Regular Graphs Research Articles

Randomly Twisted Hypercubes: Between Structure and Randomness

ABSTRACTTwisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these graphs have optimal diameter. We study twisted hypercubes in the setting where the instances can have general dependence, and also in the particular case where they are identical. We show that the resultant graph shares properties with random regular graphs, including small diameter, large vertex expansion, a semicircle law for its eigenvalues and no non‐trivial automorphisms. However, in contrast to random regular graphs, twisted hypercubes allow for short routing schemes.

Regular graphs with a cycle as a star complement

Regular graphs with a cycle as a star complement

Dynamic Graph Regularization for Multi-Stream Concept Drift Self-Adaptation

Dynamic Graph Regularization for Multi-Stream Concept Drift Self-Adaptation

Orthogonal Decompositions of Regular Graphs and Designing Tree-Hamming Codes

The paper introduces a concept of graph decomposition. That is orthogonal decompositions. Orthogonal decompositions of a graph H is a partitioning H into subgraphs of H such that any two subgraphs intersect in at most one edge. This decompositions are called G-orthogonal decompositions of H if and only if every subgraph in such decompositions is isomorphic to the graph G. Such decomposition appear in a lot of applications; statistics, information theory, in the theory of experimental design and many others. An approach of constructing orthogonal decompositions of regular graph is introduced here. Application of this approach for constructing tree -- orthogonal decompositions of complete bipartite graph is considered. Further, the use of orthogonal decompositions for designing hamming tree-codes is also discussed along with examples. The study shows that such codes have an efficient properties when are used to detect and correct the errors that may occur during a transmission of data through a network. Furthermore, we present a method for the recursive construction of orthogonal decompositions.

Graph Regularized Sparse L2,1 Semi‐Nonnegative Matrix Factorization for Data Reduction

ABSTRACTNon‐negative Matrix Factorization (NMF) is an effective algorithm for multivariate data analysis, including applications to feature selection, pattern recognition, and computer vision. Its variant, Semi‐Nonnegative Matrix Factorization (SNF), extends the ability of NMF to render parts‐based data representations to include mixed‐sign data. Graph Regularized SNF builds upon this paradigm by adding a graph regularization term to preserve the local geometrical structure of the data space. Despite their successes, SNF‐related algorithms to date still suffer from instability caused by the Frobenius norm due to the effects of outliers and noise. In this paper, we present a new SNF algorithm that utilizes the noise‐insensitive norm. We provide monotonic convergence analysis of the SNF algorithm. In addition, we conduct numerical experiments on three benchmark mixed‐sign datasets as well as several randomized mixed‐sign matrices to demonstrate the performance superiority of SNF over conventional SNF algorithms under the influence of Gaussian noise at different levels.

Counting and hardness-of-finding fixed points in cellular automata on random graphs

Abstract We study the fixed points of outer-totalistic cellular automata on sparse random regular graphs. These can be seen as constraint satisfaction problems, where each variable must adhere to the same local constraint, which depends solely on its state and the total number of its neighbors in each possible state. Examples of this setting include classical problems such as independent sets or assortative/dissasortative partitions. We analyse the existence and number of fixed points in the large system limit using the cavity method, under both the replica symmetric (RS) and one-step replica symmetry breaking (1RSB) assumption. This method allows us to characterize the structure of the space of solutions, in particular, if the solutions are clustered and whether the clusters contain frozen variables. This last property is conjectured to be linked to the typical algorithmic hardness of the problem. We bring experimental evidence for this claim by studying the performance of the belief-propagation reinforcement algorithm, a message-passing-based solver for these constraint satisfaction problems.

Reinforced graph regularization fault diagnosis network integrating multisensor rolling bearing data

Reinforced graph regularization fault diagnosis network integrating multisensor rolling bearing data

Identification of dynamic networks community by fusing deep learning and evolutionary clustering

Community detection is a critical component of network analysis and a hot topic in social computing. Detecting community structure in dynamic networks has important theoretical and practical implications for understanding the intrinsic function of networks and predicting network behavior. However, the majority of existing dynamic community detection methods adopt shallow models, which have limited ability to excavate complex non-linear structures and tend to generate undesirable community structures. In order to obtain an accurate and robust community structure in dynamic networks, we are inspired by network representation learning and utilize the deep learning to detect evolving communities in dynamic networks. In this paper, we propose a novel dynamic community detection method by fusing Deep Learning and Evolutionary Clustering (DLEC). This work attempts to combine deep learning and evolutionary clustering into a unified framework. First, we propose a matrix construction strategy to fully reveal the inherent community structures via the underlying community memberships. Then, we develop a novel multi-layer deep autoencoder framework that consists of multiple non-linear functions to extract the latent deep representation of the dynamic network. Based on the evolutionary clustering framework, a graph regularization term is introduced to ensure the smoothness of the community evolution. Finally, we employ the K-means clustering algorithm on the low-dimensional network space to obtain the community structure. Extensive experimental results on synthetic and real-world networks show that the proposed DLEC algorithm can effectively detect high-quality communities in dynamic networks.

Flip colouring of graphs

It is proved that for integers b, r such that 3≤b<r≤b+12-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3 \\le b < r \\le \\left( {\\begin{array}{c}b+1\\\\ 2\\end{array}}\\right) - 1$$\\end{document}, there exists a red/blue edge-colored graph such that the red degree of every vertex is r, the blue degree of every vertex is b, yet in the closed neighbourhood of every vertex there are more blue edges than red edges. The upper bound r≤b+12-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\le \\left( {\\begin{array}{c}b+1\\\\ 2\\end{array}}\\right) -1$$\\end{document} is best possible for any b≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b \\ge 3$$\\end{document}. We further extend this theorem to more than two colours, and to larger neighbourhoods. A useful result required in some of our proofs, of independent interest, is that for integers r, t such that 0≤t≤r22-5r3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le t \\le \\frac{r^2}{2} - 5r^{3/2}$$\\end{document}, there exists an r-regular graph in which each open neighbourhood induces precisely t edges. Several explicit constructions are introduced and relationships with constant linked graphs, (r, b)-regular graphs and vertex transitive graphs are revealed.

Polyhedra with Hexagonal and Triangular Faces and Three Faces Around Each Vertex

AbstractWe analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call “trihexes”. Trihexes are analogous to fullerenes, which are 3-regular planar graphs whose faces are all hexagons and pentagons. Every trihex can be represented as the quotient of a hexagonal tiling of the plane under a group of isometries generated by $$180^\circ $$ 180 ∘ rotations. Every trihex can also be described with either one or three “signatures”: triples of numbers that describe the arrangement of the rotocenters of these rotations. Simple arithmetic rules relate the three signatures that describe the same trihex. We obtain a bijection between trihexes and equivalence classes of signatures as defined by these rules. Labeling trihexes with signatures allows us to put bounds on the number of trihexes for a given number of vertices v in terms of the prime factorization of v and to prove a conjecture concerning trihexes that have no “belts” of hexagons.

Cardinality and Isomorphic Properties of Hajós Graphs and Hajós Fuzzy Graphs

Objectives: Fuzzy graphs allow uncertainty in the ideas characterizing vertices and edges to be included when modelling real-world scenarios into graph models. This study aims to introduce Hajós fuzzy graph, cardinality of Hajós graph and Hajós fuzzy graph and to explore many properties. Methods : Here, the Hajós construction is applied to two fuzzy graphs, and the Hajós fuzzy graph is defined by assigning membership values to all the edges and vertices of the newly formed fuzzy graph. Fuzzy graph conditions are verified for the assigned membership values. Examples are used to illustrate the concept. Findings: The Hajós fuzzy graph's size and order are established. Isomorphic property is discussed. The number of Hajós (fuzzy) graphs that exist for some combinations of two (fuzzy) graphs is calculated. Novelty: Based on Hajós construction, the Hajós fuzzy graph is defined. Based on the fact that different choices of vertices and edges produce different Hajós (fuzzy) graph, its cardinality is defined. The notion of cardinality is a novel concept in Hajos graph which helps to find the total number of Hajós (fuzzy) graphs. The cardinality of the Hajós (fuzzy) graph which arises from two (fuzzy) graphs that can be a cycle, path, regular graph, complete graph and complete bipartite graph are derived. Mathematics subject classification. (2010): 05C72, 05C76 Keywords: Hajós Graph, Hajós fuzzy graph, Order, Size, Isomorphism, Cardinality of Hajós fuzzy graph

RGP: Neural Network Pruning Through Regular Graph With Edges Swapping.

Deep learning technology has found a promising application in lightweight model design, for which pruning is an effective means of achieving a large reduction in both model parameters and float points operations (FLOPs). The existing neural network pruning methods mostly start from the consideration of the importance of model parameters and design parameter evaluation metrics to perform parameter pruning iteratively. These methods were not studied from the perspective of network model topology, so they might be effective but not efficient, and they require completely different pruning for different datasets. In this article, we study the graph structure of the neural network and propose a regular graph pruning (RGP) method to perform a one-shot neural network pruning. Specifically, we first generate a regular graph and set its node-degree values to meet the preset pruning ratio. Then, we reduce the average shortest path-length (ASPL) of the graph by swapping edges to obtain the optimal edge distribution. Finally, we map the obtained graph to a neural network structure to realize pruning. Our experiments demonstrate that the ASPL of the graph is negatively correlated with the classification accuracy of the neural network and that RGP has a strong precision retention capability with high parameter reduction (more than 90%) and FLOPs reduction (more than 90%) (the code for quick use and reproduction is available at https://github.com/Holidays1999/Neural-Network-Pruning-through-its-RegularGraph-Structure).

Rainbow Mean Index of Some Classes of Graphs

In this article, we determine the rainbow mean index of cartesianproductG◻Pn, where G is a regular graph;Pm◻Pn; Wn◻K2;total graph of paths; chain of cycles and complete graphs; triangular treeand join graphs tCs∨K1.

Sparse regularized graph pooling network optimal sensor placement method for diesel engine vibration fault perception system

Vibration sensor network optimization increases monitoring effectiveness and reduces sensor quantity and transmission burden. However, the traditional model-driven methods depend on precise finite element models, challenging for complex machinery. This paper presents a data-driven approach using the Sparse Regularized Graph Pooling Network (SRGPN), which conceptualizes sensor networks as graphs and uses graph pooling to identify optimal sensor combinations. A sparsity regularization term related to the number of sensors is included in the loss function, aiming for the minimal yet effective sensor combination. Additionally, a monitoring capability metric suited for diesel engines with multi-source impulse signals is proposed, reflecting the sensors’ monitoring capacity. Validated through simulations and tests on a diesel engine test bench, the results show that SRGPN optimally places sensors near excitation sources, balancing sensor count and monitoring needs. This approach shows potential for optimizing sensor placements in condition monitoring.

Discriminative fuzzy [formula omitted]-means clustering with local structure preservation for high-dimensional data

K-means clustering, which aims to distribute data points into K different clusters, is an effective technique for data mining and machine learning. However, several factors degrade clustering performance in real-world applications. High-dimensional data containing redundant features is one of the key issues in the clustering process. Many traditional hard or fuzzy versions of K-means clustering fail to handle high-dimensional data effectively. In this paper, we present a novel model referred to as discriminative fuzzy K-means clustering, which incorporates discriminative projection and p-Laplacian graph regularization into a joint framework. Specifically, the proposed model can reduce the negative impact of redundant features and enhance the discriminative power when projecting the original data into a lower-dimensional space. It simultaneously preserves the structural locality properties of the data by performing p-Laplacian graph regularization on the membership matrix. Consequently, clustering performance is significantly enhanced for high-dimensional data. Finally, an effective algorithm is presented to solve the formulated optimization problem. Experimental results obtained on benchmark datasets are promising and demonstrate the superiority of the proposed method.

On a family of quasi-strongly regular Cayley graphs

In this paper, we construct a family of quasi-strongly regular Cayley graphs which is defined on a finite group G with respect to a proper subgroup H of G and denoted by Γ H ( G ) . We also compute the full automorphism group and characterize various transitivity properties of Γ H ( G ) for any group G and for any subgroup H of G.

Minimal directed strongly regular Cayley graphs over generalized dicyclic groups

Minimal directed strongly regular Cayley graphs over generalized dicyclic groups

DeepASD: a deep adversarial-regularized graph learning method for ASD diagnosis with multimodal data

Autism Spectrum Disorder (ASD) is a prevalent neurological condition with multiple co-occurring comorbidities that seriously affect mental health. Precisely diagnosis of ASD is crucial to intervention and rehabilitation. A single modality may not fully reflect the complex mechanisms underlying ASD, and combining multiple modalities enables a more comprehensive understanding. Here, we propose, DeepASD, an end-to-end trainable regularized graph learning method for ASD prediction, which incorporates heterogeneous multimodal data and latent inter-patient relationships to better understand the pathogenesis of ASD. DeepASD first learns cross-modal feature representations through a multimodal adversarial-regularized encoder, and then constructs adaptive patient similarity networks by leveraging the representations of each modality. DeepASD exploits inter-patient relationships to boost the ASD diagnosis that is implemented by a classifier compositing of graph neural networks. We apply DeepASD to the benchmarking Autism Brain Imaging Data Exchange (ABIDE) data with four modalities. Experimental results show that the proposed DeepASD outperforms eight state-of-the-art baselines on the benchmarking ABIDE data, showing an improvement of 13.25% in accuracy, 7.69% in AUC-ROC, and 17.10% in specificity. DeepASD holds promise for a more comprehensive insight of the complex mechanisms of ASD, leading to improved diagnosis performance.

Periodicity of bipartite walk on biregular graphs with conditional spectra

In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover’s walks. A discrete quantum walk is given by the powers of a unitary matrix U indexed by arcs or edges of the underlying graph. The walk is periodic if U k = I for some positive integer k. Kubota has given a characterization of periodicity of Grover’s walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota’s results—if a biregular graph G has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over G in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover’s walk on regular graphs.

Cross-domain recommender system with embedding- and mapping-based knowledge correlation

A knowledge transfer-based cross-domain recommender system is currently a research hotspot. Existing research has reached a high level of maturity in mining potential knowledge and establishing transfer mechanisms. However, most of them ignore the impact of the dissimilarity of potential knowledge on the transfer performance. Herein, a cross-domain recommender system based on knowledge correlation-induced the embedding and mapping approach is proposed, denoted by KCEM-CDRS. First, we propose a knowledge correlation measure, which captures the consistency of knowledge between the target and source domains to build the bridge for knowledge transfer. Second, we construct a joint matrix triple factorization model to solve the data sparsity in the target domain while introducing graph regularization to solve the problem of negative knowledge transfer. Results of extensive experiments on real Amazon metadata indicate that compared with three existing cross-domain recommendation methods, KCEM-CDRS shows performance improvements of 0.05–9.55 % and 0.02–2.63 % on mean absolute error and root mean square error, respectively. Additionally, the results of the ablation experiments indicate that consideration of the knowledge correlation between domains is beneficial for knowledge transfer when the density of the source domain is rich.