Isomorphism Detection Research Articles

Structural Synthesis of Planar Kinematic Chains and Their Detection of Isomorphism

Structural Synthesis of Planar Kinematic Chains and Their Detection of Isomorphism

Detection of Isomorphism of Kinematic Chains Using two DOF Using CSJV Matrix

Detection of Isomorphism of Kinematic Chains Using two DOF Using CSJV Matrix

Beyond Weisfeiler–Lehman with Local Ego-Network Encodings

Identifying similar network structures is key to capturing graph isomorphisms and learning representations that exploit structural information encoded in graph data. This work shows that ego networks can produce a structural encoding scheme for arbitrary graphs with greater expressivity than the Weisfeiler–Lehman (1-WL) test. We introduce IGEL, a preprocessing step to produce features that augment node representations by encoding ego networks into sparse vectors that enrich message passing (MP) graph neural networks (GNNs) beyond 1-WL expressivity. We formally describe the relation between IGEL and 1-WL, and characterize its expressive power and limitations. Experiments show that IGEL matches the empirical expressivity of state-of-the-art methods on isomorphism detection while improving performance on nine GNN architectures and six graph machine learning tasks.

A novel compound topological invariant for isomorphism detection of planar kinematic chains

Abstract. The isomorphism identification of the kinematic chain (KC) based on graph theory definition has no advantage in efficiency, especially when the number of links in the KCs is large. The topological characteristic constants for isomorphism identification, such as the power value sequence (PVS), least distance matrix sequence (LDMS), and loop number (LN), are proposed. The fourth PVS, the LDMS, and the LN are compared and arranged in descending order, to form a strong complementary chain, which can identify KCs of at least 15 links with 4 degrees of freedom (DOF). The method is applied to the complete atlas of the following: 8-link 1 DOF, 9-link 2 DOF, 10-link 1 DOF, 12-link 1 DOF, 13-link 2 DOF, and 15-link 4 DOF planar single-joint KCs, 6-link 1 DOF and 7-link 1 DOF planetary gear trains, 8-link 1 DOF planar multiple joint KCs, and contracted graphs with up to six independent loops. All results are in agreement with the reported ones in the literature. Thus, the proposed method possesses good versatility and has been verified as being reliable and efficient.

Structural synthesis of plane kinematic chain inversions without detecting isomorphism

Abstract. A plane kinematic chain inversion refers to a plane kinematic chain with one link fixed (assigned as the ground link). In the creative design of mechanisms, it is important to select proper ground links. The structural synthesis of plane kinematic chain inversions is helpful for improving the efficiency of mechanism design. However, the existing structural synthesis methods involve isomorphism detection, which is cumbersome. This paper proposes a simple and efficient structural synthesis method for plane kinematic chain inversions without detecting isomorphism. The fifth power of the adjacency matrix is applied to recognize similar vertices, and non-isomorphic kinematic chain inversions are directly derived according to non-similar vertices. This method is used to automatically synthesize 6-link 1-degree-of-freedom (DOF), 8-link 1-DOF, 8-link 3-DOF, 9-link 2-DOF, 9-link 4-DOF, 10-link 1-DOF, 10-link 3-DOF and 10-link 5-DOF plane kinematic chain inversions. All the synthesis results are consistent with those reported in literature. Our method is also suitable for other kinds of kinematic chains.

Hierarchical Harmonization of Atom-Resolved Metabolic Reactions across Metabolic Databases.

Metabolic models have been proven to be useful tools in system biology and have been successfully applied to various research fields in a wide range of organisms. A relatively complete metabolic network is a prerequisite for deriving reliable metabolic models. The first step in constructing metabolic network is to harmonize compounds and reactions across different metabolic databases. However, effectively integrating data from various sources still remains a big challenge. Incomplete and inconsistent atomistic details in compound representations across databases is a very important limiting factor. Here, we optimized a subgraph isomorphism detection algorithm to validate generic compound pairs. Moreover, we defined a set of harmonization relationship types between compounds to deal with inconsistent chemical details while successfully capturing atom-level characteristics, enabling a more complete enabling compound harmonization across metabolic databases. In total, 15,704 compound pairs across KEGG (Kyoto Encyclopedia of Genes and Genomes) and MetaCyc databases were detected. Furthermore, utilizing the classification of compound pairs and EC (Enzyme Commission) numbers of reactions, we established hierarchical relationships between metabolic reactions, enabling the harmonization of 3856 reaction pairs. In addition, we created and used atom-specific identifiers to evaluate the consistency of atom mappings within and between harmonized reactions, detecting some consistency issues between the reaction and compound descriptions in these metabolic databases.

A New Algorithm for Unique Representation and Isomorphism Detection of Planar Kinematic Chains with Simple and Multiple Joints

In this work, a new algorithm is proposed for a unique representation for simple and multiple joint planar kinematic chains (KCs) having any degree of freedom (DOF). This unique representation of KCs enhances the isomorphism detection during the structural synthesis process of KCs. First, a new concept of joint degree is generated for all joints of a given KC based on joint configuration. Then, a unified loop array (ULA) is obtained for each independent loop. Finally, a unified chain matrix (UCM) is established as a unique representation for a KC. Three examples are presented to illustrate the proposed algorithm procedures and to test its validity. The algorithm is applied to get a UCM for planar KCs having 7–10 links. As a result, a complete atlas database is introduced for 7–10-link non-isomorphic KCs with simple or/and multiple joints and their corresponding unified chain matrix.

A new method for isomorphism identification of planetary gear trains

Abstract. Planetary gear trains (PGTs) are widely used in machinery such as vehicles, pulley blocks, wrist watches, machine tools, and robots. During the process of structural synthesis of PGTs using graph theory, isomorphism identification of graphs is an important and complicated problem. The reliability of the isomorphism detection method directly determines the accuracy of the synthesis result. In this paper, a novel isomorphism identification method for PGTs is proposed. First, a new weighted adjacent matrix is presented to describe the topological graph of PGTs, which has is unique in describing the structure of PGTs. Then, the weighted distance matrix is proposed and the sum of the matrix is obtained, which can determine whether the planetary gear trains is isomorphic or not. Eventually, the examples demonstrate that this new method can be accurately and effectively performed.

Similar Vertices and Isomorphism Detection for Planar Kinematic Chains Based on Ameliorated Multi-Order Adjacent Vertex Assignment Sequence

Isomorphism detection is fundamental to the synthesis and innovative design of kinematic chains (KCs). The detection can be performed accurately by using the similarity of KCs. However, there are very few works on isomorphism detection based on the properties of similar vertices. In this paper, an ameliorated multi-order adjacent vertex assignment sequence (AMAVS) method is proposed to seek out similar vertices and identify the isomorphism of the planar KCs. First, the specific definition of AMAVS is described. Through the calculation of the AMAVS, the adjacent vertex value sequence reflecting the uniqueness of the topology features is established. Based on the value sequence, all possible similar vertices, corresponding relations, and isomorphism discrimination can be realized. By checking the topological graph of KCs with a different number of links, the effectiveness and efficiency of the proposed method are verified. Finally, the method is employed to implement the similar vertices and isomorphism detection of all the 9-link 2-DOF(degree of freedom) planar KCs.

Event detection of different English data sources based on transfer learning

The methods of multi-modal English event detection under a single data source and isomorphic event detection of different English data sources based on transfer learning still need to be improved. In order to improve the efficiency of English and data source time detection, based on the transfer learning algorithm, this paper proposes multi-modal event detection under a single data source and isomorphic event detection based on transfer learning for different data sources. Moreover, by stacking multiple classification models, this paper makes each feature merge with each other, and conducts confrontation training through the difference between the two classifiers to further make the distribution of different source data similar. In addition, in order to verify the algorithm proposed in this paper, a multi-source English event detection data set is collected through a data collection method. Finally, this paper uses the data set to verify the method proposed in this paper and compare it with the current most mainstream transfer learning methods. Through experimental analysis, convergence analysis, visual analysis and parameter evaluation, the effectiveness of the algorithm proposed in this paper is demonstrated.

Computer-aided detection of isomorphism among planar kinematic chains using regression loop-based method

The isomorphism identification among planar kinematic chains is an essential step in the structural synthesis. This article presents some new concepts, such as the regression loop and the regression loop set. The regression loop is a combination of independent loops. According to the mathematical operation “⊗,” the regression loop set is considered to be the unique representation from kinematic chain to isomorphism identification. And the sequence and direction of labeled links have no effect on the results of isomorphism identification. Based on the inherent relation among links and joints, this article proposed a regression loop-based method to identify isomorphism among planar kinematic chains. Some classic examples are illustrated to confirm the effectiveness of the proposed method. Compared with other classic methods, the proposed method is available and there is no counter-example. It enriches the loop theory for isomorphism identification and lays a solid foundation in the structural synthesis.

Reconfigurable 3D CAD Feature Recognition Supporting Confluent n-Dimensional Topologies and Geometric Filters for Prismatic and Curved Models

Feature Recognition (FR) in Computer-aided Design (CAD) models is central for Design and Manufacturing. FR is a problem whose computational burden is intractable (NP-hard), given that its underlying task is the detection of graph isomorphism. Until now, compromises have been reached by only using FACE-based geometric information of prismatic CAD models to prune the search domain. Responding to such shortcomings, this manuscript presents an interactive FR method that more aggressively prunes the search space with reconfigurable geometric tests. Unlike previous approaches, our reconfigurable FR addresses curved EDGEs and FACEs. This reconfigurable approach allows enforcing arbitrary confluent topologic and geometric filters, thus handling an expanded scope. The test sequence is itself a graph (i.e., not a linear or total-order sequence). Unlike the existing methods that are FACE-based, the present one permits combinations of topologies whose dimensions are two (SHELL or FACE), one (LOOP or EDGE), or 0 (VERTEX). This system has been implemented in an industrial environment, using icon graphs for the interactive rule configuration. The industrial instancing allows industry based customization and itis faster when compared to topology-based feature recognition. Future work is required in improving the robustness of search conditions, treating the problem of interacting or nested features, and improving the graphic input interface.

Efficient and Exact Multigraph Matching Search

A multigraph is modeled as a bag of graphs. Exact multigraph matching search aims to find all multigraphs that are the same as the query multigraphs from the data multigraph datasets. To the best of our knowledge, works regarding exact multigraph matching search have not been reported although they have a very wide range of application scenarios. In this article, we propose an efficient algorithm to solve the problem of exact multigraph matching search. We first propose a definition of exact multigraph matching and its Basic Method (BM), called BM, which has a considerable amount of graph isomorphism detection calculations and, thus, has very high computational complexity. Obviously, it is impractical to compare the query multigraph to each data multigraph in the multigraph datasets. To reduce the search space, multiple filtering conditions are proposed to obtain a candidate result set containing all the final results, including the cardinality filter, the vertex filter, the edge filter, the size filter, and the star filter. Then, each multigraph in the candidate result set is verified with the Improved BM (IBM) algorithm. Moreover, an offline and Multilayer Inverted Index (MII), named MII, is proposed to further accelerate the search process. Finally, we propose an Exact Multigraph Matching Search (EMMS) algorithm, based on the abovementioned technologies. We also analyze its time complexity. Extensive experiments on real datasets demonstrate the effectiveness and efficiency of the proposed algorithms.

Comparison between hamming method and modified path matrix approach to identify isomorphism in PGTs

Structural Synthesis and Analysis plays a major role in identifying the Isomorphism and Symmetry of a Planetary Gear Train (PGT). In synthesis process, detection of isomorphism in PGTs with given number of links and Degree of Freedom (DOF) is essential. One gear pair and one turning pair will be added to the existing link to generate next level PGT. One isomorphic PGT is selected at random to generate higher link PGTs. Graph theory and matrix methods are used by different kinematicians to identify the symmetry and isomorphism in PGTs. Every method has its own merits and demerits. In this paper a comparison was made between the Hamming number technique and Modified Path Matrix (MPM) approach. For easy analysis, PGTs are converted in to graphs and matrices are extracted from these graphs. In this paper isomorphism is checked by using Adjacency matrix and Modified path Matrix. 4-link and 6-link PGTs were considered for analysis in both the methods and compared the results. It is observed that Hamming Number approach is simple and feasible method among the two to identify the symmetry and isomorphism in PGTs.

An Application of Modified Path Matrix Approach for Detection of Isomorphism Among Epicyclic Gear Trains

The identification of isomorphism in epicyclic gear trains has been found a lot of attention by researchers for the last few years. Various methods have been suggested by different authors for the detection of isomorphism in planer kinematic chains and epicyclic gear trains (EGTs), but everyone has found some difficulties to address new issues. In this paper, a modified path matrix approach was presented in order to compare all the distinct geared kinematic mechanisms. A new method based on the matrix approach and corresponding train values is required to identify isomorphism among epicyclic gear trains and their mechanisms. The proposed method was examined on the basis of various examples from four-link, five-link, six-link, and eight-link one-degree-of-freedom EGTs and six-link two-degree-of-freedom EGTs. All the examples have been found satisfactory results with existing literature.

A distinct matrix representation of the planar kinematic chains and isomorphism recognition

Structural synthesis of kinematic chains has been an indispensable area of the mechanism-design problem. The duplication may occur while developing kinematic chains. Therefore, an isomorphic test is required to eliminate duplication. For this purpose, the numbers of methods are proposed during recent years. However, most of the methods are complex and difficult to understand, and fulfil the only primary condition, but not the secondary conditions for isomorphism detection. In the present work, a new method is introduced to detect isomorphism in planar kinematic chains (KCs) fulfilling both primary and secondary conditions. First, KC’s are topologically transformed into skeleton diagrams, and then skeleton matrices [S] and identification strings [IS] are formulated consequently. In order to detect isomorphism, the IS is considered as an invariant string of a KC which in turn, enables the detection of isomorphism between the KCs. The proposed method accurately recognizes isomorphism up to 12 links KCs with no counter examples found in the literature. Three examples with one degree of freedom having 10 links 12 joints, 10 links 13 joints and 12 links three degree of freedom systems are introduced to reveal the reliability and strength of the proposed method.

A novel algorithm for the isomorphism detection of various kinematic chains using topological index

For mechanism design, the isomorphic kinematic chains (KCs) should be deleted to improve design efficiency. At present, most methods of isomorphism detection involve complex concepts and comparison of intermediate parameters. Moreover, when the number of components increases, it is complex and difficult to discriminate a large number of KCs in short time. However, the molecular topological index is proposed to identify isomer in organic chemistry, which shows excellent abilities for isomer recognition. In some respects, topological graph of kinematic chain is similar to chemical molecular model. Therefore, this idea is adopted to obtain extended adjacency identification index (EAID) which fits for isomorphism detection of KCs, namely KC-EAID. If two topological graphs have same KC-EAID value, then they are isomorphic. Otherwise, they are not. Three kinds of kinematic chains, including simple joints KCs, multiple joints KCs and gear-link KCs are given to verify effectiveness of this index in isomorphism detection. For KC-EAID index, it requires a few known quantities and no comparison of intermediate parameters. The algorithm only includes the calculation of matrices, which saves operation time. Therefore, KC-EAID can be used as a powerful tool for isomorphism detection in number synthesis of KCs.

An innovative approach for detection of isomorphism of epicyclic gear trains

Detection of isomorphism in epicyclic gear trains (EGTs) and their mechanisms has been a great challenge for several researchers since few years. Checking for isomorphism at the conceptual stage will avoid duplication in the generation process. In this paper, an innovative approach based on Modified gradient method (MGM) is presented for identification of isomorphism of epicyclic gear trains and inversions of mechanisms. The new concept ‘MGM’ is proposed on basis of Train values and connectivity Matrices to be a quantitative measurement of isomorphism in EGTs. This is a novel, simple algorithm and less computation required to detect the duplication of EGTs. The proposed algorithm is applied to various examples from 4-links, 5-links, 6-links, and 8-links 1-DOF of the epicyclic gear trains. All the examples have been found satisfactory results with existing literature.

Matrix-Based Operation Method for Detecting Structural Isomorphism of Planetary Gear Train Structures

AbstractPlanetary gear sets (PGSs) have been widely used in automatic transmissions (AT) and dedicated hybrid transmissions (DHTs). In this paper, a novel isomorphic detection method for planetary gear transmission structure is proposed based on matrix operation. The isomorphic detection process includes two main parts. In the first part, various components of the transmission structure are classified. In the second part, isomorphic structures of the numerous structures are detected. Through the application of the proposed detection approach, the structures obtained by different synthesis algorithms can be greatly reduced. Furthermore, by the analysis and transformation of the hybrid configuration to conventional transmission configuration, the scope of use of the algorithm can be expanded through the method. The proposed detection approach is capable of automatically detecting the isomorphic structure of the potential structures obtained by synthesis algorithm.

MapEff: An Effective Graph Isomorphism Agorithm Based on the Discrete-Time Quantum Walk.

Graph isomorphism is to determine whether two graphs have the same topological structure. It plays a significant role in areas of image matching, biochemistry, and information retrieval. Quantum walk, as a novel quantum computation model, has been employed to isomorphic mapping detection to optimize the time complexity compared with a classical computation model. However, these quantum-inspired algorithms do not perform well—and even cease to work—for graphs with inherent symmetry, such as regular graphs. By analyzing the impacts of graphs symmetry on isomorphism detection, we proposed an effective graph isomorphism algorithm (MapEff) based on the discrete-time quantum walk (DTQW) to improve the accuracy of isomorphic mapping detection, especially for regular graphs. With the help of auxiliary edges, this algorithm can distinguish the symmetric nodes efficiently and, thus, deduct the qualified isomorphic mapping by rounds of selections. The experiments tested on 1585 pairs of graphs demonstrated that our algorithm has a better performance compared with other state-of-the-art algorithms.