Directed Graph Research Articles

Distributed finite-time optimization for networked Euler–Lagrange systems under a directed graph

In this paper, we investigate the finite-time distributed optimization problem for multiple Euler–Lagrange systems. A new distributed optimization control scheme is presented to achieve the state agreement in finite time while minimizing the sum of each agent’s local cost function. The proposed algorithm has the advantage of being able to achieve distributed finite-time optimization consensus under general unbalanced connected directed communication graphs. By virtue of finite-time Lyapunov theory and convex optimization, the finite-time convergence for the algorithm is analyzed. A numerical example is also presented to illustrate the effectiveness of the obtained results.

Multi-orientation adaptive slice and path generation based on support constraint and relationship matrix

PurposeThe step effect and support structure generated by the manufacturing process of fused deposition molding parts increase the consumables cost and decrease the printing quality. Multiorientation printing helps improve the surface quality of parts and reduce support, but path interference exists between the printing layer and the layers printed. The purpose of this study is to design printing paths between different submodels to avoid interference when build orientation changed.Design/methodology/approachConsidering support constraint, build orientation sequence is designed for submodels decomposed by model topology. The minimum printing angle between printing layers is analyzed. Initial path through the oriented bounding box is planned and slice interference relationship is then detected according to the projection topology mapping. Based on the relationship matrix of multiorientation slice, feasible path is calculated by directed graph (DG). Final printing path is determined under support constraint and checked by minimum printing angle. The simulation model of the robotic arm is established to verify the accessibility of printing path under the constraint of support and slice.FindingsThe proposed method can reduce support structure, decrease volume error and effectively solve the interference problem of the printing path for multiorientation slice.Originality/valueThe method based on projection topology mapping greatly improves the efficiency of interference detection. A feasible path calculated through DGs ensures the effectiveness of the printing path with the constraint of support and slice.

The oriented chromatic number of the hexagonal grid is 6

The oriented chromatic number of a directed graph G is the minimum order of an oriented graph to which G has a homomorphism. The oriented chromatic number χo(F) of a graph family F is the maximum oriented chromatic number over any orientation of any graph in F. For the family of hexagonal grids H2, Bielak (2006) proved that 5≤χo(H2)≤6. Here we close the gap by showing that χo(H2)≥6.

Bayesian inference of graph-based dependencies from mixed-type data

Mixed data comprise measurements of different types, with both categorical and continuous variables, and can be found in various areas, such as in life science or industrial processes. Inferring conditional independencies from the data is crucial to understand how these variables relate to each other. To this end, graphical models provide an effective framework, which adopts a graph-based representation of the joint distribution to encode such dependence relations. This framework has been extensively studied in the Gaussian and categorical settings separately; on the other hand, the literature addressing this problem in presence of mixed data is still narrow. We propose a Bayesian model for the analysis of mixed data based on the notion of Conditional Gaussian (CG) distribution. Our method is based on a canonical parameterization of the CG distribution, which allows for posterior inference of parameters indexing the (marginal) distributions of continuous and categorical variables, as well as expressing the interactions between the two types of variables. We derive the limiting Gaussian distributions, centered on the correct unknown value and with vanishing variance, for the Bayesian estimators of the canonical parameters expressing continuous, discrete and mixed interactions. In addition, we implement the proposed method for structure learning purposes, namely to infer the underlying graph of conditional independencies. When compared to alternative frequentist methods, our approach shows favorable results both in a simulation setting and in real-data applications, besides allowing for a coherent uncertainty quantification around parameter estimates.

Brauer Analysis of Some Cayley and Nilpotent Graphs and Its Application in Quantum Entanglement Theory

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages.

BF-BigGraph: An efficient subgraph isomorphism approach using machine learning for big graph databases

Graph databases are flexible NoSQL databases used to efficiently store and query complex and big data. One of the most difficult problems in graph databases is the problem of subgraph isomorphism, which involves finding a matching pattern in a given graph. Subgraph isomorphism algorithms generally encounter problems in the efficient processing of complex queries based on a lack of pruning methods and the use of a matching order. In this study, we present a new subgraph isomorphism approach based on the best-first search design strategy and name it BF-BigGraph. Our approach includes a machine learning technique to efficiently find the best matching order for various complex queries. The parameters we used in our approach as heuristics to improve the performance of complex queries on graph-based NoSQL databases are database volatility, database size, type of query, and the size of the query. We utilized the Random Forest machine learning method to narrow candidate nodes to a higher level of search and effectively reduce the search space for efficient querying and retrieval. We compared BF-BigGraph with state-of-the-art approaches, namely BB-Graph, Neo4j’s Cypher, DualIso, GraphQL, TurboIso, and VF3 using publicly available databases including undirected graphs; WorldCup, Pokec, Youtube, and a big graph database of a real demographic application (a population database) with approximately 70 million nodes of a big directed graph. The performance results of our approach for different types of complex queries on all these databases are significantly better in terms of computation time and required memory than other competing approaches in the literature.

Cycle-Star Motifs: Network Response to Link Modifications

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master–slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as Rössler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties.

Dominant eigenvalue and universal winners of digraphs

Let A be a nonnegative irreducible matrix of order n and Ai(t) be the matrix obtained by increasing its ith diagonal entry by a positive number t. An index i∈[n]={1,…,n} is called a universal winner for A, if, letting ρ(⋅) denote the spectral radius, ρ(Ai(t))≥ρ(Aj(t)) for j∈[n] and for t∈(0,∞).Let G be a strongly connected digraph with vertex set [n]. Let WG be the set of all nonnegative matrices whose underlying graph structure is G. We say a vertex u structurally dominates another vertex v in G, if ρ(Au(t))≥ρ(Av(t)) for all t∈(0,∞) and for all A∈WG. We characterize the class of digraphs G that do not have a vertex that structurally dominates all other vertices. We say two vertices u and v structurally tie if they structurally dominate each other in G. We supply an equivalent graph theoretic condition for the structural tying of two vertices in G. Let S⊆[n] be nonempty. We characterize the class of strongly connected digraphs G with vertex set [n] such that S is the set of universal winners for each A∈WG. We also characterize the strongly connected digraphs whose vertex set can be partitioned into subsets P1,P2,…,Pk such that vertices inside a part Pi structurally tie with each other and vertices of Pi structurally dominate vertices of Pj strictly (without tying) for i<j.

Distributed Byzantine-Resilient Observer for High-Order Integrator Multiagent Systems on Directed Graphs: An Edge-Based Approach

Distributed Byzantine-Resilient Observer for High-Order Integrator Multiagent Systems on Directed Graphs: An Edge-Based Approach

Design and application of data security privacy protection technology in dance action sequence arrangement and display system

In view of the excessive consumption of difficult resources in traditional dance choreography, insufficient data in choreography and data leakage crisis in the system, this paper puts forward artificial intelligence technology to study the protection of dance choreography and system data security and privacy. In this paper, the directed graph neural network is used to intelligently arrange the dance movements, and the Hidden Markov Model (HMM) is used to process the dance scene data so that the output data is highly similar to the original data. In the protection of data security and privacy, the method of data encryption is used, and the encrypted data needs to be decrypted by algorithm for reading. The experimental results show that the more complete the dance movement display, the higher the dance recognition rate, and the recognition rate of the dynamic light projection algorithm is greater than 90 % regardless of several sets of dance movements, while the recognition rate of the traditional light projection algorithm is not up to the standard regardless of several sets of movements, and the recognition rate of the point cloud segmentation method is higher than 90 % only in 3–4 sets of dance and 5–6 sets of dance movements.

Fiedler value: The cumulated dynamical contribution value of all edges in a complex network.

Fiedler value, as the minimal real part of (or the minimal) nonzero Laplacian eigenvalue, garners significant attention as a metric for evaluating network topology and its dynamics. In this paper, we address the quantification relation between Fiedler value and each edge in a directed complex network, considering undirected networks as a special case. We propose an approach to measure the dynamical contribution value of each edge. Interestingly, these contribution values can be both positive and negative, which are determined by the left and right Fiedler vectors. Further, we show that the cumulated dynamical contribution value of all edges is exactly the Fiedler value. This provides a promising angle on the Fiedler value in terms of dynamics and network structure. Therefore, the percentage of contribution of each edge to the Fiedler value is quantified. Numerical results reveal that network dynamics is significantly influenced by a small fraction of edges, say, one single directed edge contributes to over 90% of the Fiedler value in the Cat Cerebral Cortex network.

A method for calculating the technical readiness of aviation refuelling vehicles

In the paper a mathematical model of the process of operating aviation refuelling vehicles supplying fuel to aircraft before flight was developed. The present work is a continuation and supplement to the model contained in [52]. The phase space of the process under study was mapped by a 7-state directed graph of the operation process. To calculate the technical readiness index (K_gt) Markov chains and processes were used. Also, in Section 3, Results and discussions, optional methods for determining the technical readiness coefficient of a vehicle were provided k_gt) , based on the total time of the object in individual operating states. This is an alternative in a situation where the analysed process cannot reach a stable average state indefinitely. Two types of measures were used to determine the readiness, i.e. border probabilities and average times of the object in individual states. In both cases, the basis was statistical databases with operational vehicle data, which enabled the calculation of the readiness index and coefficient.

Extending Undirected Graph Techniques to Directed Graphs via Category Theory

We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering.

Advancements in Soft Directed Graph Theory: New Ideas and Properties

The emergence of soft set theory represented a significant advancement in establishing a holistic mathematical framework for handling uncertain data. Currently, numerous scholars are integrating this theory to address decision-making problems. In graph theory, a directed graph consists of vertices connected by directed edges, commonly referred to as arcs. Soft directed graphs are introduced by integrating the principles of soft sets into directed graphs, offering a parameterized perspective on directed graphs. In this paper, we introduce the fundamental notions of order, size, and frequency within the context of soft directed graphs. Furthermore, we establish the concepts of vertex and arc degree sums and explore their properties. Additionally, we delve into the concepts of strong vertex and strong arc, elucidating their significance. Finally, we define the notion of complement within soft directed graphs.

Grammar-based test suite construction using coverage-directed algorithms over LR-graphs

In grammar-based testing, the test suites that drive the system under test are typically constructed from a given context-free grammar through a set of derivations that jointly satisfy some coverage criterion. In this paper, we describe and evaluate a new algorithm that instead constructs test suites from a set of valid paths that cover all edges in a labeled directed graph corresponding to an LR-automaton that accepts the language of the grammar. Vertices in this graph correspond to states in the LR-automaton; two vertices are connected by an edge iff the top of the LR-automaton’s stack can change from one state to the other, either by shifting a terminal or non-terminal symbol (push edges), or by reducing with a grammar rule (pop edges). The algorithm constructs a unique reduction path for each pop edge in the graph. These reduction paths are recursively embedded into each other, and any unresolved non-terminal push edges are substituted by shortest derivations for the non-terminal symbol. The algorithm can work with different types of LR-automata, including LR(0)- and LR(1)-automata, and can successfully generate a test suite from an LR-graph even if the underlying LR-automaton construction leads to shift/reduce or reduce/reduce conflicts. The algorithm only constructs valid paths over the LR-graphs that correspond to sentences in the language and thus generates only positive tests. We therefore also describe mutations to the positive paths that are guaranteed to generate negative tests without needing any further verification by an oracle.Our algorithm is substantially more efficient than an earlier algorithm that explores LR-graphs with two consecutive breadth-first graph traversals and our experimental evaluation shows that it scales to large production-quality grammars. It is robust against random choices made to resolve ambiguity in the construction of the tests, while the code coverage of the different test suite variants is relatively uniform. Finally, our evaluation shows that the negative test suites constructed by path mutation identify more faults in a set of student grammars than those constructed by rule mutation.

A planar graph cluster‐routing approach for optimizing medical waste collection based on spatial constraint

AbstractMedical Solid Wastes (MSWs) are major hazardous materials containing harmful biological or chemical compounds that present public and environmental health risks. The collection and transportation of waste are usually informed by optimized work‐balanced routing based on comprehensive spatial data in urban traffic networks, called a Vehicle Routing Problem (VRP). This may be unsuitable for MSWs as their special category means they impose additional complexity. The present article develops a planar graph‐based cluster‐routing approach for the optimal collection of MSWs informed by a Geospatial Information System (GIS). The problem is first formulated as a mixed integer linear program in road network spatial data, in the context of Tehran city. The work has two key aims: (i) to minimize the total routing cost of MSW collection and transfer to waste landfills; (ii) to balance workload across waste collectors. There are three main contributions of the proposed approach: (i) to simplify the large search space area by converting the road network to a planar graph based on graph theory, spatial parameters, and topological rules; (ii) to use a modified K‐means algorithm for clustering; (iii) to consider average traffic impacts in the clustering stage and momentary traffic in the route planning stage. A planar graph extraction procedure is applied to capture the network sketch (i.e., a directed graph) from the traffic roadway network. An iterative cluster‐first‐route‐second heuristic is employed to solve the proposed routing problem. This heuristic customizes a K‐means algorithm to determine the optimal number and size of clusters (i.e., routes). A Traveling Salesman Problem (TSP) algorithm is applied to regulate the optimal sequence of visits to medical centers. The experimental results show improvements in balancing collectors' workload (i.e., ~4 min reduction in the standard deviation of average travel time) with reductions in travel time (i.e., an average ~1 h for the entire fleet and ~4 min per route). These findings confirm that the proposed methodology can be considered as an approach for optimizing waste collection routes.

Properties of congruence lattices of graph inverse semigroups

From any directed graph [Formula: see text] one can construct the graph inverse semigroup [Formula: see text], whose elements, roughly speaking, correspond to paths in [Formula: see text]. Wang and Luo showed that the congruence lattice [Formula: see text] of [Formula: see text] is upper-semimodular for every graph [Formula: see text], but can fail to be lower-semimodular for some [Formula: see text]. We provide a simple characterization of the graphs [Formula: see text] for which [Formula: see text] is lower-semimodular. We also describe those [Formula: see text] such that [Formula: see text] is atomistic, and characterize the minimal generating sets for [Formula: see text] when [Formula: see text] is finite and simple.

Self-stabilizing synchronous unison in directed networks

Self-stabilization is a general paradigm that characterizes the ability of a distributed system to recover from transient faults. Since its introduction by Dijkstra in 1974, self-stabilization has been successfully applied to efficiently solve many networking tasks. However, most of the literature focuses on bidirectional networks. Now, in today's networks such as WSNs, some communication channels may be one-way only. Considering such network topologies, a.k.a. directed graphs, makes self-stabilization more complicated, and sometimes even impossible. In this paper, we investigate the gap in terms of requirements and efficiency when considering a directed graph instead of an undirected one as network topology for a self-stabilizing algorithm. Our case study is a variant of a synchronous unison algorithm proposed by Arora et al.; the synchronous unison being a clock synchronization problem.

The genus of regular languages and directed graph emulators

The article continues our study of the genus of a regular language L, defined as the minimal genus among all genera of all finite deterministic automata recognizing L. Here we define and study two closely related tools on a directed graph: directed emulators and automatic relations. A directed emulator morphism essentially encapsulates at the graph-theoretic level an epimorphism onto the minimal deterministic automaton. An automatic relation is the graph-theoretic version of the Myhill-Nerode relation. We show that an automatic relation determines a directed emulator morphism and respectively, a directed emulator morphism determines an automatic relation up to isomorphism. Consider the set S of all directed emulators of the underlying directed graph of the minimal deterministic automaton for L. We prove that the genus of L is minG∈Sg(G). We also consider the more restrictive notion of directed cover and prove that the genus of L is reached in the class of directed covers of the underlying directed graph of the minimal deterministic automaton for L. This stands in sharp contrast to undirected emulators and undirected covers which we also consider. Finally we prove that if the problem of determining the minimal genus of a directed emulator of a directed graph has a solution then the problem of determining the minimal genus of an undirected emulator of an undirected graph has a solution.

Parameterized algorithms for Steiner tree and (connected) dominating set on path graphs

AbstractChordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).