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At least 667 records · Page 37Linked to original sources
PubMed2005

Voter model on heterogeneous graphs.

We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of N nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus T(N) scales as Nmu(2)1/mu(2), where mu(k) is the kth moment of the degree distribution. For a power-law degree distribution n(k) approximately k(-nu), T(N) thus scales as N for nu > 3, as N/ln(N for nu = 3, as N((2nu-4)/(nu-1)) for 2 < nu < 3, as (lnN)2 for nu = 2, and as omicron(1) for nu < 2. These results agree with simulation data for networks with both uncorrelated and correlated node degrees.

Journal Article↗
PubMed2005

Fractal structure of high-temperature graphs of O(N) models in two dimensions.

The critical behavior of the two-dimensional O(N) model close to criticality is shown to be encoded in the fractal structure of the high-temperature graphs of the model. Based on Monte Carlo simulations and with the help of percolation theory, de Gennes' results for polymer rings, corresponding to the limit N-->0, are generalized to random loops for arbitrary -2<or=N<or=2. The loops are studied also close to their tricritical point, known as the Theta point in the context of polymers, where they collapse. The corresponding fractal dimensions are argued to be in one-to-one correspondence with those at the critical point, leading to an analytic prediction for the magnetic scaling dimension at the O(N) tricritical point.

Journal Article↗
PubMed2006

Fracture surfaces as multiscaling graphs.

Fracture paths in quasi-two-dimensional (2D) media (e.g., thin layers of materials or paper) are analyzed as self-affine graphs h(x) of height h as a function of length x. We show that these are multiscaling, in the sense that nth order moments of the height fluctuations across any distance l scale with a characteristic exponent that depends nonlinearly on the order of the moment. Having demonstrated this, one rules out a widely held conjecture that fracture in 2D belongs to the universality class of directed polymers in random media. In fact, 2D fracture does not belong to any of the known kinetic roughening models. The presence of multiscaling offers a stringent test for any theoretical model; we show that a recently introduced model of quasistatic fracture passes this test.

Journal Article↗
PubMed2006

Evolutionary dynamics on degree-heterogeneous graphs.

The evolution of two species with different fitness is investigated on degree-heterogeneous graphs. The population evolves either by one individual dying and being replaced by the offspring of a random neighbor (voter model dynamics) or by an individual giving birth to an offspring that takes over a random neighbor node (invasion process dynamics). The fixation probability for one species to take over a population of N individuals depends crucially on the dynamics and on the local environment. Starting with a single fitter mutant at a node of degree k, the fixation probability is proportional to k for voter model dynamics and to 1/k for invasion process dynamics.

Animals↗
PubMed2006

Perfect quantum state transfer with spinor bosons on weighted graphs.

A duality between the properties of many spinor bosons on a regular lattice and those of a single particle on a weighted graph reveals that a quantum particle can traverse an infinite hierarchy of networks with perfect probability in polynomial time, even as the number of nodes increases exponentially. The one-dimensional "quantum wire" and the hypercube are special cases in this construction, where the number of spin degrees of freedom is equal to one and the number of particles, respectively. An implementation of a near-perfect quantum state transfer across a weighted parallelepiped with ultracold atoms in optical lattices is discussed.

Journal Article↗
PubMed2000

Search for isotypism in crystal structures by means of the graph theory

A method for the classification of crystal structures of chemical compounds is proposed, which is based on the representation of the system of interatomic bonds in a crystal as a finite 'reduced' graph. The program IsoTest is described, allowing one to find automatically the topological similarity (isotypism) for large groups of stoichiometrically and structurally different compounds. The analysis of crystal structures of simple and double sulfates and binary inorganic compounds was carried out and numerous examples of topological isotypism of the representatives of these groups of substances were found. It is shown that in many cases the ionic sublattices, constructed according to one of the close packings, can be selected in the sulfate crystal structures.

Journal Article↗
PubMed2000

Automated assignment of graph-set descriptors for crystallographically symmetric molecules

Algorithms for the automatic assignment of graph-set notation for intermolecular networks have been extended to molecules having internal crystallographic symmetry, for patterns up to the second level. This provides a means of achieving systematic and consistent assignments for networks containing symmetric molecules. These methodologies have been implemented in the program RPLUTO. Examples are given of the application of the method to a number of molecules with hydrogen-bonded and other intermolecular networks, illustrating the diversity of the patterns that occur.

Journal Article↗
PubMed2000

Graph-set and packing analysis of hydrogen-bonded networks in polyamide structures in the Cambridge Structural Database.

The hydrogen-bond networks and crystal packing of 81 unique secondary di- and polyamides in the Cambridge Structural Database are investigated. Graph-set analysis, as implemented in the RPluto program, is used to classify network motifs. These have been rationalized in terms of the relative dispositions of the amide groups. Peptide and retropeptides exhibit significant conformational flexibility, which permits alternative hydrogen-bonding patterns. In peptides, dihedral angles of -psi approximately varphi approximately 105 degrees allow an antiparallel ladder arrangement, containing rings of either the same or alternating sizes. For retropeptides, and diamides with an odd number of CH(2) spacers, this conformation leads to a parallel ladder with rings of equal size. If varphi approaches -60 degrees and psi 180 degrees, ladders adopt a helical twist, and if the conformation is distorted further, a three-dimensional network is usually adopted. Diamides with aromatic or an even number of CH(2) spacers generally form either antiparallel ladders or sheets, although some exhibit both polymorphs. Symmetry relationships within and between hydrogen-bonded chains, ladders and sheets in the crystal packing have also been analysed. Polyamides form considerably more complex networks, although many of the structural motifs present in the diamides occur as components of these networks.

Journal Article↗
PubMed2004

Geometry of the 2-aminoheterocyclic-carboxylic acid R2(2)(8) graph set: implications for crystal engineering.

The geometry of the R2(2)(8) graph set formed between a 2-aminoheterocyclic ring containing an Nsp2 atom (in the 1-position of the ring) and a carboxylic acid has been studied. Collating data from known co-crystal structures containing five- and six-membered heterocyclic rings from the Cambridge Structural Database revealed unexpected differences between two kinds of non-hydrogen contact distances, and between specific bond distances and angles of the heterocycle. Not only were the interatomic non-hydrogen distances between the N atoms (heterocycle) and O atoms (carboxylate) asymmetric, but also the 2-amino N atom (N21) to the heterocyclic C atom (C2) bond was shorter than the C2 to N1sp2 bond. However, this shortening of the C2-N21 bond was not observed in the examples where N21 was substituted with a non-H atom. For the six-membered rings the data also showed that as the C2-N21 bond shortened the N1-C2-N21 bond angle increased.

Journal Article↗
PubMed1990

Graph-set analysis of hydrogen-bond patterns in organic crystals.

A method is presented based on graph theory for categorizing hydrogen-bond motifs in such a way that complex hydrogen-bond patterns can be disentangled, or decoded, systematically and consistently. This method is based on viewing hydrogen-bond patterns topologically as if they were intertwined nets with molecules as the nodes and hydrogen bonds as the lines. Surprisingly, very few parameters are needed to define the hydrogen-bond motifs comprising these networks. The methods for making these assignments, and examples of their chemical utility are given.

Chemical Phenomena↗
PubMed1999

Visualization and characterization of non-covalent networks in molecular crystals: automated assignment of graph-set descriptors for asymmetric molecules.

A method of visualizing intermolecular networks (for example, hydrogen-bonded networks) in the crystalline state has been developed, based on the concept of link atoms, i.e. those atoms deemed to be in contact with each unique molecule or ion in the crystal chemical unit (CCU). Extension of a structure using each of these primary links can be achieved, enabling the generation and investigation of extended networks. Algorithms have been developed for the automatic assignment of graph-set notation for patterns up to second level, i.e. those involving one or two crystallographically independent non-covalent bonds, in the absence of internal crystallographic symmetry in the unique molecules of the CCU. The self, ring, chain and discrete motifs may be displayed by highlighting the atoms and bonds comprising the pattern. These methodologies have been implemented in the Cambridge Structural Database program PLUTO.

Journal Article↗
PubMed1991

The use of modified constellation graph method for computer-aided classification of congenital heart diseases.

This paper describes a new method of data reduction and classification in a multidimensional symptom space for diagnostic aid of congenital heart diseases. The algorithm developed here is to reduce interactively a multidimensional symptom space to sectorial regions representing each disease in a semicircle using the modified constellation graph method. This method enables us to classify patients using the angle in the semicircle as a single classifying parameter with an accuracy of about 90%, that is, with little overlapping between disease sectors. Comparing this method with conventional factor analysis, we have found the former far more effective than the latter for disease region separation.

Algorithms↗
PubMed2006

Multiscale segmentation with vector-valued nonlinear diffusions on arbitrary graphs.

We propose a novel family of nonlinear diffusion equations and apply it to the problem of segmentation of multivalued images. We show that this family can be viewed as an extension of stabilized inverse diffusion equations (SIDEs) which were proposed for restoration, enhancement, and segmentation of scalar-valued signals and images in [39]. Our new diffusion equations can process vector-valued images defined on arbitrary graphs which makes them well suited for segmentation. In addition, we introduce novel ways of utilizing the shape information luring the diffusion process. We demonstrate the effectiveness of our methods on a large number of segmentation tasks.

Algorithms↗
PubMed2004

A POCS-based graph matching algorithm.

A novel Projections Onto Convex Sets (POCS) graph matching algorithm is presented. Two-way assignment constraints are enforced without using elaborate penalty terms, graduated nonconvexity, or sophisticated annealing mechanisms to escape from poor local minima. Results indicate that the presented algorithm is robust and compares favorably to other well-known algorithms.

Algorithms↗
PubMed2006

Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization.

We provide evidence that nonlinear dimensionality reduction, clustering, and data set parameterization can be solved within one and the same framework. The main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise. Our construction, which is based on a Markov random walk on the data, offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry. We show that clustering in embedding spaces is equivalent to compressing operators. The objective of data partitioning and clustering is to coarse-grain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy. We show that the quantization distortion in diffusion space bounds the error of compression of the operator, thus giving a rigorous justification for k-means clustering in diffusion space and a precise measure of the performance of general clustering algorithms.

Algorithms↗
PubMed2006

Optimal surface segmentation in volumetric images--a graph-theoretic approach.

Efficient segmentation of globally optimal surfaces representing object boundaries in volumetric data sets is important and challenging in many medical image analysis applications. We have developed an optimal surface detection method capable of simultaneously detecting multiple interacting surfaces, in which the optimality is controlled by the cost functions designed for individual surfaces and by several geometric constraints defining the surface smoothness and interrelations. The method solves the surface segmentation problem by transforming it into computing a minimum s-t cut in a derived arc-weighted directed graph. The proposed algorithm has a low-order polynomial time complexity and is computationally efficient. It has been extensively validated on more than 300 computer-synthetic volumetric images, 72 CT-scanned data sets of different-sized plexiglas tubes, and tens of medical images spanning various imaging modalities. In all cases, the approach yielded highly accurate results. Our approach can be readily extended to higher-dimensional image segmentation.

Algorithms↗
PubMed2006

Fast agglomerative clustering using a k-nearest neighbor graph.

We propose a fast agglomerative clustering method using an approximate nearest neighbor graph for reducing the number of distance calculations. The time complexity of the algorithm is improved from O(tauN2) to O(tauNlogN) at the cost of a slight increase in distortion; here, tau denotes the number of nearest neighbor updates required at each iteration. According to the experiments, a relatively small neighborhood size is sufficient to maintain the quality close to that of the full search.

Algorithms↗