Average Convergence for Directed&Undirected Graphs in Distributed Systems

Consensus control of multi-agent systems is an innovative paradigm for the development of intelligent distributed systems.This has fascinated numerous scientific groups for their promising applications as they have the freedom to achieve their local and global goals and make their own decisions.Network communication topologies based on graph and matrix theory are widely used in a various real-time applications ranging from software agents to robotics.Therefore,while sustaining the significance of both directed and undirected graphs,this research emphases on the demonstration of a distributed average consensus algorithm.It uses the harmonic mean in the domain of multi-agent systems with directed and undirected graphs under static topologies based on a control input scheme.The proposed agreement protocol focuses on achieving a constant consensus on directional and undirected graphs using the exchange of information between neighbors to update their status values and to be able to calculate the total number of agents that contribute to the communication network at the same time.The proposed method is implemented for the identical networks that are considered under the directional and non-directional communication links.Two different scenarios are simulated and it is concluded that the undirected approach has an advantage over directed graph communication in terms of processing time and the total number of iterations required to achieve convergence.The same network parameters are introduced for both orientations of the communication graphs.In addition,the results of the simulation and the calculation of various matrices are provided at the end to validate the effectiveness of the proposed algorithm to achieve consensus.

Signed Directed Graph and Qualitative Trend Analysis Based Fault Diagnosis in Chemical Industry

In the past 30 years,signed directed graph(SDG) ,one of the qualitative simulation technologies,has been widely applied for chemical fault diagnosis.However,SDG based fault diagnosis,as any other qualitative method,has poor diagnostic resolution.In this paper,a new method that combines SDG with qualitative trend analysis(QTA) is presented to improve the resolution.In the method,a bidirectional inference algorithm based on assumption and verification is used to find all the possible fault causes and their corresponding consistent paths in the SDG model.Then an improved QTA algorithm is used to extract and analyze the trends of nodes on the consis-tent paths found in the previous step.New consistency rules based on qualitative trends are used to find the real causes from the candidate causes.The resolution can be improved.This method combines the completeness feature of SDG with the good diagnostic resolution feature of QTA.The implementation of SDG-QTA based fault diagno-sis is done using the integrated SDG modeling,inference and post-processing software platform.Its application is illustrated on an atmospheric distillation tower unit of a simulation platform.The result shows its good applicability and efficiency.

Drawing Weighted Directed Graph from It's Adjacency Matrix

This paper proposes an algorithm for building weighted directed graph, defmes the weighted directed relationship matrix of the graph, and describes algorithm implementation using this matrix. Based on this algorithm, an effective way for building and drawing weighted directed graphs is presented, forming a foundation for visual implementation of the algorithm in the graph theory.

Distributed Cooperative Learning for Discrete-Time Strict-Feedback Multi Agent Systems Over Directed Graphs

This paper focuses on the distributed cooperative learning(DCL)problem for a class of discrete-time strict-feedback multi-agent systems under directed graphs.Compared with the previous DCL works based on undirected graphs,two main challenges lie in that the Laplacian matrix of directed graphs is nonsymmetric,and the derived weight error systems exist n-step delays.Two novel lemmas are developed in this paper to show the exponential convergence for two kinds of linear time-varying(LTV)systems with different phenomena including the nonsymmetric Laplacian matrix and time delays.Subsequently,an adaptive neural network(NN)control scheme is proposed by establishing a directed communication graph along with n-step delays weight updating law.Then,by using two novel lemmas on the extended exponential convergence of LTV systems,estimated NN weights of all agents are verified to exponentially converge to small neighbourhoods of their common optimal values if directed communication graphs are strongly connected and balanced.The stored NN weights are reused to structure learning controllers for the improved control performance of similar control tasks by the“mod”function and proper time series.A simulation comparison is shown to demonstrate the validity of the proposed DCL method.

A Fuzzy Directed Graph-Based QoS Model for Service Composition

Web service composition lets developers create applications on top of service-oriented computing and its native description, discovery, and communication capabilities. This paper mainly focuses on the QoS when the concrete composition structure is unknown. A QoS model of service composition is presented based on the fuzzy directed graph theory. According to the model, a recursive algorithm is also described for calculating such kind of QoS. And, the feasibility of this QoS model and the recursive algorithm is verified by a case study. The proposed approach enables customers to get a possible value of the QoS before they achieve the service.

Evolution of Word-updating Dynamical Systems (WDS) on Directed Graphs

This paper continues the research on theoretical foundations for computer simulation.We introduce the concept of word-updating dynamical systems(WDS)on directed graphs,which is a kind of generalization of sequential dynamical systems(SDS)on graphs.Some properties on WDS,especially some results on NOR-WDS,which are different from that on NOR-SDS,are obtained.

Colouring of COVID-19 Affected Region Based on Fuzzy Directed Graphs

Graph colouring is the system of assigning a colour to each vertex of a graph.It is done in such a way that adjacent vertices do not have equal colour.It is fundamental in graph theory.It is often used to solve real-world problems like traffic light signalling,map colouring,scheduling,etc.Nowadays,social networks are prevalent systems in our life.Here,the users are considered as vertices,and their connections/interactions are taken as edges.Some users follow other popular users’profiles in these networks,and some don’t,but those non-followers are connected directly to the popular profiles.That means,along with traditional relationship(information flowing),there is another relation among them.It depends on the domination of the relationship between the nodes.This type of situation can be modelled as a directed fuzzy graph.In the colouring of fuzzy graph theory,edge membership plays a vital role.Edge membership is a representation of flowing information between end nodes of the edge.Apart from the communication relationship,there may be some other factors like domination in relation.This influence of power is captured here.In this article,the colouring of directed fuzzy graphs is defined based on the influence of relationship.Along with this,the chromatic number and strong chromatic number are provided,and related properties are investigated.An application regarding COVID-19 infection is presented using the colouring of directed fuzzy graphs.

On (t, r) Broadcast Domination of Directed Graphs

A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases r = 1 and (t, r) = (2, 2), for every integer value in this interval, there exists an orientation of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.

Cooperative Output Regulation of Heterogeneous Systems over Directed Graphs:A Dynamic Adaptive Event-Triggered Strategy

This paper investigates the cooperative output regulation problem of heterogeneous linear multi-agent systems over directed graphs with the constraint of communication bandwidth.Given that there exists an exosystem whose state information is not available to all agents,the authors develop distributed adaptive event-triggered observers for the followers based on relative information between neighboring agents.It should be pointed out that,two kinds of time-varying gains are introduced to avoid relying on any global information associated with the network,and dynamic triggering conditions are designed to get rid of continuous communications.On the basis of the designed observers,the authors devise a local controller for each agent.Compared with the existing related works,the main contribution of the current paper is that the cooperative output regulation problem for general directed graphs is solved requiring neither global information nor continuous communications.

On an Invariant of Tournament Digraphs

To date, it is unknown whether it is possible to construct a complete graph invariant in polynomial time, so fast algorithms for checking non-isomorphism are important, including heuristic algorithms, and for successful implementations of such heuristics, both the tasks of some modification of previously described graph invariants and the description of new invariants remain relevant. Many of the described invariants make it possible to distinguish a larger number of graphs in the real time of a computer program. In this paper, we propose an invariant for a special kind of directed graphs, namely, for tournaments. The last ones, from our point of view, are interesting because when fixing the order of vertices, the number of different tournaments is exactly equal to the number of undirected graphs, also with fixing the order of vertices. In the invariant we are considering, all possible tournaments consisting of a subset of vertices of a given digraph with the same set of arcs are iterated over. For such subset tournaments, the places are calculated in the usual way, which are summed up to obtain the final values of the points of the vertices;these points form the proposed invariant. As we expected, calculations of the new invariant showed that it does not coincide with the most natural invariant for tournaments, in which the number of points is calculated for each participant. So far, we have conducted a small number of computational experiments, and the minimum value of the pair correlation between the sequences representing these two invariants that we found is for dimension 15.

Directed Acyclic Graph Blockchain for Secure Spectrum Sharing and Energy Trading in Power IoT

Peer-to-peer(P2P)spectrum sharing and energy trading are promising solutions to locally satisfy spectrum and energy demands in power Internet of Things(IoT).However,implementation of largescale P2P spectrum sharing and energy trading confronts security and privacy challenges.In this paper,we exploit consortium blockchain and Directed Acyclic Graph(DAG)to propose a new secure and distributed spectrum sharing and energy trading framework in power IoT,named spectrum-energy chain,where a set of local aggregators(LAGs)cooperatively confirm the identity of the power devices by utilizing consortium blockchain,so as to form a main chain.Then,the local power devices verify spectrum and energy micro-transactions simultaneously but asynchronously to form local spectrum tangle and local energy tangle,respectively.Moreover,an iterative double auction based micro transactions scheme is designed to solve the spectrum and energy pricing and the amount of shared spectrum and energy among power devices.Security analysis and numerical results illustrate that the developed spectrum-energy chain and the designed iterative double auction based microtransactions scheme are secure and efficient for spectrum sharing and energy trading in power IoT.

A Secure Microgrid Data Storage Strategy with Directed Acyclic Graph Consensus Mechanism

The wide application of intelligent terminals in microgrids has fueled the surge of data amount in recent years.In real-world scenarios,microgrids must store large amounts of data efficiently while also being able to withstand malicious cyberattacks.To meet the high hardware resource requirements,address the vulnerability to network attacks and poor reliability in the tradi-tional centralized data storage schemes,this paper proposes a secure storage management method for microgrid data that considers node trust and directed acyclic graph(DAG)consensus mechanism.Firstly,the microgrid data storage model is designed based on the edge computing technology.The blockchain,deployed on the edge computing server and combined with cloud storage,ensures reliable data storage in the microgrid.Secondly,a blockchain consen-sus algorithm based on directed acyclic graph data structure is then proposed to effectively improve the data storage timeliness and avoid disadvantages in traditional blockchain topology such as long chain construction time and low consensus efficiency.Finally,considering the tolerance differences among the candidate chain-building nodes to network attacks,a hash value update mechanism of blockchain header with node trust identification to ensure data storage security is proposed.Experimental results from the microgrid data storage platform show that the proposed method can achieve a private key update time of less than 5 milliseconds.When the number of blockchain nodes is less than 25,the blockchain construction takes no more than 80 mins,and the data throughput is close to 300 kbps.Compared with the traditional chain-topology-based consensus methods that do not consider node trust,the proposed method has higher efficiency in data storage and better resistance to network attacks.

Minimum Coverings of Complete Directed Graphs with Odd Size Circuits

Let DKv denote the symmetric complete directed graph with v vertices, the covering number C(v,m) is a minimum number of covering DKv by m-circuits. In this paper, C(v,m) is determined for any fixed odd positive integer m and positive integer v, m ≤ v ≤ m + 6.

From undirected graphs to directed graphs:a new technique makes it possible for multi-agent systems

This communique introduces a simple technique,which enables the Laplacian-like matrix of a directed graph to be diagonalisable.Thus many results on multi-agent systems over undirected graphs can be extended to directed graphs.As an example,we show how to make this extension of a popular Automatica paper.

Directed graph-based distribution network reconfiguration for operation mode adjustment and service restoration considering distributed generation

We present a directed graph-based method for distribution network reconfiguration considering distributed generation. Two reconfiguration situations are considered: operation mode adjustment with the objective of minimizing active power loss(situation Ⅰ) and service restoration with the objective of maximizing loads restored(situation Ⅱ). These two situations are modeled as a mixed integer quadratic programming problem and a mixed integer linear programming problem, respectively. The properties of the distribution network with distributed generation considered are reflected as the structure model and the constraints described by directed graph. More specifically, the concepts of "in-degree" and "out-degree"are presented to ensure the radial structure of the distribution network, and the concepts of "virtual node" and"virtual demand" are developed to ensure the connectivity of charged nodes in every independent power supply area.The validity and effectiveness of the proposed method are verified by test results of an IEEE 33-bus system and a 5-feeder system.

Semantic Relation Annotation for Biomedical Text Mining Based on Recursive Directed Graph

In this paper we propose a novel model ＂recursive directed graph＂ based on feature structure, and apply it to represent the semantic relations of postpositive attributive structures in biomedical texts. The usages of postpositive attributive are complex and variable, especially three categories： present participle phrase, past participle phrase, and preposition phrase as postpositire attributive, which always bring the difficulties of automatic parsing. We summarize these categories and annotate the semantic information. Compared with dependency structure, feature structure, being recursive directed graph, enhances semantic information extraction in biomedical field. The annotation results show that recursive directed graph is more suitable to extract complex semantic relations for biomedical text mining.

Observer-based event-triggered control for linear MASs under a directed graph and DoS attacks

In this paper,we investigate the observer-based event-triggered consensus problem for linear multi-agent systems(MASs)under a directed graph and denial-of-service(DoS)attacks.A type of DoS attacks launched by malicious attackers at irregular intervals is considered,which can cause communication channel disruption.A novel event-triggered secure control scheme based on a closed-loop observer is proposed to determine the scheduling of the controller update,and a separation method with less conservativeness is employed to design the controller and observer gains.Then,the frequency and duration of DoS attacks that can be tolerated are analysed for the observer-based secure consensus problem.In addition,a strictly positive minimal event-triggered time interval for each agent is designed with the help of the proposed eventtriggered condition to eliminate the Zeno behaviour.Finally,a numerical simulation is given to verify the theoretical analysis.

Continuous-time Distributed Heavy-ball Algorithm for Distributed Convex Optimization over Undirected and Directed Graphs

This paper proposes second-order distributed algorithms over multi-agent networks to solve the convex optimization problem by utilizing the gradient tracking strategy, with convergence acceleration being achieved. Both the undirected and unbalanced directed graphs are considered, extending existing algorithms that primarily focus on undirected or balanced directed graphs. Our algorithms also have the advantage of abandoning the diminishing step-size strategy so that slow convergence can be avoided. Furthermore, the exact convergence to the optimal solution can be realized even under the constant step size adopted in this paper. Finally, two numerical examples are presented to show the convergence performance of our algorithms.

A Fuzzy Fault Diagnosis Method for Large Radar Based on Directed Graph Model

To meet the requirement of the real-time, accuracy and multi-target diagnosis of the large radar system,a new fuzzy fault diagnosis method based on directed graph model is proposed in this paper. In this method, the large complex system model is defined using the directed graph model firstly, in which the nodes observing the fault by the hierarchical reconstruction of the directed graph are located, then the fault dependency matrix between these nodes and the fault sources are established. And then, we utilize the sensors' alarm probabilities under different situations to build the characteristic fault observation matrix in the fault observation space. Finally,the optimized corresponding diagnosis method using a fuzzy function, which describes the similarity between the actual observation vector and the fault's characteristic vector, is designed. The experimental results demonstrate that the proposed method can achieve high diagnosis efficiency and accuracy. It can be widely used in the real radar system.

Nash equilibrium seeking over directed graphs

In this paper,we aim to develop distributed continuous-time algorithms over directed graphs to seek the Nash equilibrium in a noncooperative game.Motivated by the recent consensus-based designs,we present a distributed algorithm with a proportional gain for weight-balanced directed graphs.By further embedding a distributed estimator of the left eigenvector associated with zero eigenvalue of the graph Laplacian,we extend it to the case with arbitrary strongly connected directed graphs having possible unbalanced weights.In both cases,the Nash equilibrium is proven to be exactly reached with an exponential convergence rate.An example is given to illustrate the validity of the theoretical results.