scholarly journals Shortest Directed Networks in the Plane

2020 ◽  
Vol 36 (5) ◽  
pp. 1457-1475
Author(s):  
Alastair Maxwell ◽  
Konrad J. Swanepoel
Keyword(s):  
Directed Graph ◽  
Local Structure ◽  
Euclidean Plane ◽  
Directed Network ◽  
Directed Path ◽  
Directed Networks ◽  
Sources And Sinks ◽  
Steiner Points ◽  
Unpublished Work ◽  
The Given

Abstract Given a set of sources and a set of sinks as points in the Euclidean plane, a directed network is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane. This characterization implies that these networks are constructible by straightedge and compass. Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990. Part of the proof is based on a new method that uses other norms in the plane. This approach gives more conceptual proofs of some of their results, and as a consequence, we also obtain results on shortest directed networks for these norms.

Necessary and Sufficient Conditions for Group Consensus of Agents With Third-Order Dynamics in Directed Networks

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
Onur Cihan ◽  
Mehmet Akar
Keyword(s):  
Directed Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Consensus ◽  
Directed Networks ◽  
Third Order ◽  
Systematic Method ◽  
Necessary And Sufficient ◽  
The Given ◽  
Theoretical Results

Abstract In this paper, we investigate the group consensus problem in directed networks where agents have third-order dynamics. Necessary and sufficient conditions on the controller parameters are obtained to ensure K-equilibria group consensus where K is determined by the structure of the directed graph. It is theoretically shown that, for an arbitrary directed graph, there exist controller parameters that satisfy the given conditions. A systematic method for choosing the controller parameters to guarantee group consensus is suggested and theoretical results are verified by numerical examples.

COMPUTING STEINER POINTS AND PROBABILITY STEINER POINTS IN ℓ1 AND ℓ2 METRIC SPACES

2009 ◽  
Vol 01 (04) ◽  
pp. 541-554
Author(s):  
J. F. WENG ◽  
I. MAREELS ◽  
D. A. THOMAS
Keyword(s):  
Steiner Tree ◽  
Euclidean Plane ◽  
Metric Spaces ◽  
Steiner Tree Problem ◽  
Smooth Functions ◽  
Steiner Points ◽  
New Type ◽  
Point Set ◽  
Steiner Minimum Tree ◽  
The Given

The Steiner tree problem is a well known network optimization problem which asks for a connected minimum network (called a Steiner minimum tree) spanning a given point set N. In the original Steiner tree problem the given points lie in the Euclidean plane or space, and the problem has many variants in different applications now. Recently a new type of Steiner minimum tree, probability Steiner minimum tree, is introduced by the authors in the study of phylogenies. A Steiner tree is a probability Steiner tree if all points in the tree are probability vectors in a vector space. The points in a Steiner minimum tree (or a probability Steiner tree) that are not in the given point set are called Steiner points (or probability Steiner points respectively). In this paper we investigate the properties of Steiner points and probability Steiner points, and derive the formulae for computing Steiner points and probability Steiner points in ℓ1- and ℓ2-metric spaces. Moreover, we show by an example that the length of a probability Steiner tree on 3 points and the probability Steiner point in the tree are smooth functions with respect to p in d-space.

scholarly journals Cycle and flow trusses in directed networks

2016 ◽  
Vol 3 (11) ◽  
pp. 160270 ◽  
Author(s):  
Taro Takaguchi ◽  
Yuichi Yoshida
Keyword(s):  
Truss Structures ◽  
Directed Network ◽  
Unified Framework ◽  
Directed Networks ◽  
Research Fields ◽  
Wide Range ◽  
Definition Of ◽  
Information Finding ◽  
Empirical Networks ◽  
Module Structures

When we represent real-world systems as networks, the directions of links often convey valuable information. Finding module structures that respect link directions is one of the most important tasks for analysing directed networks. Although many notions of a directed module have been proposed, no consensus has been reached. This lack of consensus results partly because there might exist distinct types of modules in a single directed network, whereas most previous studies focused on an independent criterion for modules. To address this issue, we propose a generic notion of the so-called truss structures in directed networks. Our definition of truss is able to extract two distinct types of trusses, named the cycle truss and the flow truss, from a unified framework. By applying the method for finding trusses to empirical networks obtained from a wide range of research fields, we find that most real networks contain both cycle and flow trusses. In addition, the abundance of (and the overlap between) the two types of trusses may be useful to characterize module structures in a wide variety of empirical networks. Our findings shed light on the importance of simultaneously considering different types of modules in directed networks.

scholarly journals Controllability of Weighted and Directed Networks with Nonidentical Node Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Linying Xiang ◽  
Jonathan J. H. Zhu ◽  
Fei Chen ◽  
Guanrong Chen
Keyword(s):  
Graph Theory ◽  
Control Theory ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Directed Path ◽  
Directed Networks ◽  
Leaders And Followers ◽  
Heterogenous Network

The concept of controllability from control theory is applied to weighted and directed networks with heterogenous linear or linearized node dynamics subject to exogenous inputs, where the nodes are grouped into leaders and followers. Under this framework, the controllability of the controlled network can be decomposed into two independent problems: the controllability of the isolated leader subsystem and the controllability of the extended follower subsystem. Some necessary and/or sufficient conditions for the controllability of the leader-follower network are derived based on matrix theory and graph theory. In particular, it is shown that a single-leader network is controllable if it is a directed path or cycle, but it is uncontrollable for a complete digraph or a star digraph in general. Furthermore, some approaches to improving the controllability of a heterogenous network are presented. Some simulation examples are given for illustration and verification.

scholarly journals Directed networks as a novel way to describe and analyze cardiac excitation: Directed Graph mapping

2019 ◽  
Author(s):  
Nele Vandersickel ◽  
Enid Van Nieuwenhuyse ◽  
Nico Van Cleemput ◽  
Jan Goedgebeur ◽  
Milad El Haddad ◽  
...  
Keyword(s):  
Cardiac Arrhythmias ◽  
Normal Sinus Rhythm ◽  
Social Systems ◽  
Daily Practice ◽  
Directed Network ◽  
Directed Networks ◽  
Normal Sinus ◽  
Graph Mapping ◽  
Brain Data ◽  
Cardiac Excitation

AbstractNetworks provide a powerful methodology with applications in a variety of biological, technological and social systems such as analysis of brain data, social networks, internet search engine algorithms, etc. To date, directed networks have not yet been applied to characterize the excitation of the human heart. In clinical practice, cardiac excitation is recorded by multiple discrete electrodes. During (normal) sinus rhythm or during cardiac arrhythmias, successive excitation connects neighboring electrodes, resulting in their own unique directed network. This in theory makes it a perfect fit for directed network analysis. In this study, we applied directed networks to the heart in order to describe and characterize cardiac arrhythmias. Proofof-principle was established using in-silico and clinical data. We demonstrated that tools used in network theory analysis allow to determine the mechanism and location of certain cardiac arrhythmias. We show that the robustness of this approach can potentially exceed the existing state-of-the art methodology used in clinics. Furthermore, implementation of these techniques in daily practice can improve accuracy and speed of cardiac arrhythmia analysis. It may also provide novel insights in arrhythmias that are still incompletely understood.

Circles on the Complex Plane

2021 ◽  
pp. 3-12
Author(s):  
A. Girsh
Keyword(s):  
Euclidean Space ◽  
Complex Plane ◽  
Euclidean Plane ◽  
Complex Space ◽  
Radical Center ◽  
Special Cases ◽  
Different Types ◽  
Different Dimensions ◽  
The Given ◽  
Made In

The Euclidean plane and Euclidean space themselves do not contain imaginary elements by definition, but are inextricably linked with them through special cases, and this leads to the need to propagate geometry into the area of imaginary values. Such propagation, that is adding a plane or space, a field of imaginary coordinates to the field of real coordinates leads to various variants of spaces of different dimensions, depending on the given axiomatics. Earlier, in a number of papers, were shown examples for solving some urgent problems of geometry using imaginary geometric images [2, 9, 11, 13, 15]. In this paper are considered constructions of orthogonal and diametrical positions of circles on a complex plane. A generalization has been made of the proposition about a circle on the complex plane orthogonally intersecting three given spheres on the proposition about a sphere in the complex space orthogonally intersecting four given spheres. Studies have shown that the diametrical position of circles on the Euclidean E-plane is an attribute of the orthogonal position of the circles’ imaginary components on the pseudo-Euclidean M-plane. Real, imaginary and degenerated to a point circles have been involved in structures and considered, have been demonstrated these circles’ forms, properties and attributes of their orthogonal position. Has been presented the construction of radical axes and a radical center for circles of the same and different types. A propagation of 2D mutual orthogonal position of circles on 3D spheres has been made. In figures, dashed lines indicate imaginary elements.

GENERALIZED MELZAK'S CONSTRUCTION IN THE STEINER TREE PROBLEM

2002 ◽  
Vol 12 (06) ◽  
pp. 481-488 ◽  
Author(s):  
JIA F. WENG
Keyword(s):  
Steiner Tree ◽  
Euclidean Plane ◽  
Steiner Tree Problem ◽  
Minimal Tree ◽  
Steiner Minimal Tree ◽  
The Core ◽  
Steiner Minimal Trees ◽  
Minimum Networks ◽  
Set Of Points ◽  
The Given

For a given set of points in the Euclidean plane, a minimum network (a Steiner minimal tree) can be constructed using a geometric method, called Melzak's construction. The core of the Melzak construction is to replace a pair of terminals adjacent to the same Steiner point with a new terminal. In this paper we prove that the Melzak construction can be generalized to constructing Steiner minimal trees for circles so that either the given points (terminals) are constrained on the circles or the terminal edges are tangent to the circles. Then we show that the generalized Melzak construction can be used to find minimum networks separating and surrounding circular objects or to find minimum networks connecting convex and smoothly bounded objects and avoiding convex and smoothly bounded obstacles.

Optimizing complex networks controllability by local structure information

2016 ◽  
Vol 27 (10) ◽  
pp. 1650115
Author(s):  
Houyi Yan ◽  
Lvlin Hou ◽  
Yunxiang Ling ◽  
Guohua Wu
Keyword(s):  
Numerical Simulation ◽  
Complex Networks ◽  
Network Structure ◽  
Local Structure ◽  
Global Structure ◽  
Directed Network ◽  
Structure Information ◽  
Different Types ◽  
Extensive Numerical Simulation ◽  
Network Controllability

Research in network controllability has mostly been focused on the effects of the network structure on its controllability, and some methods have been proposed to optimize the network controllability. However, they are all based on global structure information of networks. We propose two different types of methods to optimize controllability of a directed network by local structure information. Extensive numerical simulation on many modeled networks demonstrates that this method is effective. Since the whole topologies of many real networks are not visible and we only get some local structure information, this strategy is potentially more practical.

VORONOI DIAGRAMS FOR A TRANSPORTATION NETWORK ON THE EUCLIDEAN PLANE

2006 ◽  
Vol 16 (02n03) ◽  
pp. 117-144 ◽  
Author(s):  
SANG WON BAE ◽  
KYUNG-YONG CHWA
Keyword(s):  
Voronoi Diagram ◽  
Euclidean Plane ◽  
Transportation Network ◽  
Voronoi Diagrams ◽  
The Other ◽  
Constant Number ◽  
Time Path ◽  
The Given

This paper investigates geometric and algorithmic properties of the Voronoi diagram for a transportation network on the Euclidean plane. In the presence of a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm2+ m3+ nm log n) algorithm to compute the Voronoi diagram for a transportation network on the Euclidean plane, where n is the number of given sites and m is the complexity of the given transportation network. Moreover, in the case that the roads in a transportation network have only a constant number of directions and speeds, we propose two algorithms; one needs O(nm + m2+ n log n) time with O(m(n + m)) space and the other O(nm log n + m2log m) time with O(n + m) space.

scholarly journals Directed graph mapping is able to distinguish between the true and false rotors in a complex in-silico excitation pattern

EP Europace ◽  
2021 ◽  
Vol 23 (Supplement_3) ◽  
Author(s):  
E Van Nieuwenhuyse ◽  
L Martinez-Mateu ◽  
J Saiz ◽  
A V Panfilov ◽  
N Vandersickel
Keyword(s):  
Directed Graph ◽  
In Silico ◽  
Directed Network ◽  
Far Field ◽  
Field Effects ◽  
In Silico Studies ◽  
Basket Catheter ◽  
Graph Mapping ◽  
Excitation Pattern ◽  
Public Grant

Abstract Funding Acknowledgements Type of funding sources: Public grant(s) – EU funding. Main funding source(s): Supported in part by Dirección General de Polı́tica Cientı́fica de la Generalitat Valenciana PROMETEU 2020/043 Background In realistic in-silico studies (Figure1, top row) it was shown that phase mapping PM (Figure 1, A) can detect the correct rotor as well as phantom rotors as an artefact of interpolation or due to the far field (Figure 1, B). After interpretation of the LAT, the far field detections could not be distinguished from the true rotor driving the excitation pattern. This can contribute to failure in Atrial Fibrillation (AF) ablation procedures. Objective We tested if the recently developed tool Directed Graph mapping (DGM) is less prone to far-field effects and interpolation artefacts than PM on the same in-silico data. DGM represents the excitation pattern as a directed network, from which the rotational activity is detected as cycles in that network. Methods Starting from the electrograms (EGMs) of the 64 electrode basket catheter, we interpolated to 957 equidistant electrodes and calculated local activation times (LATs) of the interpolated EGMs (Figure 1, C). We varied the minimal allowed conduction velocity and calculated the corresponding networks for the complete simulation time. Detections were considered as correct if they were located in the same region of the true core of the phasemaps. The false detections were classified in multiple different regions (Figure 1, D). Results We find that by proper choice of CVs in the graphs it is possible to achieve a 80% detection of true rotors with 26% detection of false rotors. Reducing restrictions on the CVs increased the detection rate of the false rotors. False rotors due to artifacts were not detected by DGM (Figure 1, last row). Conclusion DGM is able to distinguish between true and far field rotors. False detections due to interpolation artifacts as seen in the PM protocol were not observed. The velocity limits in the graph construction play a keyrole in eliminating the far field effects. Abstract Figure 1

Sign in / Sign up

Export Citation Format

Share Document