Undirected Models Research Articles

Packing non-zero A-paths in an undirected model of group labeled graphs

Let Γ be an abelian group, and let γ : E ( G ) → Γ be a function assigning values in Γ to every edge of a graph G. For a subgraph H of G, let γ ( H ) = ∑ e ∈ E ( H ) γ ( e ) . For a set A of vertices of G, an A-path is a path with both endpoints in A and otherwise disjoint from A. In this article, we show that either there exist k vertex disjoint A-paths P 1 , P 2 , … , P k such that γ ( P i ) ≠ 0 for all 1 ⩽ i ⩽ k , or there exists a set X of vertices such that G − X does not contain a non-zero A-path with | X | ⩽ 50 k 4 .

The distant-2 chromatic number of random proximity and random geometric graphs

We are interested in finding bounds for the distant-2 chromatic number of geometric graphs drawn from different models. We consider two undirected models of random graphs: random geometric graphs and random proximity graphs for which sharp connectivity thresholds have been shown. We are interested in a.a.s. connected graphs close just above the connectivity threshold. For such subfamilies of random graphs we show that the distant-2-chromatic number is Θ ( log n ) with high probability. The result on random geometric graphs is extended to the random sector graphs defined in [J. Díaz, J. Petit, M. Serna. A random graph model for optical networks of sensors, IEEE Transactions on Mobile Computing 2 (2003) 143–154].

Critical behavior of sandpile models with sticky grains

We revisit the question whether the critical behavior of sandpile models with sticky grains is in the directed percolation universality class. Our earlier theoretical arguments in favor, supported by evidence from numerical simulations [P.K. Mohanty, D. Dhar, Phys. Rev. Lett. 89 (2002) 104303], have been disputed by Bonachela et al. [Phys. Rev. E 74 (2006) 050102] for sandpiles with no preferred direction. We discuss possible reasons for the discrepancy. Our new results of longer simulations of the one-dimensional undirected model fully support our earlier conclusions.

Directed and undirected multiurn models in a one-dimensional ring

The flea model by Ehrenfest describes the jumps of a fixed number of fleas between two dogs. In each time step a randomly selected flea jumps on the other dog. We study directed and undirected multiurn models in a one-dimensional ring. The introduced models represent generalizations of three recently proposed multiurn models which themselves are generalizations of Ehrenfest's model. The models are solved analytically. For the directed case we find oscillations of the average number of balls or fleas in a certain urn before the system reaches its equilibrium state. The discussed models may serve as basic models of dynamics of granular media in connected periodic compartment systems.

Generic sandpile models have directed percolation exponents.

We study sandpile models with stochastic toppling rules and having sticky grains so that with a nonzero probability no toppling occurs, even if the local height of pile exceeds the threshold value. Dissipation is introduced by adding a small probability of particle loss at each toppling. Generically for the models with a preferred direction, the avalanche exponents are those of critical directed percolation clusters. For undirected models, avalanche exponents are those of directed percolation clusters in one higher dimension.

Single Step Undirected Reconfigurable Networks

The reconfigurable mesh (RN-MESH) can solve a large class of problems in constant time, including problems that require logarithmic time by other, even shared memory, models such as the PRAM with a similar number of processors [3]. In this work we show that for the RN-MESH these constants can always be reduced to one, still using a polynomial number of processors. Given a reconfigurable mesh that computes a set of values in constant time, we show that it can be simulated by a single step reconfigurable mesh with maximum size that is polynomial in the size of the original mesh. The proof is constructive, where the construction of the single step RN-MESH holds for the relatively weak undirected RN-MESH model. In this model broadcasts made on buses arrive at all nodes that belong to the undirected connected component of the transmitting processor. A result similar to the one that is obtained in this work was previously obtained for the directed reconfigurable mesh model (DRN) [5]. However, the construction for the DRN-MESH relies on the fact that the buses are directed, and thus cannot be applied to the undirected case. In addition, the construction presented here is simpler and uses significantly fewer processors than the one obtained for the DRN-MESH.

Efficient estimation and model selection in large graphical models

We develop a computationally efficient method to determine the interaction structure in a multidimensional binary sample. We use an interaction model based on orthogonal functions, and give a result on independence properties in this model. Using this result we develop an efficient approximation algorithm for estimating the parameters in a given undirected model. To find the best model, we use a heuristic search algorithm in which the structure is determined incrementally. We also give an algorithm for reconstructing the causal directions, if such exist. We demonstrate that together these algorithms are capable of discovering almost all of the true structure for a problem with 121 variables, including many of the directions.

Modeling and Heuristic Worst-Case Performance Analysis of the Two-Level Network Design Problem

This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We are given an undirected network containing two types of nodes—primary and secondary—and fixed costs for installing either a primary or a secondary facility on each edge. Primary nodes require higher grade interconnections than secondary nodes, using the more expensive primary facilities. The TLND problem seeks a minimum cost connected design that spans all the nodes, and connects primary nodes via edges containing primary facilities; the design can use either primary or secondary edges to connect the secondary nodes. The TLND problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem (e.g., directed and undirected models), and analyze the worst-case performance for a composite TLND heuristic based upon solving embedded subproblems (e.g., minimum spanning tree and either Steiner tree or shortest path subproblems). When the ratio of primary to secondary costs is the same for all edges and when we solve the embedded subproblems optimally, the worst-case performance ratio of the composite TLND heuristic is 4/3. This result applies to the hierarchical network design problem with constant primary-to-secondary cost ratio since its subproblems are shortest path and minimum spanning tree problems. For more general situations, we express the TLND heuristic worst-case ratio in terms of the performance of any heuristic used to solve the embedded Steiner tree subproblem. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of a multi-level extension of this problem.

Prediction of magnetic flux-controlled gate voltage in superconducting field-effect transistors

The undirectional model to the superconducting field-effect transistor (SFET) is shown to be thermodynamically unsound. A gate voltage which is controlled by the magnetic flux difference in a Josephson weak link is predicted by energy arguments. For a passive SFET model to be consistent with recent experimental observations of a charge-controlled critical current, a back-reaction from the DC drain-to-source flux (phase difference) to the DC gate voltage is required. As this effect is important in large devices and occurs at V/sub DS/=0, it does not appear to be directly related to charge-space energy bands or quasiparticle interference.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Procedure for testing kinetic models of the photocycle of bacteriorhodopsin

Given some simple kinetic models of the photocycle of bacteriorhodopsin (bR) and data taken at many wavelengths and under conditions that avoid photoselection and steady-state cycling complications, it is shown how to extract the apparent rate constants and the spectra of the intermediates. Special consideration was given to establishing the range of error of these results. There are many criteria, which we explicitly discuss, that the spectra should satisfy in order that the kinetic model be acceptable. New data for the photocycle of purple membrane fragments in dilute buffer at pH 7.0 has been obtained at 15 measuring wavelengths and four temperatures. The procedure, which can be generalized to more complex models, has been applied to these data to test two kinds of kinetic models: the unidirectional unbranched model and the undirectional model with simple branching straight back to bR from any intermediate. In these models the spectrum of the O intermediate is highly temperature sensitive, even with branching, and/or has two broad maxima. Moreover, the spectrum of the M intermediate has a secondary maximum and two M-like states appear to be required. Thus, neither model satisfies the physical criteria.

Series expansions for the directed-bond percolation problem

Series expansion methods have been used to estimate the critical probability, pc, and to investigate percolation behaviour for the directed-bond problem on most of the usual lattices. High-density series expansions of P(p) and low-density series expansions of S(p) were obtained for the acyclic directed square (SQ), triangular (T), SC, BCC and FCC lattices, and also the cyclic directed triangular lattice. The results suggest that directed models need not fall into the same universality class as the corresponding undirected models. The results for the acyclic and cyclic directed triangular lattices further suggest that different directings of the same lattice may yield models falling into different universality classes.