Core Vertices Research Articles

Bounds on Molecular Properties of π Systems: First-order Properties

Wederive upper and lower bounds to first-order properties in the H¨uckel model: π charge, bond order, bond number, and matrices of higher spectral moments. Bounds that depend on the electronic configuration, and on the molecular graph alone are derived. A ladder of relations between higher and lower spectral moments leads to bounds via the Cauchy-Schwarz theorem. The old square-root degree bound on bond number implicit in the work of Coulson, Moffitt and Longuet-Higgins is sharpened. Key to this development is the distinction between core and core-forbidden vertices of a graph.

Stable structural clustering in uncertain graphs

The uncertain graph is widely used to model and analyze graph data in which the relation between objects is uncertain. We here study the structural clustering in uncertain graphs. As an important method in graph clustering, structural clustering can not only discover the densely connected core vertices, but also the hub vertices and the outliers. We propose a new clustering model named stable structural clustering, to solve the problem existing in previous models that the mined core vertex is a ‘real’ core one in only a small amount of possible worlds of the uncertain graph. Specifically, we first propose the concept of probability core reliability which measures the probability of a vertex being a core vertex in the uncertain graph. On the basis of probability core reliability, we propose the definition of stable core vertex and formulate the stable structural clustering problem. Comparing with other structural clustering models, the proposed stable structural clustering performs better in crucial indicators that reflect the quality of clustering. We develop two algorithms to calculate stable core vertex, a precise dynamic programming based algorithm and a sampling based algorithm with some effective pruning techniques, based on which we give our structural clustering algorithm. Extensive experiments show that comparing with other structural clustering algorithms in uncertain graphs, the stable structural clustering algorithms proposed can get better clustering to a certain extent.

λ–Core distance partitions

The λ–core vertices of a graph correspond to the non–zero entries of some eigenvector of λ for a universal adjacency matrix U of the graph. We define a partition of the vertex set V based on the λ–core vertex set and its neighbourhoods at a distance r, and give a number of results relating the structure of the graph to this partition. For such partitions, we also define an entropic measure for the information content of a graph, related to every distinct eigenvalue λ of U, and discuss its properties and potential applications.

Nullspace vertex partition in graphs

The core vertex set of a graph is an invariant of the graph. It consists of those vertices associated with the non-zero entries of the nullspace vectors of a $$\{0,1\}$$ -adjacency matrix. The remaining vertices of the graph form the core-forbidden vertex set. For graphs with independent core vertices, such as bipartite minimal configurations and trees, the nullspace induces a well defined three part vertex partition. The parts of this partition are the core vertex set, their neighbours and the remote core-forbidden vertices. The set of the remote core-forbidden vertices are those not adjacent to any core vertex. We show that this set can be removed, leaving the nullity unchanged. For graphs with independent core vertices, we show that the submatrix of the adjacency matrix defining the edges incident to the core vertices determines the nullity of the adjacency matrix. For the efficient allocation of edges in a network graph without altering the nullity of its adjacency matrix, we determine which perturbations make up sufficient conditions for the core vertex set of the adjacency matrix of a graph to be preserved in the process.

The complexity of finding small separators in temporal graphs

Temporal graphs have time-stamped edges. Building on previous work, we study the problem of finding a small vertex set (the separator) whose removal destroys all temporal paths between two designated terminal vertices. Herein, we consider two models of temporal paths: those that pass through arbitrarily many edges per time step (non-strict) and those that pass through at most one edge per time step (strict). Regarding the number of time steps of a temporal graph, we show a complexity dichotomy (NP-completeness versus polynomial-time solvability) for both problem variants. Moreover, we prove both problem variants to be NP-complete even on temporal graphs whose underlying graph is planar. Finally, we introduce the notion of a temporal core (vertices whose incident edges change over time) and prove that the non-strict variant is fixed-parameter tractable when parameterized by the temporal core size, while the strict variant remains NP-complete, even for constant-size temporal cores.

GA-ADE: a novel approach based on graph algorithm to improves the detection of adverse drug events

The adverse drug event (ADE) is an unexpected and harmful consequence of drug usege. Identifying the association between the use of drugs and adverse events from biomedical literature can contribute a lot to drug safety supervision. Such identification can not only assist drug safety monitoring, but also correct known dependencies among events. In this paper,we propose a novel approach based on graph algorithm to detect adverse drug events(GA-ADE). In our approach, we first construct a graph using candidate ADE extracted from biomedical literature. We then propose a method to select important vertices from the graph as core vertices, and design a Personal Rank algorithm using the core vertices for clustering to build subgraphs. Lastly, the correlation between the drug and the event is calculated based on the subgraphs. Experiments show that our approach is feasible.

Constructing Three-Dimensional Mesoporous Bouquet-Posy-like TiO2 Superstructures with Radially Oriented Mesochannels and Single-Crystal Walls.

Constructing three-dimensional (3-D) hierarchical mesostructures with unique morphology, pore orientation, single-crystal nature, and functionality remains a great challenge in materials science. Here, we report a confined microemulsion self-assembly approach to synthesize an unprecedented type of 3-D highly ordered mesoporous TiO2 superstructure (Level-1), which consists of 1 spherical core and 12 symmetric satellite hemispheres epitaxially growing out of the core vertices. A more complex and asymmetric TiO2 superstructure (Level-2) with 13 spherical cores and up to 44 symmetric satellite hemispheres can also be well manipulated by increasing the size or content of impregnated TiO2 precursor emulsion droplets. The obtained 3-D mesoporous TiO2 superstructures have well-defined bouquet-posy-like topologies, oriented hexagonal mesochannels, high accessible surface area (134-148 m2/g), large pore volume (0.48-0.51 cm3/g), and well single-crystalline anatase walls with dominant (001) active facets. More interestingly, all cylindrical mesopore channels are highly interconnected and radially distributed within the whole superstructures, and all TiO2 nanocrystal building blocks are oriented grown into a single-crystal anatase wall, making them ideal candidates for various applications ranging from catalysis to optoelectronics. As expected, the bouquet-posy-like mesoporous TiO2 superstructure supported catalysts show excellent catalytic activity (≥99.7%) and selectivity (≥96%) in cis-semihydrogenation of various alkynes, exceeding that of commercial TiO2 (P25) supported catalyst by a factor of 10. No decay in the activity was observed for 25 cycles, revealing a high stability of the mesoporous TiO2 superstructure supported catalyst.

Core–periphery models for graphs based on their δ-hyperbolicity: An example using biological networks

Hyperbolicity is a global property of graphs that measures how close their structures are to trees in terms of their distances. It embeds multiple properties that facilitate solving several problems that found to be hard in the general graph form. In this paper, we investigate the hyperbolicity of graphs not only by considering Gromov’s notion of δ-hyperbolicity but also by analyzing its relationship to other graph’s parameters. This new perspective allows us to classify graphs with respect to their hyperbolicity and to show that many biological networks are hyperbolic. Then we introduce the eccentricity-based bending property which we exploit to identify the core vertices of a graph by proposing two models: the maximum-peak model and the minimum cover set model. In this extended version of the paper, we include some new theorems, as well as proofs of the theorems proposed in the conference paper. Also, we present the algorithms we used for each of the proposed core identification models, and we provide more analysis, explanations, and examples.

Coalescing Fiedler and core vertices

The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex of unique type in the coalescence. Moreover, the nullity of subgraphs obtained by perturbations of the coalescence G is determined relative to the nullity of G. This has direct applications in spectral graph theory as well as in the construction of certain ipso-connected nano-molecular insulators.

Detecting and refining overlapping regions in complex networks with three-way decisions

The identification of communities is crucial to an understanding of the structural and functional properties of a complex network. Many methods and algorithms have been developed to detect overlapping communities. A problem that has not been addressed satisfactorily is the relationship or difference between vertices in overlapping regions during the formation and growth of communities. This paper investigates methods that not only detect the overlapping communities but also refine the overlapping regions. We give a three-way representation of a community by using interval sets and re-formalize the problem of community detection as three-way clustering. We suggest four macro types and eight micro types of vertices to characterize members in overlapping regions for their refinement. We propose an overlapping community detection algorithm by classifying the vertices into core vertices, bone vertices, and trivial vertices. The main strategy of this algorithm is to find an initial cluster core, to expand the core to a preliminary community according to a new fitness function, and to merge trivial vertices based on three-way decision strategies. The experimental results on both real-world social networks and computer-generated artificial networks show the effectiveness and efficiency of the proposed methods.

Detection of core–periphery structure in networks using spectral methods and geodesic paths

We introduce several novel and computationally efficient methods for detecting “core–periphery structure” in networks. Core–periphery structure is a type of mesoscale structure that consists of densely connected core vertices and sparsely connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that that vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which we express as a perturbation of a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically generated networks and a variety of networks constructed from real-world data sets.

Network analysis of terrorist activities

This paper uses an extensive network approach to “East Turkistan” activities by building both the one-mode and the bipartite networks for these activities. In the one-mode network, centrality analysis and spectrum analysis are used to describe the importance of each vertex. On this basis, two types of core vertices — The center of communities and the intermediary vertices among communities — are distinguished. The weighted extreme optimization (WEO) algorithm is also applied to detect communities in the one-mode network. In the “terrorist-terrorist organization” bipartite network, the authors adopt centrality analysis as well as clustering analysis based on the original bipartite network in order to calculate the importance of each vertex, and apply the edge clustering coefficient algorithm to detect the communities. The comparative and empirical analysis indicates that this research has been proved to be an effective way to identify the core members, key organizations, and communities in the network of “East Turkistan” terrorist activity. The results can provide a scientific basis for the analysis of “East Turkistan” terrorist activity, and thus provide decision support for the real work of “anti-terrorism”.

Controllability of undirected graphs

In control theory, networked dynamical systems have a wide range of engineering applications. In a relational graph among followers (F) and leaders (R), new necessary and sufficient conditions for a pair (F,R) to be controllable are presented. The choice of leader vertices for controllability is shown to be facilitated by identifying the core vertices associated with the eigenvectors of a matrix S related to a graph. We present new necessary and sufficient conditions for a graph to be controllable relative to its adjacency matrix or to its signless Laplacian without having to evaluate any eigenspaces, which is the criterion usually employed. The symmetries of the system graph represented by S are also shown to aid in the choice of a potential leader vertex that is able to control the follower subgraph on its own. Moreover, we define k-omnicontrollable graphs for controllability by any k leaders and show that simple 1-omnicontrollable graphs have only two possible automorphism groups.

Correlated percolation and tricriticality

The recent proliferation of correlated percolation models--models where the addition of edges and/or vertices is no longer independent of other edges and/or vertices--has been motivated by the quest to find discontinuous percolation transitions. The leader in this proliferation is what is known as explosive percolation. A recent proof demonstrates that a large class of explosive percolation-type models does not, in fact, exhibit a discontinuous transition [Riordan and Warnke, Science, 333, 322 (2011)]. Here, we discuss two lesser known correlated percolation models--the k ≥ 3-core model on random graphs and the counter-balance model in two-dimensions--both exhibiting discontinuous transitions. To search for tricriticality, we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition. Using a powerful rate equation approach, we demonstrate that a mixture of k = 2-core and k = 3-core vertices on the random graph exhibits a tricritical point. However, for a mixture of k-core and counter-balance vertices in two dimensions, as the fraction of counter-balance vertices is increased, numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition, i.e., when all vertices are counter-balanced. Interestingly, these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions. In addition, our results may have potential implications for glassy and jamming systems.

The role of core documents in bibliometric network analysis and their relation with h-type indices

The notion of core documents and their application is discussed in the context of scientometric networks. An interesting solution of the problem of the arbitrariness of thresholds emerges from the application of Hirsch-type indices to dense networks as are typically observed in local clustering. Examples from several disciplines in the sciences and social sciences illustrate how these core vertices can be determined using this approach, and visualise how core documents are applied to represent the internal structure of the complete network or of parts of it.

Identifying the overlapping complexes in protein interaction networks

Identification of protein complexes in large interaction networks is crucial to understanding principles of cellular organisation and predict protein functions. In this paper, a new algorithm of Identifying Protein Complexes based on Maximal Clique Extension (IPC-MCE) is proposed. The maximal clique is considered as the core of the protein complex. Proteins in a complex are classed into core vertices and peripheral vertices. The relation between the core vertices and peripheral vertices is measured by the Interaction Probability. The algorithm IPC-MCE is applied to the protein interaction network of Saccharomyces cerevisiae. Many well-known protein complexes are detected.

Nonbonding Orbitals in Fullerenes: Nuts and Cores in Singular Polyhedral Graphs

A zero eigenvalue in the spectrum of the adjacency matrix of the graph representing an unsaturated carbon framework indicates the presence of a nonbonding pi orbital (NBO). A graph with at least one zero in the spectrum is singular; nonzero entries in the corresponding zero-eigenvalue eigenvector(s) (kernel eigenvectors) identify the core vertices. A nut graph has a single zero in its adjacency spectrum with a corresponding eigenvector for which all vertices lie in the core. Balanced and uniform trivalent (cubic) nut graphs are defined in terms of (-2, +1, +1) patterns of eigenvector entries around all vertices. In balanced nut graphs all vertices have such a pattern up to a scale factor; uniform nut graphs are balanced with scale factor one for every vertex. Nut graphs are rare among small fullerenes (41 of the 10 190 782 fullerene isomers on up to 120 vertices) but common among the small trivalent polyhedra (62 043 of the 398 383 nonbipartite polyhedra on up to 24 vertices). Two constructions are described, one that is conjectured to yield an infinite series of uniform nut fullerenes, and another that is conjectured to yield an infinite series of cubic polyhedral nut graphs. All hypothetical nut fullerenes found so far have some pentagon adjacencies: it is proved that all uniform nut fullerenes must have such adjacencies and that the NBO is totally symmetric in all balanced nut fullerenes. A single electron placed in the NBO of a uniform nut fullerene gives a spin density distribution with the smallest possible (4:1) ratio between most and least populated sites for an NBO. It is observed that, in all nut-fullerene graphs found so far, occupation of the NBO would require the fullerene to carry at least 3 negative charges, whereas in most carbon cages based on small nut cubic polyhedra, the NBO would be the highest occupied molecular orbital (HOMO) for the uncharged system.

Core-periphery organization of complex networks

Networks may, or may not, be wired to have a core that is both itself densely connected and central in terms of graph distance. In this study we propose a coefficient to measure if the network has such a clear-cut core-periphery dichotomy. We measure this coefficient for a number of real-world and model networks and find that different classes of networks have their characteristic values. Among other things we conclude that geographically embedded transportation networks have a strong core-periphery structure. We proceed to study radial statistics of the core, i.e., properties of the neighborhoods of the core vertices for increasing n. We find that almost all networks have unexpectedly many edges within n neighborhoods at a certain distance from the core suggesting an effective radius for nontrivial network processes.

A Simple Algorithm for Unique Representation of Chemical StructuresCyclic/Acyclic Functionalized Achiral Molecules

An algorithm, based on "vertex priority values" has been proposed to uniquely sequence and represent connectivity matrices of chemical structures of cyclic/acyclic functionalized achiral hydrocarbons and their derivatives. In this method, "vertex priority values" have been assigned in terms of atomic weights, subgraph lengths, loops, and heteroatom contents. Subsequently, the terminal vertices have been considered upon completing the sequencing of the core vertices. This approach provides a multilayered connectivity graph, which can be put to use in comparing two or more structures or parts thereof for any given purpose. Furthermore, the basic vertex connection tables generated here are useful in the computation of characteristic matrices/topological indices and automorphism groups and in storing, sorting, and retrieving chemical structures from databases.