphous solids subjected to external shear remains poorly understood because of the absence of well-defined topological defects mediating the plastic deformation. The notion of soft spots has emerged as a useful tool to characterize the onset of irreversible rearrangements and plastic flow, but these entities are not clearly defined in terms of geometry and topology. In this study, we unveil the phenomenology of recently discovered, precisely defined topological defects governing the microscopic mechanical and yielding behavior of a model 3D glass under shear deformation. We identify the existence of vortex-like and antivortex-like topological defects within the 3D nonaffine displacement field. The number density of these defects exhibits a significant anticorrelation with the plastic events, with defect proliferation-annihilation cycles matching the alternation of elastic-like segments and catastrophic plastic drops, respectively. Furthermore, we observe collective annihilation of these point-like defects via plastic events, with large local topological charge fluctuations in the vicinity of regions that feature strong nonaffine displacements. We reveal that plastic yielding is driven by several large sized clusters of net negative topological charge, the massive annihilation of which triggers the onset of plastic flow. These findings suggest a geometric and topological characterization of soft spots and pave the way for the mechanistic understanding of topological defects as mediators of plastic deformation in glassy materials.

Singular value decomposition for single-phase flow and cluster identification in heterogeneous pore networks

Pore networks play a key role in understanding and quantifying flow and transport processes in complex porous media. Realistic pore-spaces may be characterized by singular regions, that is, isolated subnetworks that do not connect inlet and outlet, resulting from unconnected porosity or multiphase configurations. The robust identification of these features is critical for the characterization of network topology and for the solution of the set of linear equations of flow and transport. We propose a robust method based on singular value decomposition to solve for network flow and locate isolated subnetworks simultaneously. The performance of the method is demonstrated for pore networks of different complexity.

Topological Characterization for Triangular, RegularTriangular Oxides and Silicates Networks

Chemical graph theory can be studied with the aid of mathematical tools called m-polynomials. M-Polynomials offer a potent tool for computing different topological indices associated with vertex degrees and analyzing degree-based structural information in graphs. By counting specific substructure types within them, they are able to encode information about the structure of molecules or networks. In this article, we have developed M-Polynomials with the help of different topological invariants such as first Zagreb (M1(β)), second Zagreb (M2(β)), second modified Zagreb (Mm2(β)), inverse sum (I(β)), harmonic index (H(β)) and Randic index (Rα0(β)) for the molecular structures of Triangular oxide TOX(r), Regular triangular oxide RTOX(r), Triangularsilicate TSL(r) & Regular triangular silicate RTSL(r) networks to introduce new closed formulas to get better understanding the applications of M-Polynomials and topological indices in mathematical chemistry especially in the field of QSAR and QSPR study with the help of some software like MATLAB. We have also discussed the graphical behaviors of the above-mentioned structures.

Lower density soft operators and density soft topologies

The classical density topology is derived from the Lebesgue density points of Lebesgue measurable sets. Wilczynski proposed a more general scheme for defining density points. It turns out that his scheme works in the category sense in addition to the measure-theoretic one. The sets of density points necessitate some properties of the operator of density points. Such an operator is named the lower density operator. In this theme, we introduce lower density soft operators on (abstract) chargeable soft spaces along with their basic properties. Among others, we show that each lower density soft operator defines a system of parametric lower density operators. Then, we study density soft topologies, the soft topologies generated by lower density soft operators. The main attributes and topological properties of density soft topologies are analyzed. We finalize this work by demonstrating several characterizations of (simple) soft density topologies. In particular, we prove that in density soft topologies, the Borel soft σ-algebra coincides with the soft σ-algebra of soft sets with the Baire property.

Symmetry constraints and spectral crossing in a Mott insulator with Green's function zeros

Lattice symmetries are central to the characterization of electronic topology. Recently, it was shown that Green's function eigenvectors form a representation of the space group. This formulation has allowed the identification of gapless topological states even when quasiparticles are absent. Here we demonstrate the profundity of the framework in the extreme case, when interactions lead to a Mott insulator, through a solvable model with long-range interactions. We find that both Mott poles and zeros are subject to the symmetry constraints, and relate the symmetry-enforced spectral crossings to degeneracies of the original noninteracting eigenstates. Our results lead to new understandings of topological quantum materials and highlight the utility of interacting Green's functions toward their symmetry-based design. Published by the American Physical Society 2024

The structural evolution of temporal hypergraphs through the lens of hyper-cores

The richness of many complex systems stems from the interactions among their components. The higher-order nature of these interactions, involving many units at once, and their temporal dynamics constitute crucial properties that shape the behaviour of the system itself. An adequate description of these systems is offered by temporal hypergraphs, that integrate these features within the same framework. However, tools for their temporal and topological characterization are still scarce. Here we develop a series of methods specifically designed to analyse the structural properties of temporal hypergraphs at multiple scales. Leveraging the hyper-core decomposition of hypergraphs, we follow the evolution of the hyper-cores through time, characterizing the hypergraph structure and its temporal dynamics at different topological scales, and quantifying the multi-scale structural stability of the system. We also define two static hypercoreness centrality measures that provide an overall description of the nodes aggregated structural behaviour. We apply the characterization methods to several data sets, establishing connections between structural properties and specific activities within the systems. Finally, we show how the proposed method can be used as a model-validation tool for synthetic temporal hypergraphs, distinguishing the higher-order structures and dynamics generated by different models from the empirical ones, and thus identifying the essential model mechanisms to reproduce the empirical hypergraph structure and evolution. Our work opens several research directions, from the understanding of dynamic processes on temporal higher-order networks to the design of new models of time-varying hypergraphs.

Elemental evolution characteristics and influencing factors of green infrastructure network in karst mountain cities: a case study of Qianzhong urban agglomeration in Southwest China

BackgroundThe urban green infrastructure (GI) network is an important conduit for ecological flows and plays a crucial role in improving regional habitats, especially in karst areas that are highly ecologically fragile and sensitive. However, the existing research only focuses on the construction of GI network in karst mountain cities, and the evolution characteristics of its elements and driving mechanism are not clear, which is of great significance for guiding urban land use planning and comprehensively improving the quality of the ecological environment. In view of this, this study took Qianzhong urban agglomeration as the study area, based on multi-source data, and identified ecological sources through ecological resilience analysis. Considering the special geographic environment, the rock exposure rate factor was added to correct the resistance surface, and the minimum cumulative resistance (MCR) and gravity model were coupled to extract the GI network. The complex network topology characterization parameter was introduced to assess the spatial and temporal variations of ecological sources and corridors. Finally, the geographical detector was used to identify the dominant influencing factors and interactions of the spatial distribution of the GI network.ResultsThe results showed that from 2000 to 2020, the condition of GI network elements in the study area presented a decreasing and then an increasing trend. The ecological sources or corridors in highly urbanized areas were critical for ecological flow transport and the overall structural stability of the GI network. The influence of natural factors on the spatial distribution of the GI network gradually weakened, and the influence of human factors continuously increased. The spatial distribution of the GI network was influenced by multiple factors, and the interaction between all the factors was enhanced, which gradually changed from the interaction of natural factors to the interaction of human factors during the study period.ConclusionsThe research results will provide scientific references for the construction of an ecologically safe environment and sustainable development of karst mountain cities.

Topological Characterization of Metal-Organic Frameworks: A Perspective.

Metal-organic frameworks (MOFs) began to emerge over two decades ago, resulting in the deposition of 120 000 MOF-like structures (and counting) into the Cambridge Structural Database (CSD). Topological analysis is a critical step toward understanding periodic MOF materials, offering insight into the design and synthesis of these crystals via the simplification of connectivity imposed on the complete chemical structure. While some of the most prevalent topologies, such as face-centered cubic (fcu), square lattice (sql), and diamond (dia), are simple and can be easily assigned to structures, MOFs that are built from complex building blocks, with multiple nodes of different symmetry, result in difficult to characterize topological configurations. In these complex structures, representations can easily diverge where the definition of nodes and linkers are blurred, especially for cases where they are not immediately obvious in chemical terms. Currently, researchers have the option to use software such as ToposPro, MOFid, and CrystalNets to aid in the assignment of topology descriptors to new and existing MOFs. These software packages are readily available and are frequently used to simplify original MOF structures into their basic connectivity representations before algorithmically matching these condensed representations to a database of underlying mathematical nets. These approaches often require the use of in-built bond assignment algorithms alongside the simplification and matching rules. In this Perspective, we discuss the importance of topology within the field of MOFs, the methods and techniques implemented by these software packages, and their availability and limitations and review their uptake within the MOF community.

Protein complex heterogeneity and topology revealed by electron capture charge reduction and surface induced dissociation.

We illustrate the utility of native mass spectrometry (nMS) combined with a fast, tunable gas-phase charge reduction, electron capture charge reduction (ECCR), for the characterization of protein complex topology and glycoprotein heterogeneity. ECCR efficiently reduces the charge states of tetradecameric GroEL, illustrating Orbitrap m/z measurements to greater than 100,000 m/z. For pentameric C-reactive protein and tetradecameric GroEL, our novel device combining ECCR with surface induced dissociation (SID) reduces the charge states and yields more topologically informative fragmentation. This is the first demonstration that ECCR yields more native-like SID fragmentation. ECCR also significantly improved mass and glycan heterogeneity measurements of heavily glycosylated SARS-CoV-2 spike protein trimer and thyroglobulin dimer. Protein glycosylation is important for structural and functional properties and plays essential roles in many biological processes. The immense heterogeneity in glycosylation sites and glycan structure poses significant analytical challenges that hinder a mechanistic understanding of the biological role of glycosylation. Without ECCR, average mass determination of glycoprotein complexes is available only through charge detection mass spectrometry or mass photometry. With narrow m/z selection windows followed by ECCR, multiple glycoform m/z values are apparent, providing quick global glycoform profiling and providing a future path for glycan localization on individual intact glycoforms.

Topological Characterization and Typical Topologies of Disruption Aggregates in Asphalt Mixture

Topological Characterization and Typical Topologies of Disruption Aggregates in Asphalt Mixture