Orthogonal Graph Research Articles

Power System Connectivity Visualization Using an Orthogonal Graph Layout Algorithm Based on the Space-Filling Technique

In this study, a novel orthogonal graph layout algorithm was proposed to efficiently represent the connection relationship between each node in power system data. The proposed layout algorithm is a four-stage algorithm. First, clustering was performed based on the connection relationship between nodes, and the obtained clusters were placed close to squares by using the squarified treemap technique to fully utilize the given space. Next, adjacent nodes were arranged in a snakelike order in each cluster according to the characteristics of the IEEE test system data. The links were then arranged according to the positional relationship of the pairs of connected nodes. Each node had several ports so that links could be distributed evenly according to the direction of the links. A case in which all nodes were arranged orthogonally in an arbitrary manner without performing clustering; a case in which adjacent nodes within each cluster were arranged in arbitrary order after only performing clustering; and a case where adjacent nodes within each cluster were arranged in row-major order after only performing clustering were compared. The results revealed that the proposed method considerably improved edge crossing, edge bending, and edge length.

Multi-view clustering with orthogonal mapping and binary graph

The multi-view clustering problem has attracted considerable attention over recent years for the remarkable clustering performance due to exploiting complementary information from multiple views. Most existing related research work processes data in the decimal real value space that is not the most compatible space for computers. Binary code learning, also known as hashing technology, is well-known for fast Hamming distance computation, less storage requirement and accurate calculation results. The Hamming space is most enjoyed by computers because of binary/hash codes. Several studies combine multi-view clustering with binary code learning for improving clustering performance. However, there is much redundant information contained in the learned binary codes, which negatively affects the clustering performance, but these studies ignore eliminating redundant information for learning compact codes. In addition, they don’t give a unified (one-step) clustering framework with binary graph structure, which doesn’t lead to the optimal clustering result due to the information loss during the two-step process. In this paper, to cope with the two issues, we propose an orthogonal mapping binary graph method (OMBG) for the multi-view clustering problem, which makes the mapping matrix of every view orthogonalize for eliminating redundant information and embeds a binary graph structure into the unified binary multi-view clustering framework for extracting local geometric structure information of binary codes and achieving the optimal clustering result. Furthermore, we design an effective optimization algorithm based on alternating direction minimization to solve the model of OMBG. Extensive experiments performed on four frequently-used benchmark multi-view datasets illustrate the superiority of OMBG which is compared with some state-of-the-art clustering baselines.

Orthogonal Graph Neural Networks

Graph neural networks (GNNs) have received tremendous attention due to their superiority in learning node representations. These models rely on message passing and feature transformation functions to encode the structural and feature information from neighbors. However, stacking more convolutional layers significantly decreases the performance of GNNs. Most recent studies attribute this limitation to the over-smoothing issue, where node embeddings converge to indistinguishable vectors. Through a number of experimental observations, we argue that the main factor degrading the performance is the unstable forward normalization and backward gradient resulted from the improper design of the feature transformation, especially for shallow GNNs where the over-smoothing has not happened. Therefore, we propose a novel orthogonal feature transformation, named Ortho-GConv, which could generally augment the existing GNN backbones to stabilize the model training and improve the model's generalization performance. Specifically, we maintain the orthogonality of the feature transformation comprehensively from three perspectives, namely hybrid weight initialization, orthogonal transformation, and orthogonal regularization. By equipping the existing GNNs (e.g. GCN, JKNet, GCNII) with Ortho-GConv, we demonstrate the generality of the orthogonal feature transformation to enable stable training, and show its effectiveness for node and graph classification tasks.

On the Generalized $\vartheta$-Number and Related Problems for Highly Symmetric Graphs

This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound for both the $k$-multichromatic number of a graph and the maximum $k$-colorable subgraph problem. We present various properties of $\vartheta_k(G)$, such as that the sequence $(\vartheta_k(G))_k$ is increasing and bounded from above by the order of the graph $G$. We study $\vartheta_k(G)$ when $G$ is the strong, disjunction, or Cartesian product of two graphs. We provide closed form expressions for the generalized $\vartheta$-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs, and orthogonality graphs. Our paper provides bounds on the product and sum of the $k$-multichromatic number of a graph and its complement graph, as well as lower bounds for the $k$-multichromatic number on several graph classes including the Hamming and Johnson graphs.

On Graph-Transversal Designs and Graph-Authentication Codes Based on Mutually Orthogonal Graph Squares

Combinatorial designs have many interesting and genuine wide applications in areas including analysis and design of algorithms, cryptography, analysis and design of experiments, storage system design, tournament scheduling, optical communications, and computer networks to mention just a few areas. In this paper, we are concerned with the transversal designs and authentication codes as direct applications of combinatorial designs. The novelty of the current paper is demonstrated by the fact that it is the first to introduce the transversal designs and authentication codes by the mutually orthogonal graph squares (MOGS); we call them graph-transversal designs and graph-authentication codes, respectively. Here, the major contributions are the constructions of graph-transversal designs and graph-authentication codes based on several classes of graphs. Also, we present several results such as path-transversal designs, cycle-transversal designs, and disjoint unions of stars-transversal designs.

The automorphism group and fixing number of the orthogonality graph of the full matrix ring

Let [Formula: see text], [Formula: see text] be the set of all [Formula: see text] matrices over a finite field [Formula: see text], and [Formula: see text] the subset of [Formula: see text] consisting of all rank one matrices. In this paper, we first determine the automorphism group and the fixing number of the orthogonality graph of [Formula: see text], and then characterize the automorphism group and the fixing number of the orthogonality graph of [Formula: see text].

Synthesis of combinational circuits by means of bi-decomposition of Boolean functions

O b j e c t i v e s . The problem of synthesis of combinational circuits in the basis of two-input gates is considered. Those gates are AND, OR, NAND and NOR. The objective of the paper is to investigate the possibilities of application of bi-decomposition of Boolean functions to the synthesis of combinational circuits.M e t h o d s . The method for bi-decomposition is reduced to the search in a graph for a weighted two-block cover with complete bipartite subgraphs (bi-cliques).R e s u l t s . The initial Boolean function is given as two ternary matrices, one of which represents the domain of Boolean space where the function has the value 1, and the other is the domain of Boolean space where the function has the value 0. The orthogonality graph of rows of ternary matrices representing the given function is considered. The method for two-bi-clique covering the orthogonality graph of rows of ternary matrices is described. Every bi-clique in the obtained cover is assigned in a certain way with а set of variables that are the arguments of the function. This set is the weight of the bi-clique. Each of those bi-cliques defines a Boolean function whose arguments are the variables assigned to it. The functions obtained in such a way constitute the required decomposition.Co n c l u s i o n . The process of synthesis of a combinational circuit consists of a successive application of bi-decomposition to obtained functions. The suggested method allows obtaining the circuits with short delay.

Multi-View Clustering with Orthogonal Mapping and Binary Graph

Multi-View Clustering with Orthogonal Mapping and Binary Graph

A novel application on mutually orthogonal graph squares and graph-orthogonal arrays

<abstract> <p>Security of personal information has become a major concern due to the increasing use of the Internet by individuals in the digital world. The main purpose here is to prevent an unauthorized person from gaining access to confidential information. The solution to such a problem is by authentication of users. Authentication has a very important role in achieving security. Mutually orthogonal graph squares (MOGS) are considered the generalization of mutually orthogonal Latin squares (MOLS). Also, MOGS are generated from edge decompositions of complete bipartite graphs by isomorphic graphs. Graph-orthogonal arrays can be constructed by MOGS. In this paper, graph-orthogonal arrays are used for constructing authentication codes. These arrays are the encoding matrices of authentication tags. We introduce the concepts and basic theorems of MOGS, graph-orthogonal arrays, and authentication codes. After constructing graph-orthogonal arrays by MOGS, then there is an established mapping between graph-orthogonal arrays and message set. This manages us to construct perfect non-splitting and splitting Cartesian authentication codes. In both cases, we calculate the probabilities of successful impersonation attacks and substitution attacks. Besides that, the performance of constructed non-splitting and splitting authentication codes is analyzed. In the end, optimal authentication codes and secure authentication codes are constructed.</p> </abstract>

Construction of Mutually Orthogonal Graph Squares Using Novel Product Techniques

Sets of mutually orthogonal Latin squares prescribe the order in which to apply different treatments in designing an experiment to permit effective statistical analysis of results, they encode the incidence structure of finite geometries, they encapsulate the structure of finite groups and more general algebraic objects known as quasigroups, and they produce optimal density error‐correcting codes. This paper gives some new results on mutually orthogonal graph squares. Mutually orthogonal graph squares generalize orthogonal Latin squares interestingly. Mutually orthogonal graph squares are an area of combinatorial design theory that has many applications in optical communications, wireless communications, cryptography, storage system design, algorithm design and analysis, and communication protocols, to mention just a few areas. In this paper, novel product techniques of mutually orthogonal graph squares are considered. Proposed product techniques are the half‐starters’ vectors Cartesian product, half‐starters’ function product, and tensor product of graphs. It is shown that by taking mutually orthogonal subgraphs of complete bipartite graphs, one can obtain enough mutually orthogonal subgraphs in some larger complete bipartite graphs. Also, we try to find the minimum number of mutually orthogonal subgraphs for certain graphs based on the proposed product techniques. As a direct application to the proposed different product techniques, mutually orthogonal graph squares for disjoint unions of stars are constructed. All the constructed results in this paper can be used to generate new graph‐orthogonal arrays and new authentication codes.

Statistical Multimode Accounting in the Problem of Optimal Reactive Load Compensation when Constructing Smart Grids

The article deals with the development prospects for smart grids. An approach to optimizing the modes of electric power systems in terms of reactive power is presented in detail. The solution of this problem considers the multimodality, determination of the integral characteristics of the modes, etc. Analytical modeling of load changes by means of the method of factor analysis allows to reduce the amount of information drastically without a significant loss of the accuracy of the obtained solutions. For this purpose, the actually correlated electrical loads of various nodes of electric power systems are represented in the form of a linear combination of independent random variables, namely, generalized load graphs. It is shown that the inclusion of multimode by orthogonal graphs results in a significant simplification of the solution of multimode problems. The choice of the dependent and independent variables composition when solving the optimization problem with the power consumption modes has a fundamental effect both on the modeling of constraints, the formation of the reduced gradient and the main calculation expressions, and on the speed of the optimization search as a whole

On automorphism group of orthogonality graph of finite semisimple rings

The orthogonality graph of a ring R, written as O(R), is an undirected simple graph with vertex set the set of all nonzero two-sides zero-divisors of R, and for distinct there exists an edge joining x and y if and only if In this paper, we describe the automorphism group of O(R) when R is a finite semisimple ring.

Space Layout Method for Carton Conveying System Based on SOSG Matrix

The traditional cigarette conveying system of cigarette industry adopts a one-to-one hard connection method, which has problems such as poor spatial scalability and high equipment coupling. A spatial layout method for carton conveying system based on the SOSG (Solid Orthogonal Structure Graph) matrix is proposed. Through this method, a matrix cigarette conveying system is constructed. Compared with the traditional mode, its space utilization rate is increased by 67%, equipment coupling is reduced, and the downtime rate of carton conveying line due to the shutdown of sealing machine can be reduced from 100% to 0%.

The automorphism group and fixing number of orthogonality graph over a vector space

Let [Formula: see text] be a field, and [Formula: see text] the [Formula: see text]-dimensional row vector space over [Formula: see text]. The orthogonality graph [Formula: see text] of [Formula: see text] is an undirected simple graph which has [Formula: see text] as its vertex set, and for distinct [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] is the transpose of [Formula: see text]. When [Formula: see text] is finite, it is shown that any automorphism of [Formula: see text] can be decomposed into the product of a row-orthogonal automorphism and either a permutation automorphism or a field automorphism; moreover, the fixing number and metric dimension of [Formula: see text] are considered.

Distinguishing orthogonality graphs

AbstractA graph is said to be ‐distinguishable if there is a labeling of the vertices with labels so that only the trivial automorphism preserves the labels. The smallest such is the distinguishing number, . A set of vertices is a determining set for if every automorphism of is uniquely determined by its action on . The size of a smallest determining set for is called the determining number, . The orthogonality graph has vertices which are bitstrings of length with an edge between two vertices if they differ in precisely bits. This paper shows that and that, if when is odd or when is even, then .

Orthogonality graphs of real Cayley–Dickson algebras. Part II: The subgraph on pairs of basis elements

We consider zero divisors of an arbitrary real Cayley–Dickson algebra such that their components are both standard basis elements. We construct inductively the orthogonality graph on these elements. Then we show that, if we restrict our attention to at least [Formula: see text]-dimensional algebras, two algebras are isomorphic if and only if their graphs are isomorphic. We also provide an algorithm to retrieve the Cayley–Dickson parameters of an algebra from its graph.

Orthogonality graphs of real Cayley–Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons

We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley–Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the multiplication table of the vertices of a double hexagon. Then we find a sufficient condition for three elements to generate an alternative subalgebra of an arbitrary Cayley–Dickson algebra. Finally, we consider those zero divisors whose components are both standard basis elements up to sign. We classify them and determine necessary and sufficient conditions under which two such elements are orthogonal.

Generalization of MacNeish’s Kronecker product theorem of mutually orthogonal Latin squares

The subject of mutually orthogonal Latin squares (MOLSs) has fascinated researchers for many years. Although there is a number of intriguing results in this area, many open problems remain to which the answers seem as elusive as ever. Mutually orthogonal graph squares (MOGSs) are considered a generalization to MOLS. MOLS are considered an area of combinatorial design theory which has many applications in optical communications, cryptography, storage system design, wireless communications, communication protocols, and algorithm design and analysis, to mention just a few areas. In this paper, we introduce a technique for constructing the mutually orthogonal disjoint union of graphs squares and the generalization of the Kronecker product of MOGS as a generalization to the MacNeish’s Kronecker product of MOLS. These are useful for constructing many new results concerned with the MOGS.

Orthogonal Dual Graph-Regularized Nonnegative Matrix Factorization for Co-Clustering

Nonnegative Matrix Factorization (NMF) has received great attention in the era of big data, owing to its roles in efficiently reducing data dimension and producing feature-based data representation. In this paper, we first propose two new NMF optimization models, called an orthogonal dual graph regularized nonnegative matrix factorization (ODGNMF) method and its modified version: an orthogonal dual graph regularized nonnegative matrix tri-factorization (ODGNMTF) method. Compared with the existing models, our models can preserve the geometrical structures of data manifold and feature manifold by constructing two graphs, and ensure the orthogonality of factor matrices such that they have better NMF performance. Then, two efficient algorithms are developed to solve the models, and the convergence theory of the algorithms is established. Numerical tests by applying our algorithms to mine randomly generated data sets and well-known public databases demonstrate that ODGNMF and ODGNMTF have better numerical performance than the state-of-the-art algorithms in view of computational cost, robustness, sensitivity and sparseness.

On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

During the last two centuries, after the question asked by Euler concerning mutually orthogonal Latin squares (MOLS), essential advances have been made. MOLS are considered as a construction tool for orthogonal arrays. Although Latin squares have numerous helpful properties, for some factual applications these structures are excessively prohibitive. The more general concepts of graph squares and mutually orthogonal graph squares (MOGS) offer more flexibility. MOGS generalize MOLS in an interesting way. As such, the topic is attractive. Orthogonal arrays are essential in statistics and are related to finite fields, geometry, combinatorics and error-correcting codes. Furthermore, they are used in cryptography and computer science. In this paper, our current efforts have concentrated on the definition of the graph-orthogonal arrays and on proving that if there are k MOGS of order n, then there is a graph-orthogonal array, and we denote this array by G-OA(n2,k,n,2). In addition, several new results for the orthogonal arrays obtained from the MOGS are given. Furthermore, we introduce a recursive construction method for constructing the graph-orthogonal arrays.