Showing papers on "Spanning tree published in 2022"

Minimum spanning tree hierarchical clustering algorithm: A new Pythagorean fuzzy similarity measure for the analysis of functional brain networks

Abstract: Clustering structures are one of the most important aspects of complex networks. Minimum spanning tree (MST), the tree that connects all vertices with minimum total weight, can be considered as a fundamental unit of original weighted graphs. There are different types of algorithms that identify clusters in a network, but the existing theories and algorithms for searching trees have not been investigated for uncertain scenarios. This paper tackles the situations where network parameters may be uncertain. Rather, we permit the parameters to take the form of Pythagorean fuzzy numbers (PFNs). Moreover, to represent qualitative aspects of uncertainty, the use of linguistic variables (LVs) has effective means for experts in expressing their views. The current study proposes a graph theory-based agglomerative hierarchical clustering technique for Pythagorean fuzzy sets. We first define generalized Pythagorean fuzzy numbers (GPFNs) and LR-type PFNs. Then we compute the PF distance between two GPFNs and also LR-type PFNs, and formulate the expressions for PF similarity measure. This paper mainly examines the use of PF distance and similarity measures in a minimum spanning tree agglomerative hierarchical clustering method by considering PFLVs. Since the structural and functional systems of brain are characterized by complex networks, we apply the proposed algorithm on a functional brain network to prove its practicality and efficiency. We discuss the hierarchical clustering consequences of the proposed algorithm in the shape of a dendrogram. Finally, we compare the clustering results obtained by different similarity measures.

Asynchronous Tracking Control of Leader–Follower Multiagent Systems With Input Uncertainties Over Switching Signed Digraphs

Abstract: Signed digraphs with both positive and negative weighted edges are widely applied to explain cooperative and competitive interactions arising from various social, biological, and physical systems. This article formulates and solves the asynchronous tracking control problem of multiagent systems with input uncertainties on switching signed digraphs. In the interaction setting, we assume that the leader moves at a time-varying acceleration that cannot be measured by the followers accurately, and further suppose that each agent receives its neighbors' states information at certain instants determined by its own clock, which is not necessary to be synchronized with those of other agents. Using dynamically changing spanning subdigraphs of signed digraphs to describe graphically asynchronous interactions, the asynchronous tracking problem is equivalently transformed into a convergence problem of products of general substochastic matrices (PGSSM), in which the matrix elements are not necessarily non-negative and the row sums are less than or equal to 1. With the help of the matrix analysis technique and the composition of binary relations, we propose a new and original method to deal with the convergence problem of PGSSM, and further establish a spanning tree condition for asynchronous tracking control. Finally, the validity of the theoretical findings is verified through several numerical examples.

Adaptive Bipartite Time-Varying Output Formation Control for Multiagent Systems on Signed Directed Graphs

Abstract: The issue of bipartite time-varying formation (BTVF) tracking for linear multiagent systems (MASs) with a leader of unknown input on signed digraphs is investigated. An adaptive nonsmooth protocol is taken in this article that utilizes only the local output feedback information among neighbors and, thus, can avoid employing the eigenvalue information of the Laplacian matrix of the graph. It is proven that if the interaction network of agents containing a spanning tree is structurally balanced, the BTVF tracking can be achieved with a leader of the bounded input via the proposed scheme. This leader-following BTVF includes two time-varying subformations, whose relationship is antagonistic. A convergence analysis of the proposed protocol for MASs is reflected by the Lyapunov method. Finally, the validly numerical simulations are illustrated to show the performance of the proposed schemes.

The Kemeny’s Constant and Spanning Trees of Hexagonal Ring Network

Abstract: Spanning tree () has an enormous application in computer science and chemistry to determine the geometric and dynamics analysis of compact polymers. In the field of medicines, it is helpful to recognize the epidemiology of hepatitis C virus (HCV) infection. On the other hand, Kemeny’s constant () is a beneficial quantifier characterizing the universal average activities of a Markov chain. This network invariant infers the expressions of the expected number of time-steps required to trace a randomly selected terminus state since a fixed beginning state . Levene and Loizou determined that the Kemeny’s constant can also be obtained through eigenvalues. Motivated by Levene and Loizou, we deduced the Kemeny’s constant and the number of spanning trees of hexagonal ring network by their normalized Laplacian eigenvalues and the coefficients of the characteristic polynomial. Based on the achieved results, entirely results are obtained for the Möbius hexagonal ring network.

Optimal learning of Markov <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e86" altimg="si603.svg"><mml:mi>k</mml:mi></mml:math>-tree topology

Abstract: The seminal work of Chow and Liu (1968) shows that approximation of a finite probabilistic system by Markov trees can achieve the minimum information loss with the topology of a maximum spanning tree. Our current paper generalizes the result to Markov networks of tree-width ≤k, for every fixed k≥2. In particular, we prove that approximation of a finite probabilistic system with such Markov networks has the minimum information loss when the network topology is achieved with a maximum spanning k-tree. While constructing a maximum spanning k-tree is intractable for even k=2, we show that polynomial algorithms can be ensured by a sufficient condition accommodated by many meaningful applications. In particular, we show an efficient algorithm for learning the optimal topology of higher order correlations among random variables that belong to an underlying linear structure. As an application, we demonstrate effectiveness of this efficient algorithm applied to biomolecular 3D structure prediction.

Bioinspired Bare Bones Mayfly Algorithm for Large-Scale Spherical Minimum Spanning Tree

Abstract: Mayfly algorithm (MA) is a bioinspired algorithm based on population proposed in recent years and has been applied to many engineering problems successfully. However, it has too many parameters, which makes it difficult to set and adjust a set of appropriate parameters for different problems. In order to avoid adjusting parameters, a bioinspired bare bones mayfly algorithm (BBMA) is proposed. The BBMA adopts Gaussian distribution and Lévy flight, which improves the convergence speed and accuracy of the algorithm and makes better exploration and exploitation of the search region. The minimum spanning tree (MST) problem is a classic combinatorial optimization problem. This study provides a mathematical model for solving a variant of the MST problem, in which all points and solutions are on a sphere. Finally, the BBMA is used to solve the large-scale spherical MST problems. By comparing and analyzing the results of BBMA and other swarm intelligence algorithms in sixteen scales, the experimental results illustrate that the proposed algorithm is superior to other algorithms for the MST problems on a sphere.

On Multi-Objective Minimum Spanning Tree Problem under Uncertain Paradigm

Abstract: Minimum spanning tree problem (MSTP) has allured many researchers and practitioners due to its varied range of applications in real world scenarios. Modelling these applications involves the incorporation of indeterminate phenomena based on their subjective estimations. Such phenomena can be represented rationally using uncertainty theory. Being a more realistic variant of MSTP, in this article, based on the principles of the uncertainty theory, we have studied a multi-objective minimum spanning tree problem (MMSTP) with indeterminate problem parameters. Subsequently, two uncertain programming models of the proposed uncertain multi-objective minimum spanning tree problem (UMMSTP) are developed and their corresponding crisp equivalence models are investigated, and eventually solved using a classical multi-objective solution technique, the epsilon-constraint method. Additionally, two multi-objective evolutionary algorithms (MOEAs), non-dominated sorting genetic algorithm II (NSGAII) and duplicate elimination non-dominated sorting evolutionary algorithm (DENSEA) are also employed as solution methodologies. With the help of the proposed UMMSTP models, the practical problem of optimizing the distribution of petroleum products was solved, consisting in the search for symmetry (balance) between the transportation cost and the transportation time. Thereafter, the performance of the MOEAs is analyzed on five randomly developed instances of the proposed problem.

Event-Triggered Consensus Control for Networked Underactuated Robotic Systems

Abstract: In this article, the consensus of networked underactuated robotic systems subject to fixed and switched communication networks is discussed by developing some novel event-triggered control algorithms, which can synchronously guarantee the convergence of the active states, the boundedness of the velocities of passive actuators, and the exclusion of Zeno behaviors. In the cases of fixed networks, the sufficient criteria are established for the presented distributed event-triggered mechanisms with and without using neighbors' velocities, in order to achieve a better tradeoff between the communication load and system performance. Besides, in the situation of switched networks, the sufficient criterion is established by assuming that the union of the network has a spanning tree. A distributed sampled-data rule is constructed to decide when to update its own and neighbors' estimated positions, and thus further reduces the unnecessary control cost. Finally, by further extending the main results to three other sampled-data control algorithms, several examples with performance comparisons are provided to validate the efficiency and advantages of the theoretical results.

Enumeration of spanning trees in a chain of diphenylene graphs

Abstract: Abstract Cheminformatics is a modern field of chemistry information science and mathematics that is very much helpful in keeping the data and getting information about chemicals. A new two-dimensional carbon known as diphenylene was identified and synthesized. It is considered one of the materials that have many applications in most fields such as catalysis. The number of spanning trees of a graph G, also known as the complexity of a graph G, denoted by τ(G), is an important, well-studied quantity in graph theory, and appears in a number of applications. In this paper, we introduce a new chemical compound that is a chain of diphenylene where any two diphenylene intersect by one edge. We derive two formulas for the number of spanning trees in a chain of diphenylene planar graphs that have connected intersection of one edge but where the diphenylenes have same sizes.

Consensusability of First-Order Multiagent Systems Under Distributed PID Controller With Time Delay

Abstract: This article analyzes the consensus of first-order multiagent systems under the network topology with a directed spanning tree. A distributed PID controller with time delay is designed. D-parameterization approach is used and the crossing set consisting of frequencies such that at least one characteristic root is on the imaginary axis is identified. It is proven that the rightward crossings of the characteristic roots are always guaranteed. The exact delay margin is then determined. Numerical simulation is proposed to demonstrate the theoretical analysis.

Link Expiration Time and Minimum Distance Spanning Trees based Distributed Data Gathering Algorithms for Wireless Mobile Sensor Networks

Abstract: The high-level contributions of this paper are the design and development of two distributed spanning tree-based data gathering algorithms for wireless mobile sensor networks and their exhaustive simulation study to investigate a complex stability vs. node-network lifetime tradeoff that has been hitherto not explored in the literature. The topology of the mobile sensor networks changes dynamically with time due to random movement of the sensor nodes. Our first data gathering algorithm is stability-oriented and it is based on the idea of finding a maximum spanning tree on a network graph whose edge weights are predicted link expiration times (LET). Referred to as the LET-DG tree, the data gathering tree has been observed to be more stable in the presence of node mobility. However, stability-based data gathering coupled with more leaf nodes has been observed to result in unfair use of certain nodes (the intermediate nodes spend more energy compared to leaf nodes), triggering pre-mature node failures eventually leading to network failure (disconnection of the network of live nodes). As an alternative, we propose an algorithm to determine a minimum-distance spanning tree (MST) based data gathering tree that is more energy-efficient and prolongs the node and network lifetimes, at the cost frequent tree reconfigurations.

Brief Announcement: Distributed MST Computation in the Sleeping Model: Awake-Optimal Algorithms and Lower Bounds

Abstract: We study the distributed minimum spanning tree (MST) problem, a fundamental problem in distributed computing. It is well-known that distributed MST can be solved in Õ(D+√n) rounds in the standard CONGEST model (where n is the network size and D is the network diameter) and this is essentially the best possible round complexity (up to logarithmic factors). However, in resource-constrained networks such as wireless ad hoc and sensor networks, nodes spending so much time can lead to significant spending of resources such as energy. Motivated by the above consideration, we study distributed algorithms for MST under the sleeping model [Chatterjee et al., PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. In the sleeping model, a node can be in one of two modes in any round --- sleeping or awake (unlike the traditional model where nodes are always awake). Only the rounds in which a node is awake are counted, while sleeping rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the awake complexity of a distributed algorithm, the worst-case number of rounds any node is awake. We present distributed MST algorithms that have optimal awake complexity with a matching lower bound. We also show that our awake-optimal algorithms have essentially the best possible round complexity by presenting a lower bound on the product of the awake and round complexity of any distributed algorithm (including randomized).

Group Consensus of Linear Multiagent Systems Under Nonnegative Directed Graphs

Abstract: Group consensus implies reaching multiple groups where agents belonging to the same cluster reach state consensus. This article focuses on linear multiagent systems under nonnegative directed graphs. A new necessary and sufficient condition for ensuring group consensus is derived, which requires the spanning forest of the underlying directed graph and that of its quotient graph induced with respect to a clustering partition to contain equal minimum number of directed trees. This condition is further shown to be equivalent to containing cluster spanning trees, a commonly used topology for the underlying graph in the literature. Under a designed controller gain, lower bound of the overall coupling strength for achieving group consensus is specified. Moreover, the pattern of the multiple consensus states formed by all clusters is characterized when the overall coupling strength is large enough.

Deep Q-Learning-based Distribution Network Reconfiguration for Reliability Improvement

Abstract: Distribution network reconfiguration (DNR) has proved to be an economical and effective way to improve the reliability of distribution systems. As optimal network configuration depends on system operating states (e.g., loads at each node), existing analytical and population-based approaches need to repeat the entire analysis and computation to find the optimal network configuration with a change in system operating states. Contrary to this, if properly trained, deep reinforcement learning (DRL)-based DNR can determine optimal or nearoptimal configuration quickly even with changes in system states. In this paper, a Deep Q Learning-based framework is proposed for the optimal DNR to improve reliability of the system. An optimization problem is formulated with an objective function that minimizes the average curtailed power. Constraints of the optimization problem are radial topology constraint and all nodes traversing constraint. The distribution network is modeled as a graph and the optimal network configuration is determined by searching for an optimal spanning tree. The optimal spanning tree is the spanning tree with the minimum value of the average curtailed power. The effectiveness of the proposed framework is demonstrated through several case studies on 33-node and 69node distribution test systems.

Udwadia-Kalaba approach based distributed consensus control for multi-mobile robot systems with communication delays

Abstract: In this paper, a distributed consensus algorithm for multi-mobile robot systems (MMRSs) with communication delays is proposed based on the Udwadia-Kalaba (UK) approach. The key feature of the proposed algorithm is that the consensus requirement is configured as a second-order constraint, and then a concise and explicit equation of motion for the constrained mechanical systems is formulated. Furthermore, the necessary and sufficient conditions for achieving the consensus of MMRSs with or without communication delays are developed under the network topology possessing a directed spanning tree. Finally, some numerical simulations are performed to verify the validity of the proposed consensus algorithm.

Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion

Abstract: We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a tileable surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the partition function of cycle-rooted spanning forests weighted by the monodromy of the unitary connection on the vector bundle, to the corresponding zeta-regularized determinants. As a consequence, we establish open problems 2 and 4, formulated by Kenyon in 2000. The spectral theory on discretizations of flat surfaces, Fourier analysis on discrete square and the analytic methods used in the proof of Ray–Singer conjecture lie in the core of our approach.

Leader‐following consensus control of second‐order nonlinear multi‐agent systems with applications to slew rate control

Abstract: This paper addresses the distributed leader‐following consensus control of second‐order strict‐feedback nonlinear multi‐agent systems. By employing mean value theorem, variable separation technique, and backstepping methodology, a fully distributed adaptive control law is designed using only local relative state information. The proposed control law solves the leader‐following consensus problem for any directed communication graph that contains a spanning tree with the root node being the leader agent. The application to hovercraft slew rate control system is given to verify the effectiveness of the theoretical results.

Graph Theory Algorithms of Hamiltonian Cycle from Quasi-Spanning Tree and Domination Based on Vizing Conjecture

Abstract: In this study, from a tree with a quasi-spanning face, the algorithm will route Hamiltonian cycles. Goodey pioneered the idea of holding facing 4 to 6 sides of a graph concurrently. Similarly, in the three connected cubic planar graphs with two-colored faces, the vertex is incident to one blue and two red faces. As a result, all red-colored faces must gain 4 to 6 sides, while all obscure-colored faces must consume 3 to 5 sides. The proposed routing approach reduces the constriction of all vertex colors and the suitable quasi-spanning tree of faces. The presented algorithm demonstrates that the spanning tree parity will determine the arbitrary face based on an even degree. As a result, when the Lemmas 1 and 2 theorems are compared, the greedy routing method of Hamiltonian cycle faces generates valuable output from a quasi-spanning tree. In graph idea, a dominating set for a graph S = V , E is a subset D of V . The range of vertices in the smallest dominating set for S is the domination number ( S ). Vizing’s conjecture from 1968 proves that the Cartesian fabricated from graphs domination variety is at least as big as their domination numbers production. Proceeding this work, the Vizing’s conjecture states that for each pair of graphs S , L .

Fast Labeled Spanning Tree in Binary Irregular Graph Pyramids

Abstract: : Irregular Pyramids are powerful hierarchical structures in pattern recognition and image processing. They have high potential of parallel processing that makes them useful in processing of a huge amount of digital data generated every day. This paper presents a fast method for constructing an irregular pyramid over a binary image where the size of the images is more than 2000 in each of 2/3 dimensions. Selecting the contraction kernels (CKs) as the main task in constructing the pyramid is investigated. It is shown that the proposed fast labeled spanning tree (FLST) computes the equivalent contraction kernels (ECKs) in only two steps. To this purpose, first, edges of the corresponding neighborhood graph of the binary input image are classified. Second, by using a total order an efficient function is defined to select the CKs. By defining the redundant edges, further edge classification is performed to partition all the edges in each level of the pyramid. Finally, two important applications are presented : connected component labeling (CCL) and distance transform (DT) with lower parallel complexity 𝒪( 𝑙𝑜𝑔 ( 𝛿 )) where the 𝛿 is the diameter of the largest connected component in the image.

Forest formulas of discrete Green's functions

Abstract: The discrete Green's functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this article, we will give combinatorial interpretations of Green's functions in terms of enumerating trees and forests in a graph that will be used to derive further formulas for several graph invariants. For example, we show that the trace of the Green's function G ${\bf{G}}$ associated with the combinatorial Laplacian of a connected simple graph Γ ${\rm{\Gamma }}$ on n $n$ vertices satisfies Tr ( G ) = ∑ λ i ≠ 0 1 λ i = 1 n τ F 2 * , $\,\text{Tr}\,({\bf{G}})=\sum _{{\lambda }_{i} e 0}\frac{1}{{\lambda }_{i}}=\frac{1}{n\tau }\left|{{\mathbb{F}}}_{2}^{* }\right|,$ where λ i ${\lambda }_{i}$ denotes the eigenvalues of the combinatorial Laplacian, τ $\tau $ denotes the number of spanning trees, and F 2 * ${{\mathbb{F}}}_{2}^{* }$ denotes the set of rooted spanning 2-forests in Γ ${\rm{\Gamma }}$ . We will prove forest formulas for discrete Green's functions for directed and weighted graphs and apply them to study random walks on graphs and digraphs. We derive a forest expression of the hitting time for digraphs, which gives combinatorial proofs to old and new results about hitting times, traces of discrete Green's functions, and other related quantities.

Optimal Communication Spanning Tree

Abstract: In this work we address the Optimal Communication Spanning Tree (OCST) problem. An instance of this problem consists of a tuple ( G, c, R, w ) composed of a connected graph G = ( V, E ) , a nonnegative cost function c defined on E , a set R of pairs of vertices in V , and a nonnegative function w , called demand, defined on R . Each pair ( u, v ) of R is called a requirement, the vertex u is called origin, and the vertex v is called destination of the pair. For a given spanning tree T of G , the communication cost of a requirement pair r = ( u, v ) is defined as the demand w ( r ) multiplied by the distance between u and v in T (the distance being the sum of the costs of the edges in the path from u to v ). In the Optimal Communication Spanning Tree (OCST) problem, we are given an instance ( G, c, R, w ) and we seek a spanning tree in G that minimizes the overall sum of the communication costs of all requirements in R . This problem was introduced by T. C. Hu in 1974 and is known to be NP-hard. Some of its special cases, not so trivial, can be solved in polynomial time. We address such special cases of the OCST problem, both restricted to complete graphs. The first one is the Optimum Requirement Spanning Tree (ORST) problem, in which all edges have the same cost constant). In an optimal solution is given by a Gomory-Hu tree of a certain associated The second is a special case of the OCST problem, in which all requirements the demand. problem is called Minimum Routing Cost Spanning tree the tree problem, some purely combinatorial and some based on flows (leading to mixed formulations). Furthermore, we exhibit the computational results of the experiments we conducted with our implementation of a branch-and-cut approach for the different MILP formulations that we studied.

Divide-and-conquer based all spanning tree generation algorithm of a simple connected graph

Abstract: All spanning tree generation of a simple connected graph is a well-approached problem in graph theory. In this paper, an algorithm based on a new technique, namely divide-and-conquer, has been proposed. The performance of the proposed algorithm has also been benchmarked against several existing algorithms, including one algorithm implemented in parallel, in this domain. The basis of comparison is on the number of circuits generated and CPU time taken by each of the algorithms compared, using a set of randomly generated graph instances on a common platform.

Connectedness of the Free Uniform Spanning Forest as a function of edge weights

Abstract: Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in [4] that the Free Uniform Spanning Forest (FSF) of this graph may not be connected, but the dependence of this connectedness on H remains somewhat mysterious. We study the case when a positive weight w is put on the edges of the H-copies in G, and conjecture that the connectedness of the FSF exhibits a phase transition. For large enough w we show that the FSF is connected, while for a wide family of H and T, the FSF is disconnected when w is small (relying on [4]). Finally, we prove that when H is the graph of one edge, then for any w, the FSF is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.