Directed Tree Research Articles

On Tight Spans for Directed Distances

An extension (V, d) of a metric space (S, μ) is a metric space with \({{S \subseteq V}}\) and \({{d\mid{_S} = \mu}}\) , and is said to be tight if there is no other extension (V, d′) of (S, μ) with d′ ≤ d . Isbell and Dress independently found that every tight extension embeds isometrically into a certain metrized polyhedral complex associated with (S, μ), called the tight span. This paper develops an analogous theory for directed metrics, which are “not necessarily symmetric” distance functions satisfying the triangle inequality. We introduce a directed version of the tight span and show that it has a universal embedding property for tight extensions. Also we introduce a new natural class of extensions, called cyclically tight extensions, and we show that there also exists a certain polyhedral complex having a universal property relative to cyclically tightness. This polyhedral complex coincides with (a fiber of) the tropical polytope spanned by the column vectors of –μ, which was earlier introduced by Develin and Sturmfels. Thus this gives a tight-span interpretation to the tropical polytope generated by a nonnegative square matrix satisfying the triangle inequality. As an application, we prove the following directed version of the tree metric theorem: A directed metric μ is a directed tree metric if and only if the tropical rank of –μ is at most two. Also we describe how tight spans and tropical polytopes are applied to the study of multicommodity flows in directed networks.

A non-hyponormal operator generating Stieltjes moment sequences

A linear operator S in a complex Hilbert space H for which the set D∞(S) of its C∞-vectors is dense in H and {‖Snf‖2}n=0∞ is a Stieltjes moment sequence for every f∈D∞(S) is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator S which generates Stieltjes moment sequences. What is more, D∞(S) is a core of any power Sn of S. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of indeterminate Stieltjes moment sequences. As a consequence, it is shown that there exists a non-hyponormal composition operator in an L2-space (over a σ-finite measure space) which is injective, paranormal and which generates Stieltjes moment sequences. The independence assertion of Barry Simonʼs theorem which parameterizes von Neumann extensions of a closed real symmetric operator with deficiency indices (1,1) is shown to be false.

Weighted shifts on directed trees

A new class of (not necessarily bounded) operators related to (mainly innite) directed trees is introduced and investigated. Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and of weighted adjacency operators, on the other; they are called weighted shifts on directed trees. The basic properties of such op- erators, including closedness, adjoints, polar decomposition and moduli are studied. Circularity and the Fredholmness of weighted shifts on directed trees are discussed. The relationships between domains of a weighted shift on a directed tree and its adjoint are described. Hyponormality, cohyponormality, subnormality and complete hyperexpansivity of such operators are entirely characterized in terms of their weights. Related questions that arose during the study of the topic are solved as well. Particular trees with one branching vertex are intensively studied mostly in the context of subnormality and com- plete hyperexpansivity of weighted shifts on them. A strict connection of the latter with k-step backward extendibility of subnormal as well as completely hyperexpansive unilateral classical weighted shifts is established. Models of subnormal and completely hyperexpansive weighted shifts on these particular trees are constructed. Various illustrative examples of weighted shifts on di- rected trees with the prescribed properties are furnished. Many of them are simpler than those previously found on occasion of investigating analogical properties of other classes of operators.

A proof of Sumnerʼs universal tournament conjecture for large tournaments

Abstract Sumnerʼs universal tournament conjecture from 1971 states that any tournament on 2 n − 2 vertices contains any directed tree on n vertices. We prove that this conjecture holds for all sufficiently large n.

Retracts of trees and free left adequate semigroups

AbstractRecent research of the author has studied edge-labelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting algebraic structure is a free object in the class of left adequate semigroups. Through this correspondence we establish some structural and algorithmic properties of free left adequate semigroups and monoids, and consequently of the category of all left adequate semigroups.

Normal Extensions Escape from the Class of Weighted Shifts on Directed Trees

A formally normal weighted shift on a directed tree is shown to be a bounded normal operator. The question of whether a normal extension of a subnormal weighted shift on a directed tree can be modeled as a weighted shift on some, possibly different, directed tree is answered.

Improved approximations for the hotlink assignment problem

Let G =( V,E ) be a graph representing a Web site, where nodes correspond to pages and arcs to hyperlinks. In this context, hotlinks are defined as shortcuts (new arcs) added to Web pages of G in order to reduce the time spent by users to reach their desired information. In this article, we consider the problem where G is a rooted directed tree and the goal is minimizing the expected time spent by users by assigning at most k hotlinks to each node. For the most studied version of this problem where at most one hotlink can be added to each node, we prove the existence of two FPTAS's which optimize different objectives considered in the literature: one minimizes the expected user path length and the other maximizes the expected reduction in user path lengths. These results improve over a constant factor approximation for the expected length and over a PTAS for the expected reduction, both obtained recently in Jacobs [2007]. Indeed, these FPTAS's are essentially the best possible results one can achieve under the assumption that P ≠ NP . Another contribution we give here is a 16-approximation algorithm for the most general version of the problem where up to k hotlinks can be assigned from each node. This algorithm runs in O (| V | log | V |) time and it turns to be the first algorithm with constant approximation for this problem.

An approximate version of Sumnerʼs universal tournament conjecture

Sumnerʼs universal tournament conjecture states that any tournament on 2 n − 2 vertices contains a copy of any directed tree on n vertices. We prove an asymptotic version of this conjecture, namely that any tournament on ( 2 + o ( 1 ) ) n vertices contains a copy of any directed tree on n vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed Δ, any tournament on ( 1 + o ( 1 ) ) n vertices contains a copy of any directed tree on n vertices with maximum degree at most Δ.

Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree

Let T = (V, A) be a directed tree. Given a collection \({\mathcal{P}}\) of dipaths on T, we can look at the arc-intersection graph \({I(\mathcal{P},T)}\) whose vertex set is \({\mathcal{P}}\) and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs) and proved that they are perfect. DE graphs find one of their applications in the context of optical networks. For instance, assigning wavelengths to set of dipaths in a directed tree network consists in finding a proper coloring of the arc-intersection graph. In the present paper, we give a simple algorithm finding a minimum proper coloring of the paths. a faster algorithm than previously known ones finding a minimum multicut on a directed tree. It runs in \({O(|V||\mathcal{P}|)}\) (it corresponds to the minimum clique cover of \({I(\mathcal{P},T)}\)). a polynomial algorithm computing a kernel in any DE graph whose edges are oriented in a clique-acyclic way. Even if we know by a theorem of Boros and Gurvich that such a kernel exists for any perfect graph, it is in general not known whether there is a polynomial algorithm (polynomial algorithms computing kernels are known only for few classes of perfect graphs).

Sample classification for improved performance of PLS models applied to the quality control of deep-frying oils of different botanic origins analyzed using ATR-FTIR spectroscopy

The selection of an appropriate calibration set is a critical step in multivariate method development. In this work, the effect of using different calibration sets, based on a previous classification of unknown samples, on the partial least squares (PLS) regression model performance has been discussed. As an example, attenuated total reflection (ATR) mid-infrared spectra of deep-fried vegetable oil samples from three botanical origins (olive, sunflower, and corn oil), with increasing polymerized triacylglyceride (PTG) content induced by a deep-frying process were employed. The use of a one-class-classifier partial least squares-discriminant analysis (PLS-DA) and a rooted binary directed acyclic graph tree provided accurate oil classification. Oil samples fried without foodstuff could be classified correctly, independent of their PTG content. However, class separation of oil samples fried with foodstuff, was less evident. The combined use of double-cross model validation with permutation testing was used to validate the obtained PLS-DA classification models, confirming the results. To discuss the usefulness of the selection of an appropriate PLS calibration set, the PTG content was determined by calculating a PLS model based on the previously selected classes. In comparison to a PLS model calculated using a pooled calibration set containing samples from all classes, the root mean square error of prediction could be improved significantly using PLS models based on the selected calibration sets using PLS-DA, ranging between 1.06 and 2.91% (w/w).

A proof of Sumner's universal tournament conjecture for large tournaments

Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains any directed tree on $n$ vertices. In this paper we prove that this conjecture holds for all sufficiently large $n$. The proof makes extensive use of results and ideas from a recent paper by the same authors, in which an approximate version of the conjecture was proved.

Characterizing directed path graphs by forbidden asteroids

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it does not contain an asteroidal triple. In this paper, we prove an analogous theorem for directed path graphs which are the intersection graphs of directed paths in a directed tree. For this purpose, we introduce the notion of a special connection. Two non-adjacent vertices are linked by a special connection if either they have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain conditions. A special asteroidal triple is an asteroidal triple such that each pair is linked by a special connection. We prove that a chordal graph is a directed path graph if and only if it does not contain a special asteroidal triple. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:103-112, 2011

Limit theorems for random spatial drainage networks

Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of Rd, d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1)d. The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables.

Limit theorems for random spatial drainage networks

Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of R d , d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1) d . The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables.

Reply to “Comments on “Consensus and Cooperation in Networked Multi-Agent Systems””

There are essentially four points that Dr. Chebotarev's raises in [1]. Point 1. Chebotarev claims that Lemma 2 is not correct as stated and gives a counter example consisting of a simple directed tree. This counterexample points out two issues with the lemma as stated. The second portion of Lemma 2, referring to the case in which there are c components, only applies to graphs in which there are disjoint components of the graph (no edges between the components). This is clear from the proof of this fact (which simply consists of separating the nodes so that the Laplacian is block diagonal, implying a disjoint set of nodes), but is ambiguous in the statement of the lemma.

Weak Hyponomal Composition Operators Induced by a Tree

Let G = (V;E; ) be a weighted directed tree, where V is a vertex set, E is an edge set, and is a -nite measure on V. The tree G induces a composition operator C on the Hilbert space l 2 (V ): Hand-type directed trees are dened and characterized the weak hyponormalities of such C in this note. Also some additional related properties are discussed. In addition, some examples related to directed hand-type trees are provided to separate classes of weak-hyponormal operators.

The skew energy of a digraph

We are interested in the energy of the skew-adjacency matrix of a directed graph D , which is simply called the skew energy of D in this paper. Properties of the skew energy of D are studied. In particular, a sharp upper bound for the skew energy of D is derived in terms of the order of D and the maximum degree of its underlying undirected graph. An infinite family of digraphs attaining the maximum skew energy is constructed. Moreover, the skew energy of a directed tree is independent of its orientation, and interestingly it is equal to the energy of the underlying undirected tree. Skew energies of directed cycles under different orientations are also computed. Some open problems are presented.

Kalman Filtering Over Graphs: Theory and Applications

In this technical note we consider the problem of distributed discrete-time state estimation over sensor networks. Given a graph that represents the sensor communications, we derive the optimal estimation algorithm at each sensor. We further provide a closed-form expression for the steady-state error covariance matrices when the communication graph reduces to a directed tree. We then apply the developed theoretical tools to compare the performance of two sensor trees and convert a random packet-delay model to a random packet-dropping model. Examples are provided throughout the technical note to support the theory.

Integrative Analysis of Microarray Data with Gene Ontology to Select Perturbed Molecular Functions using Gene Ontology Functional Code

A systems biology approach for the identification of perturbed molecular functions is required to understand the complex progressive disease such as breast cancer. In this study, we analyze the microarray data with Gene Ontology terms of molecular functions to select perturbed molecular functional modules in breast cancer tissues based on the definition of Gene ontology Functional Code. The Gene Ontology is three structured vocabularies describing genes and its products in terms of their associated biological processes, cellular components and molecular functions. The Gene Ontology is hierarchically classified as a directed acyclic graph. However, it is difficult to visualize Gene Ontology as a directed tree since a Gene Ontology term may have more than one parent by providing multiple paths from the root. Therefore, we applied the definition of Gene Ontology codes by defining one or more GO code(s) to each GO term to visualize the hierarchical classification of GO terms as a network. The selected molecular functions could be considered as perturbed molecular functional modules that putatively contributes to the progression of disease. We evaluated the method by analyzing microarray dataset of breast cancer tissues; i.e., normal and invasive breast cancer tissues. Based on the integration approach, we selected several interesting perturbed molecular functions that are implicated in the progression of breast cancers. Moreover, these selected molecular functions include several known breast cancer-related genes. It is concluded from this study that the present strategy is capable of selecting perturbed molecular functions that putatively play roles in the progression of diseases and provides an improved interpretability of GO terms based on the definition of Gene Ontology codes.

Minimum rank problems

A graph describes the zero–nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303–316, C.R. Johnson, A. Leal Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999) 139–144], trees allowing loops [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness. Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl. 418 (2006) 389–415], and directed trees allowing loops [F. Barioli, S. Fallat, D. Hershkowitz, H.T. Hall, L. Hogben, H. van der Holst, B. Shader, On the minimum rank of not necessarily symmetric matrices: a preliminary study, Electron. J. Linear Algebra 18 (2000) 126–145]. We survey these results from a unified perspective and solve the minimum rank problem for simple directed trees.