Projective Graphs Research Articles

Polarity-based graph neural network for sign prediction in signed bipartite graphs

As a fundamental data structure, graphs are ubiquitous in various applications. Among all types of graphs, signed bipartite graphs contain complex structures with positive and negative links as well as bipartite settings, on which conventional graph analysis algorithms are no longer applicable. Previous works mainly focus on unipartite signed graphs or unsigned bipartite graphs separately. Several models are proposed for applications on the signed bipartite graphs by utilizing the heuristic structural information. However, these methods have limited capability to fully capture the information hidden in such graphs. In this paper, we propose the first graph neural network on signed bipartite graphs, namely Polarity-based Graph Convolutional Network (PbGCN), for sign prediction task with the help of balance theory. We introduce the novel polarity attribute to signed bipartite graphs, based on which we construct one-mode projection graphs to allow the GNNs to aggregate information between the same type nodes. Extensive experiments on five datasets demonstrate the effectiveness of our proposed techniques.

Two strong 3-flow theorems for planar graphs

In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo $3$) between the inflow and outflow. They conjectured that all $5$-edge-connected graphs with a valid prescription function have a nowhere zero $3$-flow meeting that prescription (we call this the Strong $3$-Flow Conjecture). Kochol (2001) showed that replacing $4$-edge-connected with $5$-edge-connected would suffice to prove the $3$-Flow Conjecture and Lovasz et al.(2013) showed that the $3$-Flow and Strong $3$-Flow Conjectures hold if the edge connectivity condition is relaxed to $6$-edge-connected. Both problems are still open for $5$-edge-connected graphs. The $3$-Flow Conjecture was known to hold for planar graphs, as it is the dual of Grotzsch's Colouring Theorem. Steinberg and Younger (1989) provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al.(2016) provided the first direct proof using flows of the Strong $3$-Flow Conjecture for planar graphs. We provide two extensions to their result, that we developed in order to prove the Strong $3$-Flow Conjecture for projective planar graphs.

Covering projective planar graphs with three forests

It is known that all planar graphs and all projective planar graphs have an edge partition into three forests. Gonçalves proved that every planar graph has an edge partition into three forests, one having maximum degree at most four [5]. In this paper, we prove that every projective planar graph has an edge partition into three forests, one having maximum degree at most four.

A characterization of the Mathieu–Conway–Monster series of locally projective graphs

Let I=(V,L) be a vertex-line incidence system which is said to be thin or thick if every line contains two or three vertices, respectively. In either case any two lines can intersect in at most one vertex. Let Γ be the collinearity graph of I and let G be a group of automorphisms of I. Then Γ is said to be locally projective in dimension n with respect to G if (a) G is flag-transitive on I; (b) the stabilizer G(x) of a vertex x∈V in G induces on the set of lines containing x, the group Ln(2) for some n≥2 in its natural doubly transitive action of degree 2n−1. Thus a locally projective graph in dimension n has valency 2n−1 or 2(2n−1) depending on whether it is thin or thick. The G(x)-invariant structure of a projective GF(2)-space of linear dimension n on the set of lines containing x, will be denoted by Πx. In the thin case we further assume that (c) the action of G on Γ is 2- but not 3-arc transitive and (d) the action is of collineation type, meaning that for an edge {x,y} of Γ an element in G which swaps x and y can be chosen to centralize G(x)∩G(y) modulo O2(G(x)∩G(y)). The first examples of locally projective graphs come from affine (thin) and projective (thick) GF(2)-spaces. Less trivial classical examples are the dual polar GF(2)-spaces of orthogonal (thin) and symplectic (thick) types. Further examples coming from Petersen and Tilde geometries are associated with some sporadic simple groups. The fundamental problem is to establish an upper bound on |G(x)| in terms of the dimension n, by finding an absolute bound d such that the vertex-wise stabilizer in G of the ball of radius d in Γ centred at a vertex x is always trivial (independently of the graph): Gd(x)=1. In the classical examples G2(x)=1. The first examples of Petersen and Tilde geometries were constructed in the early 1980s to disprove the belief that the classical bound holds in general. In fact, for the vertex-transitive graph of the Petersen geometry of the Baby Monster group the smallest d with Gd(x)=1 is 5. In 1990s V.I. Trofimov in a brilliant series of papers proved that G6(x)=1 in the thin case without assuming (c) and (d). In 2002 it has been proved by S.V. Shpectorov and the author that in the thin case under consideration all the examples with the critical parameter d above the classical bound indeed come from Petersen geometries (one way or another). An ongoing project is to extend the classification of locally projective amalgams to the thick case, which is considerably more complicated. Until recently only a characterization of an important class of 3-dimensional locally projective graphs [3] was available. In a recent paper [6] the author has shown that if the standard geometrization of a thick locally projective graph contains planes locally isomorphic to the generalized quadrangle of order (2,2), then Γ contains a family of thin locally projective subgraphs which are densely embedded in Γ in the sense that the stabilizer in G(x) of such a subgraph containing x has index 2n. This result places in a general setting the famous embeddingsG(M22)↪G(M24),G(Co2)↪G(Co1),G(BM)↪G(M) among Petersen and Tilde geometries. In the present paper we characterize the graphs of the geometries on the right hand side of these embeddings in the class of thick locally projective graphs with planes of the above specified type and with critical parameter d above the classical bound.

Nowhere-zero 3-flows in toroidal graphs

Tutte's 3-flow conjecture states that every 4-edge-connected graph admits a nowhere-zero 3-flow. The planar case of Tutte's 3-flow conjecture is the classical Grötzsch's Theorem (1959). Steinberg and Younger (1989) further verified Tutte's 3-flow conjecture for projective planar graphs. In this paper we confirm Tutte's 3-flow conjecture for all toroidal graphs.

Semi-supervised label distribution learning via projection graph embedding

Label distribution learning (LDL) is a new machine learning paradigm that addresses label ambiguity by emphasizing the relevance of each label to a particular instance. In supervised learning, many LDL algorithms have been proposed, which often require a large amount of well-annotated training data to achieve good performance. However, annotating a label distribution is more complicated and expensive than annotating a single label or multiple labels with logical values of 0 and 1. Thus, we propose a projection graph embedding algorithm for semi-supervised label distribution learning (PGE-SLDL). Specifically, we seek a potential space by orthogonal neighborhood preserving projections, named capture space. This capture space is used to select more valuable features and construct a graph that contains more accurate data structure information. We utilize the sample correlation information contained between graph nodes to recover the unknown label distribution. In addition, compared with fixed graphs in traditional semi-supervised learning, we carry out projection and graph construction simultaneously to obtain a self-updating projection graph, which is more helpful to learn label distribution. The experimental results validate the effectiveness of the proposed algorithm.

The projective 3-annihilating-ideal hypergraphs

In this paper, we investigate the crosscap of 3-annihilating-ideal hypergraph [Formula: see text] of a commutative ring [Formula: see text] and the topological embedding of [Formula: see text] to the nonorientable compact surfaces. Furthermore, we determine all Artinian commutative non-local rings R (up to isomorphism) such that [Formula: see text] is a projective graph.

A General Block Stability Analysis Algorithm for Arbitrary Block Shapes

In rock engineering, block theory is a fundamental theory that aims to analyze the finiteness, removability, and mechanical stability of convex blocks under different engineering conditions. In practice, the possible combinations of the fractures and joint sets that may generate key blocks can be identified by stereographic projection graphs of block theory. However, classic key block theory does not provide solutions for nonconvex blocks, which are very common in civil projects, such as those with underground edges, corners, and portals. To enhance the availability of block theory, a general algorithm that can analyze the removability and stability of blocks of arbitrary shapes is proposed in this paper. In the proposed algorithm, the joint pyramid for blocks of arbitrary shapes can be computed, and the faces of the blocks are grouped according to their normal vectors such that parallel or nonadjacent sliding faces with the same normal vector can be immediately identified when the sliding mode is determined. With this algorithm, blocks of arbitrary shapes can be analyzed, and users do not need to have experience interpreting graphs of block theory to take advantage of its accuracy and effectiveness. The proposed algorithm was verified by several benchmarking examples, and it was further applied to investigate the stability of the left bank rock slope of a dam. The results showed that the proposed algorithm is correct, effective, and feasible for use in the design and support of excavation in complex rock masses.

Secure and smart autonomous multi-robot systems for opinion spammer detection

Reviews of social media have become very important reference indicators for people shopping. However, in order to attract more consumers, some bad merchants organize many fraudulent reviewers to conduct malicious reviews aim at misleading consumers, which is called opinion spammer group. Previous studies have shown that there is an implicit community between reviewers In this paper, we try to use community discovery approach in secure and smart autonomous multi-robot systems for opinion spammer detection, and improve the prediction precision. We propose a novel approach, named SPClique, which is based on the Clique Percolation Method (CPM) to detect opinion spammer groups by modeling the review dataset as reviewer projection graph. During the process, we introduce approximate computing to get a valuable smaller reviewer-projection graph and expand the computing power. It first executes the CPM algorithm in the reviewer-projection graph to find all k-clique clusters, and then innovatively describes an opinion spammer group based on k-clique clusters. Second, the group-based spam and individual-based spams indicators are used to measure the suspiciousness of each opinion spammer group. Finally, it outputs the suspect ranking of opinion spammer groups. The prediction precision of our proposed method is better than four advanced comparison methods, and our approach can detect more real opinion spammers in large-scale review datasets.

Directed Projection Graph of N-Dimensional Hypercube and Subhypercube Decomposition of Balanced Linearly Separable Boolean Functions

A directed projection graph of the [Formula: see text]-dimensional hypercube on the two-dimensional plane is successfully created. Any [Formula: see text]-variable Boolean function can be easily transformed to an induced subgraph of the projection. Therefore, the discussions on [Formula: see text]-variable Boolean functions only need to focus on a two-dimensional planar graph. Some mathematical theories on the projection graph and the induced subgraph are established, and some properties and characteristics of a balanced linearly separable Boolean function (BLSBF) are uncovered. In particular, the sub-hypercube decompositions of BLSBF is easily represented on the projection, and meanwhile, the enumeration scheme for counting the number of [Formula: see text]-variable BLSBFs is developed by using equivalence classification and conformal transformation. With the aid of the directed projection grap constructed in this paper, one can further study many difficult problems in some fields such as Boolean functions and artificial neural networks.

Research on Machine Vision Effect Based on Graph Neural Network Decision

With the continuous expansion of the scale of the network, modeling the complex graph structure is a major challenge for the recommendation task. There are conflicts between these complex information, which will directly affect the recommendation results. For this reason, a graph neural network recommendation method based on multi-branch decision is proposed. The algorithm models the social network graph and the user project graph through the neural network, connects the two graphs intrinsically, and learns the feature vectors of the target users in the social space and the project space. Then two feature vectors are connected in series by MLP to extract the potential feature vector of the user. Finally, the prediction score is generated by integrating the probability matrix decomposition model. In this method, the graph neural network is used to model the program data and dependence graph, and the effective program features are automatically extracted from the source code, and then the extracted features are input into the downstream model for loop vectorization parameter prediction. Finally, the attention mechanism is used for information fusion to get the final session representation and predict the next interactive item. Comparative experiments are carried out under the two scenarios of e-commerce and civil aviation respectively. The experimental results show that, compared with the optimal benchmark model, the improvement of MGSP model on each index of e-commerce data set is more than 1%, and that of civil aviation data set is about 3%, which verifies the effectiveness of MGSP model. On the LLVM cyclic vector test set, the proposed method achieves a speedup of 2.08 times and improves the performance by 12% compared with the existing methods. In order to solve the problem that the existing deep learning model considers the program code as a serial sequence and misses a large performance optimization space, a new program heuristic method based on depth map network is proposed to achieve optimization.

Data Visualization to solve COVID-19

This electronic document describes how data visualization can be used to solve problems related to Coronavirus using graphical representation of information and data. This article mentions several well-known methods to contain the coronavirus pandemic and discusses how these methods can be effectively implemented across the country. The focus has been given more to the implementation in local level than a country wide implementation. The article describes a comprehensive way to predict the next hotspot in a district wise level using projection graphs. The article also describes The Delhi model using data projection and analysis. The conclusions derived from theses analyses can be implemented effectively in district wise level to flatten the Coronavirus curve and develop herd immunity in category wise zones.

On Distance in Some Finite Planes and Graphs Arising from Those Planes

In this paper, affine and projective graphs are obtained from affine and projective planes of order p r by accepting a line as a path. Some properties of these affine and projective graphs are investigated. Moreover, a definition of distance is given in the affine and projective planes of order p r and, with the help of this distance definition, the point or points having the most advantageous (central) position in the corresponding graphs are determined, with some examples being given. In addition, the concepts of a circle, ellipse, hyperbola, and parabola, which are well known for the Euclidean plane, are carried over to these finite planes. Finally, the roles of finite affine and projective Klingenberg planes in all the results obtained are considered and their equivalences in graph applications are discussed.

Graph hypersurfaces with torus action and a conjecture of Aluffi

Generalizing the star graphs of M\"uller-Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable graph. This disproves a conjecture of Aluffi in a strong sense.

Locally projective graphs and their densely embedded subgraphs

The article contributes to the classification project of locally projective graphs and their locally projective groups of automorphisms outlined in Chapter 10 of Ivanov (The Mathieu Groups, Cambridge University Press, Cambridge, 2018). We prove that a simply connected locally projective graph Gamma of type (n, 3) for n ge 3 contains a densely embedded subtree provided (a) it contains a (simply connected) geometric subgraph at level 2 whose stabiliser acts on this subgraph as the universal completion of the Goldschmidt amalgam G_3^1cong {S_4 times 2,S_4 times 2} having S_6 as another completion, (b) for a vertex x of Gamma the group G_{frac{1}{2}}(x) which stabilizes every line passing through x induces on the neighbourhood Gamma (x) of x the (dual) natural module 2^n of G(x)/G_{frac{1}{2}}(x) cong L_n(2), (c) G(x) splits over G_{frac{1}{2}}(x), (d) the vertex-wise stabilizer G_1(x) of the neighbourhood of x is a non-trivial group, and (e) n ne 4.

COGNITIVE MODELING OF KNOWLEDGE MANAGEMENT MECHANISMS IN THE TRAINING OF SPECIALISTS

The subject of research in this article is management in the training of specialists in project-oriented organizations, in particular, higher education institutions. The purpose of the work is to develop a cognitive model of management for the formation of educational programs in the training system. Objectives of the study: to consider the concepts and mechanisms of management of organizations; to form a cognitive model of the management mechanism; to consider system indicators of cognitive model; to analyze the structural features of the cognitive model. The following methods are used in the work: project management methodology (knowledge management areas), system approach, cognitive and graph models. Results. The article provides a definition of knowledge management, discusses the conditions under which the necessary and information will be available to perform tasks. The issue of consideration the specifics and features of the educational institution is discussed. Knowledge management methods are used in the construction of the educational process at the university. The main functions in the practical activities of the organization are given: mediation, export, internalization, cognition. Forms of practical implementation of management are also considered: educational organization, library and strategic awareness. The article also considers the possibility of the main provisions of the theory of management in the educational process, as improving the quality of training is the goal of any educational system. The article develops a cognitive model of the management mechanism, on the basis of cognitive analysis revealed useful patterns that reflect the influence of factors of different physical nature on the results of training. Conclusions. Cognitive modeling can be the key to rapidly improving the effectiveness of management in training. The classical approach to management requires a high level of employee involvement in relevant processes, as they need to identify knowledge, share it, record it and apply it. Cognitive modeling can reduce this requirement as it streamlines the processes of identifying, capturing, storing, and sharing knowledge. In addition, they increase the applicability of by offering appropriate context for the student. The accumulated experience of cognitive modeling of complex organizational management systems allows on a fundamental basis to approach the study of management mechanisms in the training of specialists. It is proposed to use the obtained results to configure management mechanisms in order to improve the quality of training.

On Efficient Adjustment in Causal Graphs

We consider estimation of a total causal effect from observational data via covariate adjustment. Ideally, adjustment sets are selected based on a given causal graph, reflecting knowledge of the underlying causal structure. Valid adjustment sets are, however, not unique. Recent research has introduced a graphical criterion for an 'optimal' valid adjustment set (O-set). For a given graph, adjustment by the O-set yields the smallest asymptotic variance compared to other adjustment sets in certain parametric and non-parametric models. In this paper, we provide three new results on the O-set. First, we give a novel, more intuitive graphical characterisation: We show that the O-set is the parent set of the outcome node(s) in a suitable latent projection graph, which we call the forbidden projection. An important property is that the forbidden projection preserves all information relevant to total causal effect estimation via covariate adjustment, making it a useful methodological tool in its own right. Second, we extend the existing IDA algorithm to use the O-set, and argue that the algorithm remains semi-local. This is implemented in the R-package pcalg. Third, we present assumptions under which the O-set can be viewed as the target set of popular non-graphical variable selection algorithms such as stepwise backward selection.

An Arabic Grammar Auditor Based on Dependency Grammar

The Arabic language has many complex grammar rules that may seem complicated to the average user or learner. Automatic grammar checking systems can improve the quality of the text, reduce the costs of the proofreading process, and play a role in grammar teaching. This paper presents an initiative toward developing a novel and comprehensive Arabic auditor that can address vowelized texts. We called the “Arabic Grammar Detector” (AGD-أَجِــدْ). AGD was successfully implemented based on a dependency grammar and decision tree classifier model. Its purpose is to extract patterns of grammatical rules from a projective dependency graph in order to designate the appropriate syntax dependencies of a sentence. The current implementation covers almost all regular Arabic grammar rules for nonvowelized texts as well as partially or fully vowelized texts. AGD was evaluated using the Tashkeela corpus. It can detect more than 94% of grammatical errors and hint at their causes and possible corrections.

Camshift tracking method based on correlation probability graph for model pig

The identification and tracking for model pigs, as a vital research content for studying the habits of model pigs, drawed more and more considerable attention. To fulfill people requirements for the effectiveness of the non-significant model pig tracking in breeding environment, a Camshift tracking approach based on correlation probability graph, i.e., CamTracor−PG, is proposed in this paper, in which the correlation probability graph is introduced to achieve target positioning and tracking. Technically, acquiring images through a vision sensor, according to the circular arrangement of pixels in the inverse probability projection graph, and multiplying the inverse projection probability value of a pixel by its surrounding pixels could obtain the weighted sum. Then, the target projection grayscale graph is established by utilizing the correlation probability value for positioning, identification, and tracking of model pigs. Finally, extensive experiments are conducted to validate reliability and efficiency of our approach.

Edge proximity and matching extension in projective planar graphs

AbstractA graph with at least vertices is said to be distance ‐extendable if, for any matching of with edges in which the edges lie at distance at least pairwise, there exists a perfect matching of containing . In this paper we prove that every 5‐connected triangulation on the projective plane of even order is distance 3 7‐extendable and distance 4 ‐extendable for any .