Parametrical embeddings of static spherically symmetric spacetimes

An analysis for a direct calculation of the embeddings in flat spacetimes of static spherically symmetric manifolds with Lorentz metric is worked out. For each manifold with non-constant curvature we arrive at a parametrical embedding which represents an infinite geometrical multiplicity of the embedded surface. The embeddings of manifolds with constant curvature are not parametrical and can be determined univocally. Examples concerning Schwarzschild, Reissner-Weyl, de Sitter and anti-de Sitter spacetimes are considered.     

Essential Self-adjointness for Combinatorial Schrödinger Operators II-Metrically non Complete Graphs

We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schrödinger operators in the metrically non complete case.     

The use of dynamical networks to detect the hierarchical organization of financial market sectors

Two kinds of filtered networks: minimum spanning trees (MSTs) and planar maximally filtered graphs (PMFGs) are constructed from dynamical correlations computed over a moving window. We study the evolution over time of both hierarchical and topological properties of these graphs in relation to market fluctuations. We verify that the dynamical PMFG preserves the same hierarchical structure as the dynamical MST, providing in addition a more significant and richer structure, a stronger robustness and dynamical stability. Central and peripheral stocks are differentiated by using a combination of different topological measures. We find stocks well connected and central; stocks well connected but peripheral; stocks poorly connected but central; stocks poorly connected and peripheral. It results that the Financial sector plays a central role in the entire system. The robustness, stability and persistence of these findings are verified by changing the time window and by performing the computations on different time periods. We discuss these results and the economic meaning of this hierarchical positioning.     

Generalized Goodman–Harpe–Jones Construction of Subfactors, I

We construct some exceptional finite depth subfactors and determine their principal graphs from some exceptional integrable lattice models. Some of these subfactors are conjectured to be the same as those coming from certain conformal embeddings (, , and ) for which the principal graphs are previously unknown. Received: 24 November 1995 / Accepted: 14 May 1996     

Scale free effects in world currency exchange network

A large collection of daily time series for 60 world currencies' exchange rates is considered. The correlation matrices are calculated and the corresponding Minimal Spanning Tree (MST) graphs are constructed for each of those currencies used as reference for the remaining ones. It is shown that multiplicity of the MST graphs' nodes to a good approximation develops a power like, scale free distribution with the scaling exponent similar as for several other complex systems studied so far. Furthermore, quantitative arguments in favor of the hierarchical organization of the world currency exchange network are provided by relating the structure of the above MST graphs and their scaling exponents to those that are derived from an exactly solvable hierarchical network model. A special status of the USD during the period considered can be attributed to some departures of the MST features, when this currency (or some other tied to it) is used as reference, from characteristics typical to such a hierarchical clustering of nodes towards those that correspond to the random graphs. Even though in general the basic structure of the MST is robust with respect to changing the reference currency some trace of a systematic transition from somewhat dispersed – like the USD case – towards more compact MST topology can be observed when correlations increase.     

Routing of 2-D switching networks by their embedding into cubes

The proposed all-optical 2-D switching networks are (i) M×N-gon prism switches (M2, N3) and (ii) 3-D grids of any geometry N3. For the routing we assume (1) the projection of the spatial architectures onto plane graphs (2) the embedding of the latter guest graphs into (in)complete host hypercubes (N=4) and generally, into N-cube networks (N3) and (3) routing by means of the cube algorithms of the host. By the embedding mainly faulty cubes (synonyms: injured cubes, incomplete cubes) arise which complicate the routing and analysis. The application of N-cube networks (i) extend the hypercube principles to any N3 (ii) increase the number of plane host graphs and (iii) reduce the incompleteness of the host cubes. Several different embeddings of the intersection graphs (IGs) of 2-D switching networks and several different routings are explained for N=4 and 6 by various examples. By the expansion of the grids (enlargement) internal waveguides (WGs) and internal switches are introduced which interact with the switches of the original 3-D grid without increasing the number of stages (NS). The embeddings by expansion apply to interconnection networks whereas dilation-2 embeddings (dilation ≡ distance of the nearest-neighbour nodes of the guest graph at the host) are rather suitable for the emulation of algorithms. Concepts for fault-tolerant routing and algorithm mapping are briefly explained.     

From 2D hyperbolic forests to 3D Euclidean entangled thickets

A method is developed to construct and analyse a wide class of graphs embedded in Euclidean 3D space, including multiply-connected and entangled examples. The graphs are derived via embeddings of infinite families of trees (forests) in the hyperbolic plane, and subsequent folding into triply periodic minimal surfaces, including the P, D, gyroid and H surfaces. Some of these graphs are natural generalisations of bicontinuous topologies to bi-, tri-, quadra- and octa-continuous forms. Interwoven layer graphs and periodic sets of finite clusters also emerge from the algorithm. Many of the graphs are chiral. The generated graphs are compared with some organo-metallic molecular crystals with multiple frameworks and molecular mesophases found in copolymer melts. Received 10 December 1999     

The Yamada polynomial of spacial graphs and knit algebras

The Yamada polynomial for embeddings of graphs is widely generalized by using knit semigroups and polytangles. To construct and investigate them, we use a diagrammatic method combined with the theory of algebrasH _N,M(a,q), which are quotients of knit semigroups and are generalizations of Iwahori-Hecke algebrasH _n(q). Our invariants are versions of Turaev-Reshetikhin's invariants for ribbon graphs, but our construction is more specific and computable.This research was supported in part by NSF grant DMS-9100383     

Vulnerability of labeled networks

We propose a metric for vulnerability of labeled graphs that has the following two properties: (1) when the labeled graph is considered as an unlabeled one, the metric reduces to the corresponding metric for an unlabeled graph; and (2) the metric has the same value for differently labeled fully connected graphs, reflecting the notion that any arbitrarily labeled fully connected topology is equally vulnerable as any other. A vulnerability analysis of two real-world networks, the power grid of the European Union, and an autonomous system network, has been performed. The networks have been treated as graphs with node labels. The analysis consists of calculating characteristic path lengths between labels of nodes and determining largest connected cluster size under two node and edge attack strategies. Results obtained are more informative of the networks’ vulnerability compared to the case when the networks are modeled with unlabeled graphs.     

Generalized Goodman–Harpe–Jones Construction of Subfactors, II

We build some exceptional representations of Birman–Wenzl algebras from the data of certain conformal embeddings. As a result we construct new finite depth subfactors whose principal graphs are completely determined. Received: 16 February 1996 / Accepted: 25 July 1996     

Percolation in self-similar networks

We provide a simple proof that graphs in a general class of self-similar networks have zero percolation threshold. The considered self-similar networks include random scale-free graphs with given expected node degrees and zero clustering, scale-free graphs with finite clustering and metric structure, growing scale-free networks, and many real networks. The proof and the derivation of the giant component size do not require the assumption that networks are treelike. Our results rely only on the observation that self-similar networks possess a hierarchy of nested subgraphs whose average degree grows with their depth in the hierarchy. We conjecture that this property is pivotal for percolation in networks.     

Complex Embedding with Type Constraints for Link Prediction

Large-scale knowledge graphs not only store entities and relations but also provide ontology-based information about them. Type constraints that exist in this information are of great importance for link prediction. In this paper, we proposed a novel complex embedding method, CHolE, in which complex circular correlation was introduced to extend the classic real-valued compositional representation HolE to complex domains, and type constraints were integrated into complex representational embeddings for improving link prediction. The proposed model consisted of two functional components, the type constraint model and the relation learning model, to form type constraints such as modulus constraints and acquire the relatedness between entities accurately by capturing rich interactions in the modulus and phase angles of complex embeddings. Experimental results on benchmark datasets showed that CHolE outperformed previous state-of-the-art methods, and the impartment of type constraints improved its performance on link prediction effectively.     

From CFT to graphs

In this paper, we pursue the discussion of the connections between rational conformal field theories (CFT) and graphs. We generalise our recent work on the relations of operator product algebra (OPA) structure constants of sl(2) theories with the Pasquier algebra attached to the graph. We show that in a variety of CFT's built on sl(n) (typically conformal embeddings and orbifolds), similar considerations enable one to write a linear system satisfied by the matrix elements of the Pasquier algebra in terms of conformal data (quantum dimensions and fusion coefficients). In some cases this provides sufficient information for the determination of all the eigenvectors of an adjacency matrix, and hence of a graph.     

A spatial model for social networks

We study spatial embeddings of random graphs in which nodes are randomly distributed in geographical space. We let the edge probability between any two nodes to be dependent on the spatial distance between them and demonstrate that this model captures many generic properties of social networks, including the “small-world” properties, skewed degree distribution, and most distinctively the existence of community structures.     

Diffusion on Delone sets

We consider graphs associated to Delone sets in Euclidean space. Such graphs arise in various ways from tilings. Here, we provide a unified framework. In this context, we study the associated Laplace operators and show Gaussian heat kernel bounds for their semigroups. These results apply to both metric and discrete graphs.     

Derivation of the polyakov string theory from planar φ3 field theory

A rigorous derivation is given of the Polyakov surface theory by summing over all planar Feynman graphs of a φ³ scalar field theory. It uses a discretization of surfaces through the triangulation induced by the dual graphs of the Feynman graphs. No approximation is made for the propagators and the metric is shown to be locally defined. The Green's functions of both theories are shown to coincide for any finite external momenta.     

Localization on Quantum Graphs with Random Edge Lengths

The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple ${\mathbb Z^d}$ -lattice with δ-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization near the spectral edges situated outside a certain forbidden set.     

Chaos synchronization in networks of coupled maps with time-varying topologies

Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex networks both analytically and numerically. The temporal variation is rather general and formalized as being driven by a metric dynamical system. Four network models are discussed in detail in which the interconnections between vertices vary through time randomly. These models are: 1) i.i.d. sequences of random graphs with fixed wiring probability, 2) groups of graphs with random switches between the individual graphs, 3) graphs with temporary random failures of nodes, and 4) the meet-for-dinner model where the vertices are randomly grouped. We show that the temporal variation and randomness of the connection topology can enhance synchronizability in many cases; however, there are also instances where they reduce synchronizability. In analytical terms, the Hajnal diameter of the coupling matrix sequence is presented as a measure for the synchronizability of the graph topology. In topological terms, the decisive criterion for synchronization of coupled chaotic maps is that the union of the time-varying graphs contains a spanning tree.     

Dirac fermions on a disclinated flexible surface

A self-consisting gauge-theory approach to describe Dirac fermions on flexible surfaces with a disclination is formulated. The elastic surfaces are considered as embeddings into R ³ and a disclination is incorporated through a topologically nontrivial gauge field of the local SO(3) group which generates the metric with conical singularity. A smoothing of the conical singularity on flexible surfaces is naturally accounted for by regarding the upper half of two-sheet hyperboloid as an elasticity-induced embedding. The availability of the zeromode solution to the Dirac equation is analyzed.