Probability Distribution of Edge in Adjacent Matrix of Aviation Network of China and Algorithm of Searching Non-overlap Community Structure Based on Complex Network

In order to discover the probability distribution feature of edge in aviation network adjacent matrix of China and on the basis of this feature to establish an algorithm of searching non-overlap community structure in network to reveal the inner principle of complex network with the feature of small world in aspect of adjacent matrix and community structure,aviation network adjacent matrix of China was transformed according to the node rank and the matrix was arranged on the basis of ascending node rank with the center point as original point.Adjacent probability from the original point to extension around in approximate area was calculated.Through fitting probability distribution curve,power function of probability distribution of edge in adjacent matrix arranged by ascending node rank was found.According to the feature of adjacent probability distribution,deleting step by step with node rank ascending algorithm was set up to search non-overlap community structure in network and the flow chart of algorithm was given.A non-overlap community structure with 10 different scale communities in aviation network of China was found by the computer program written on the basis of this algorithm.

A Graph with Adaptive AdjacencyMatrix for Relation Extraction

The relation is a semantic expression relevant to two named entities in a sentence.Since a sentence usually contains several named entities,it is essential to learn a structured sentence representation that encodes dependency information specific to the two named entities.In related work,graph convolutional neural networks are widely adopted to learn semantic dependencies,where a dependency tree initializes the adjacency matrix.However,this approach has two main issues.First,parsing a sentence heavily relies on external toolkits,which can be errorprone.Second,the dependency tree only encodes the syntactical structure of a sentence,which may not align with the relational semantic expression.In this paper,we propose an automatic graph learningmethod to autonomously learn a sentence’s structural information.Instead of using a fixed adjacency matrix initialized by a dependency tree,we introduce an Adaptive Adjacency Matrix to encode the semantic dependency between tokens.The elements of thismatrix are dynamically learned during the training process and optimized by task-relevant learning objectives,enabling the construction of task-relevant semantic dependencies within a sentence.Our model demonstrates superior performance on the TACRED and SemEval 2010 datasets,surpassing previous works by 1.3%and 0.8%,respectively.These experimental results show that our model excels in the relation extraction task,outperforming prior models.

A new matrix-based mathematical model for determining unidirectional circuits in a ventilation network

The occurrence of local circulating ventilation can be caused by many factors, such as the airflow reversion during mine fire,the improper arrangement of local fan or underground fan station and the man-made error input of raw data before network solving. Once circulating ventilations occur,the corresponding branches in the ventilation network corresponding to the relevant airways in ventilation system form circuits,and all the direc- tions of the branches in the circuits are identical,which is the unidirectional problem in ventilation network.Based on the properties of node adjacent matrix,a serial of mathe- matical computation to node adjacent matrix were performed,and a mathematical model for determining unidirectional circuits based on node adjacent matrix was put forward.

ADJACENT MARTIX METHOD OF IDENTIFYING ISOMORPHISM TO PLANAR KINEMATIC CHAIN WITH MULTIPLE JOINTS AND HIGHER PAIRS

The adjacent matrix method for identifying isomorphism to planar kinematic chain with multiple joints and higher pairs is presented. The topological invariants of the planar kinematic chain can be calculated and compared by adjacent matrix. The quantity of calculation can be reduced effectively using the several divisions of bars and the reconfiguration of the adjacent matrix. As two structural characteristics of adjacent matrix, the number of division and division code are presented. It can be identified that two kinematic chains are isomorphic or not by comparing the structural characteristics of their adjacent matrixes using a method called matching row-to-row. This method may be applied to the planar linkage chain too. So, the methods of identifying isomorphism are unified in the planar kinematic chain that has or hasn＇t higher pairs with or without multiple joints. And it has some characters such as visual, simple and convenient for processing by computer, and so on.

AFSTGCN:Prediction for multivariate time series using an adaptive fused spatial-temporal graph convolutional network

The prediction for Multivariate Time Series(MTS)explores the interrelationships among variables at historical moments,extracts their relevant characteristics,and is widely used in finance,weather,complex industries and other fields.Furthermore,it is important to construct a digital twin system.However,existing methods do not take full advantage of the potential properties of variables,which results in poor predicted accuracy.In this paper,we propose the Adaptive Fused Spatial-Temporal Graph Convolutional Network(AFSTGCN).First,to address the problem of the unknown spatial-temporal structure,we construct the Adaptive Fused Spatial-Temporal Graph(AFSTG)layer.Specifically,we fuse the spatial-temporal graph based on the interrelationship of spatial graphs.Simultaneously,we construct the adaptive adjacency matrix of the spatial-temporal graph using node embedding methods.Subsequently,to overcome the insufficient extraction of disordered correlation features,we construct the Adaptive Fused Spatial-Temporal Graph Convolutional(AFSTGC)module.The module forces the reordering of disordered temporal,spatial and spatial-temporal dependencies into rule-like data.AFSTGCN dynamically and synchronously acquires potential temporal,spatial and spatial-temporal correlations,thereby fully extracting rich hierarchical feature information to enhance the predicted accuracy.Experiments on different types of MTS datasets demonstrate that the model achieves state-of-the-art single-step and multi-step performance compared with eight other deep learning models.

A New Proof on the Bipartite Turán Number of Bipartite Graphs

The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t, the problem of determining the value of ex(m,n;Km−t,n−t)has been solved by Balbuena et al. in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of ex(m,n;Km−t,n−t)by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using { 0,1 }-matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].

Structural Synthesis of Compliant Metamorphic Mechanisms Based on Adjacency Matrix Operations

A compliant metamorphic mechanism attributes to a new type of metamorphic mechanisms evolved from rigid metamorphic mechanisms. The structural characteristics and representations of a compliant metamorphic mechanism are different from its rigid counterparts, so does the structural synthesis method. In order to carry out its structural synthesis, a constraint graph representation for topological structure of compliant metamorphic mechanisms is introduced, which can not only represent the structure of a compliant metamorphic mechanism, but also describe the characteristics of its links and kinematic pairs. An adjacency matrix representation of the link relationships in a compliant metamorphic mechanism is presented according to the constraint graph. Then, a method for structural synthesis of compliant metamorphic mechanisms is proposed based on the adjacency matrix operations. The operation rules and the operation procedures of adjacency matrices are described through synthesis of the initial configurations composed of s＋1 links from an s-link mechanism （the final configuration）. The method is demonstrated by synthesizing all the possible four-link compliant metamorphic mechanisms that can transform into a three-link mechanism through combining two of its links. Sixty-five adjacency matrices are obtained in the synthesis, each of which corresponds to a compliant metamorphic mechanism having four links. Therefore, the effectiveness of the method is validated by a specific compliant metamorphic mechanism corresponding to one of the sixty-five adjacency matrices. The structural synthesis method is put into practice as a fully compliant metamorphic hand is presented based on the synthesis results. The synthesis method has the advantages of simple operation rules, clear geometric meanings, ease of programming with matrix operation, and provides an effective method for structural synthesis of compliant metamorphic mechanisms and can be used in the design of new compliant metamorphic mechanisms.

Configuration Analysis of Metamorphic Mechanisms Based on Extended Adjacency Matrix Operations

The adjacency matrix operations,which connect with configuration transformation correspondingly,can be used for analysis of configuration transformation of metamorphic mechanisms and the corresponding algorithm can easily be simulated by computer.But the adjacency matrix based on monochrome topological graph is not suitable for the topological representation of mechanisms with multiple joints.The method of adjacency matrix operations has its own limitations for analysis of configuration transformation of metamorphic mechanisms because it can only be used in the topological representation of mechanisms with single joints.In order to overcome the drawback of the adjacency matrix,a kind of new matrix named as extended adjacency matrix is proposed to express topological structures of all mechanisms.The extended adjacency matrix is not only suitable for the topological representation of mechanisms with single joints,but also can be used in that of mechanisms with multiple joints.On this basis,a method of matrix operations based on the extended adjacency matrix is proposed to analyze the configuration transformation of metamorphic mechanisms.The method is not only suitable for configuration analysis of metamorphic mechanisms with single joints as well as metamorphic mechanisms with multiple joints.The method is evaluated by calculating two examples representing metamorphic mechanisms with single joint and multiple joints respectively.It can be concluded that the method is effective and correct for analysis of configuration transformation of all metamorphic mechanisms.The proposed method is simple and easy to be achieved by computer programming.It provides a basis for structural synthesis of all metamorphic mechanisms.

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.

两类四全图的奇异性

Let G be a finite simple graph and A(G)be its adjacency matrix.Then G is singular if A(G)is singular.The graph obtained by bonding the starting ver-tices and ending vertices of three paths Pa1,Pa2,Pa3 is calledθ-graph,represented byθ(a1,a2,a3).The graph obtained by bonding the two end vertices of the path Ps to the vertices of theθ(a1,a2,a3)andθ(b1,b2,b3)of degree three,respectively,is denoted byα(a1,a2,a3,s,b1,b2,b3)and calledα-graph.β-graph is denoted whenβ(a1,a2,a3,b1,b2,b3)=α(a1,a2,a3,1,b1,b2,b3).In this paper,we give the necessary and sufficient conditions for the singularity ofα-graph andβ-graph,and prove that the probability that a random givenα-graph andβ-graph is a singular graph is equal to 14232048 and 733/1024,respectively.

Using adjacency matrix to explore remarkable associations in big and small mineral data

Data exploration,usually the first step in data analysis,is a useful method to tackle challenges caused by big geoscience data.It conducts quick analysis of data,investigates the patterns,and generates/refines research questions to guide advanced statistics and machine learning algorithms.The background of this work is the open mineral data provided by several sources,and the focus is different types of associations in mineral properties and occurrences.Researchers in mineralogy have been applying different techniques for exploring such associations.Although the explored associations can lead to new scientific insights that contribute to crystallography,mineralogy,and geochemistry,the exploration process is often daunting due to the wide range and complexity of factors involved.In this study,our purpose is implementing a visualization tool based on the adjacency matrix for a variety of datasets and testing its utility for quick exploration of association patterns in mineral data.Algorithms,software packages,and use cases have been developed to process a variety of mineral data.The results demonstrate the efficiency of adjacency matrix in real-world usage.All the developed works of this study are open source and open access.

An adaptive improvement on PageRank algorithm

In this article, we introduce the Google＇s method for quality ranking of web page in a formal mathematical format, use the power iteration to improve the PageRank, and also discuss the effect of different q to the PageRank, as well as how a PageRank will be changed if more links are added to one page or removed from some pages.

The Angles and Main Angles of Some Special Graphs

The eigenvalues of the adjacency matrix of a graph are called the eigenvalues of the graph. Let the vector ej =(0, … , 1, … , 0)T and the all -1 vector j =(1, 1, …,1)T, the cosine of the (acute) angle formed by the vector ej and the eigensubspace is called an angle of the graph. The cosine of the (acute) angle formed by the vector j and the eigensubspace is called a main angle of the graph. The angles and main angles are all important parameters on the graph, and they can be combined with the eigenvalues of the graph to determine the degree sequence of the graph, the number of triangles, quadrilaterals and pentagons on the graph, and the characteristic polynomials of the complement graph, but there is little study on the angles and main angles of the graph. In this paper, we determine the angles and main angles of the complete graph, the cube graph, the Petersen graph, the cycle and the complete bipartite graph.

Computer generation of detailed reaction networks in hydrocracking of Fischer-Tropsch wax

Fischer-Tropsch synthesis(FTS)wax is a mixture of linear hydrocarbons with carbon number from C7 to C70+.Converting FTS wax into high-quality diesel(no sulfur and nitrogen contents)by hydrocracking technology is attractive in economy and practicability.Kinetic study of the hydrocracking of FTS wax in elementary step level is very challenging because of the huge amounts of reactions and species involved.Generation of reaction networks for hydrocracking of FTS wax in which the chain length goes up to C70 is described on the basis of Boolean adjacency matrixes.Each of the species(including paraffins,olefins and carbenium ions)involved in the elementary steps is represented digitally by using a(N+3)N matrix,in which a group of standardized numbering rules are designed to guarantee the unique identity of the species.Subsequently,the elementary steps are expressed by computer-aided matrix transformations in terms of proposed reaction rules.Dynamic memory allocation is used in species storage and a characteristic vector with nine elements is designed to store the key information of a(N+3)N matrix,which obviously reduces computer memory consumption and improves computing efficiency.The detailed reaction networks of FTS wax hydrocracking can be generated smoothly and accurately by the current method.The work is the basis of advanced elementary-step-level kinetic modeling.

A Formula of Average Path Length for Unweighted Networks

In this paper,based on the adjacency matrix of the network and its powers,the formulas are derived for theshortest path and the average path length,and an effective algorithm is presented.Furthermore,an example is providedto demonstrate the proposed method.

Topological Configuration and Rotation Analysis of the Three⁃Order Rubik􀆳s Cube

The mechanism of three⁃order Rubik􀆳s Cube(RC)has the characteristics of recombination and variable degree of freedom,and it is difficult to accurately describe the degree of its freedom.This paper takes RC as the research object,and the adjacency matrix is constructed based on topology and graph theory in order to describe the variation rule of topological configuration in the single layer rotation of RC.In this paper,the degree of freedom of the RC in any shape can be described by defining the concept of entanglement degree of freedom,establishing a set of adjacency matrix,and determining the degree of freedom of the RC which is attributed to the number of non⁃zero elements in the set of adjacent matrix.The prime number is proposed to describe the rotation of the RC combined with the rotation recognition of RC,which is simple and convenient for computer processing.The research contents in this paper are beneficial to the study of RC from the perspective of mechanism science.Meanwhile,it is of great significance to the study of other complex mechanisms with variable degrees of freedom.

A Criterion of Decomposable Elements in V^(2)

In this paper, a criterion of decomposable element in V(2) is obtained , i. e.,Z = p(i, j)eij is decomposable if and only if its adjacent coordinate matrices are all "row 1≤i≤j≤ncommute" equivalent.

The Spectral Radii of Some Adhesive Graphs

The spectral radius of a graph is the maximum eigenvalues of its adjacency matrix. In this paper, using the property of quotient graph, the sharp upper bounds for the spectral radii of some adhesive graphs are determined.

Review and Simulation of a Novel Formation Building Algorithm While Enabling Obstacle Avoidance

Technically, a group of more than two wheeled mobile robots working collectively towards a common goal are known as a multi-robot system. An increasing number of industries have implemented multi-robot systems to eliminate the risk of human injuries while working on hazardous tasks, and to improve productivity. Globally, engineers are continuously researching better, simple, and faster cooperative Control algorithms to provide a Control strategy where each agent in the robot formation can communicate effectively and achieve a consensus in their position, orientation and speed. This paper explores a novel Formation Building Algorithm and its global stability around a configuration vector. A simulation in MATLAB? was carried out to examine the performance of the Algorithm for two geometric formations and a fixed number of robots. In addition, an obstacle avoidance technique was presented assuming that all robots are equipped with range sensors. In particular, a uniform rounded obstacle is used to analyze the performance of the technique with the use of detailed geometric calculations.

On the Number of Cycles in a Graph

In this paper, we obtain explicit formulae for the number of 7-cycles and the total number of cycles of lengths 6 and 7 which contain a specific vertex vi in a simple graph G, in terms of the adjacency matrix and with the help of combinatorics.