Regular Graphs Research Articles

Parallel Maximum Cardinality Matching for General Graphs on GPUs.

The matching problem formulated as Maximum Cardinality Matching in General Graphs (MCMGG) finds the largest matching on graphs without restrictions. The Micali-Vazirani algorithm has the best asymptotic complexity for solving MCMGG when the graphs are sparse. Parallelizing matching in general graphs on the GPU is difficult for multiple reasons. First, the augmenting path procedure is highly recursive, and NVIDIA GPUs use registers to store kernel arguments, which eventually spill into cached device memory, with a performance penalty. Second, extracting parallelism from the matching process requires partitioning the graph to avoid any overlapping augmenting paths. We propose an implementation of the Micali-Vazirani algorithm which identifies bridge edges using thread-parallel breadth-first search, followed by block-parallel path augmentation and blossom contraction. Augmenting path and Union-find methods were implemented as stack-based iterative methods, with a stack allocated in shared memory. Our experimentation shows that compared to the serial implementation, our approach results in up to 15-fold speed-up for very sparse regular graphs, up to 5-fold slowdown for denser regular graphs, and finally a 50-fold slowdown for power-law distributed Kronecker graphs. This implementation has been open-sourced for further research on developing combinatorial graph algorithms on GPUs.

Linear Codes Constructed from Two Weakly Regular Plateaued Functions with Index (p - 1)/2.

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting p≡1(mod4), we constructed an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index (p-1)/2. Their weight distributions were completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal.

Shared-Memory Parallel Edmonds Blossom Algorithm for Maximum Cardinality Matching in General Graphs.

The Edmonds Blossom algorithm is implemented here using depth-first search, which is intrinsically serial. By streamlining the code, our serial implementation is consistently three to five times faster than the previously fastest general graph matching code. By extracting parallelism across iterations of the algorithm, with coarse-grain locking, we are able to further reduce the run time on random regular graphs four-fold and obtain a two-fold reduction of run time on real-world graphs with similar topology. Solving very sparse graphs (average degree less than four) exhibiting community structure with eight threads led to a slow down of three-fold, but this slow down is replaced by marginal speed up once the average degree is greater than four. We conclude that our parallel coarse-grain locking implementation performs well when extracting parallelism from this augmenting-path-based algorithm and may work well for similar algorithms.

Fractional matchings on regular graphs

Fractional matchings on regular graphs

Hyperspectral Anomaly Detection via Low-Rank Representation with Dual Graph Regularizations and Adaptive Dictionary

In a hyperspectral image, there is a close correlation between spectra and a certain degree of correlation in the pixel space. However, most existing low-rank representation (LRR) methods struggle to utilize these two characteristics simultaneously to detect anomalies. To address this challenge, a novel low-rank representation with dual graph regularization and an adaptive dictionary (DGRAD-LRR) is proposed for hyperspectral anomaly detection. To be specific, dual graph regularization, which combines spectral and spatial regularization, provides a new paradigm for LRR, and it can effectively preserve the local geometrical structure in the spectral and spatial information. To obtain a robust background dictionary, a novel adaptive dictionary strategy is utilized for the LRR model. In addition, extensive comparative experiments and an ablation study were conducted to demonstrate the superiority and practicality of the proposed DGRAD-LRR method.

Regularisation constrained denoising discriminant least squares regression for image classification

Discriminative Least Squares Regression (DLSR) is an algorithm that utilizes ε-draggings to expand the distance between classes to better solve multi-classification problems. However, this method increases the differences in regression targets within classes during the classification process, and noise in the samples adversely affects its performance. To address the above problems, we propose a novel method called Regularisation Constrained Denoising Discriminant Least Squares Regression (RCDDLSR). Firstly, the idea of matrix decomposition is introduced and sparse constraints are imposed on the noise matrix for noise elimination, thus helping to maintain the manifold structure of the raw data. Secondly, the neighbourhood relationship between samples of the same category is ensured through graph regularisation to avoid overfitting. In addition, we introduce low-rank constraints in the labelling space to better capture the local structure of the regression target within the class. Finally, our experiments show that the method performs better in various types of image datasets compared to other different algorithms.

Eigenvalues of complex unit gain graphs and gain regularity

Abstract A complex unit gain graph (or T {\mathbb{T}} -gain graph) Γ = ( G , γ ) \Gamma =\left(G,\gamma ) is a gain graph with gains in T {\mathbb{T}} , the multiplicative group of complex units. The T {\mathbb{T}} -outgain in Γ \Gamma of a vertex v ∈ G v\in G is the sum of the gains of all the arcs originating in v v . A T {\mathbb{T}} -gain graph is said to be an a a - T {\mathbb{T}} -regular graph if the T {\mathbb{T}} -outgain of each of its vertices is equal to a a . In this article, it is proved that a a - T {\mathbb{T}} -regular graphs exist for every a ∈ R a\in {\mathbb{R}} . This, in particular, means that every real number can be a T {\mathbb{T}} -gain graph eigenvalue. Moreover, denoted by Ω ( a ) \Omega \left(a) the class of connected T {\mathbb{T}} -gain graphs whose largest eigenvalue is the real number a a , it is shown that Ω ( a ) \Omega \left(a) is nonempty if and only if a a belongs to { 0 } ∪ [ 1 , + ∞ ) \left\{0\right\}\cup \left[1,+\infty ) . In order to achieve these results, non-complete extended p p -sums and suitably defined joins of T {\mathbb{T}} -gain graphs are considered.

Constrained Symmetric Non-Negative Matrix Factorization with Deep Autoencoders for Community Detection

Recently, community detection has emerged as a prominent research area in the analysis of complex network structures. Community detection models based on non-negative matrix factorization (NMF) are shallow and fail to fully discover the internal structure of complex networks. Thus, this article introduces a novel constrained symmetric non-negative matrix factorization with deep autoencoders (CSDNMF) as a solution to this issue. The model possesses the following advantages: (1) By integrating a deep autoencoder to discern the latent attributes bridging the original network and community assignments, it adeptly captures hierarchical information. (2) Introducing a graph regularizer facilitates a thorough comprehension of the community structure inherent within the target network. (3) By integrating a symmetry regularizer, the model’s capacity to learn undirected networks is augmented, thereby facilitating the precise detection of symmetry within the target network. The proposed CSDNMF model exhibits superior performance in community detection when compared to state-of-the-art models, as demonstrated by eight experimental results conducted on real-world networks.

Joint learning of latent subspace and structured graph for multi-view clustering

Most existing multi-view clustering methods rely solely on subspace clustering or graph-based clustering. Subspace clustering reduces the redundant information in high dimensional data, but it neglects the intrinsic structural dependencies among samples. Graph-based clustering can model the similarity among samples but tends to suffer from redundant information. In this paper, a novel framework jointing subspace learning and structured graph learning for multi-view clustering (SSMC) is proposed, which benefits from the merits of both subspace learning and structured graph learning. SSMC utilizes graph regularized subspace learning to obtain low dimensional consensus features, where the embedded features are ensured to have maximized correlation to reduce the redundant information, and the graph regularization forces embedded features to preserve their sample similarities. Meanwhile, an adaptive structured graph is learned based on the consensus features in the embedded feature space, avoiding the curse of dimensionality in the graph learning procedure. A rank constraint forces the learned graph to have exactly the same number of connected components as the number of clusters, to obtain a more reliable structured graph. Moreover, an effective algorithm is proposed to optimize the SSMC, where the graph regularized subspace learning part and the structured graph learning part are jointly optimized in a mutual reinforcement manner. The experimental results on real-world benchmark datasets show that the SSMC outperforms the state-of-the-arts in multi-view clustering tasks.

On minimum vertex bisection of random d-regular graphs

Minimum vertex bisection is a graph partitioning problem in which the aim is to find a partition of the vertices into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. In this work we are interested in providing asymptotically almost surely upper bounds on the minimum vertex bisection of random d-regular graphs, for constant values of d. Our approach is based on analyzing a greedy algorithm by using the differential equation method. In this way, we obtain the first known non-trivial upper bounds for the vertex bisection number in random regular graphs. The numerical approximations of these theoretical bounds are compared with the emprical ones, and with the lower bounds from Kolesnik and Wormald (2014) [30].

Finite-element assembly approach of optical quantum walk networks

We present a finite-element approach for computing the aggregate scattering matrix of a network of linear coherent scatterers. These might be optical scatterers or more general scattering coins studied in quantum walk theory. While techniques exist for two-dimensional lattices of feed-forward scatterers, the present approach is applicable to any network configuration of any collection of scatterers. Unlike traditional finite-element methods in optics, this method does not directly solve Maxwell’s equations; instead it is used to assemble and solve a linear, coupled scattering problem that emerges after Maxwell’s equations are abstracted within the scattering matrix method. With this approach, a global unitary is assembled corresponding to one time step of the quantum walk on the network. After applying the relevant boundary conditions to this global matrix, the problem becomes non-unitary and possesses a steady-state solution that is the output scattering state. We provide an algorithm to obtain this steady-state solution exactly using a matrix inversion, yielding the scattering state without requiring a direct calculation of the eigenspectrum. The approach is then numerically validated on a coupled-cavity interferometer example that possesses a known, closed-form solution. Finally, the method is shown to be a generalization of the Redheffer star product, which describes scatterers on one-dimensional lattices (2-regular graphs) and is often applied to the design of thin-film optics, making the current approach an invaluable tool for the design and validation of high-dimensional phase-reprogrammable optical devices and study of quantum walks on arbitrary graphs.

Ant Mill: an adversarial traffic pattern for low-diameter direct networks

Since today’s HPC and data center systems can comprise hundreds of thousands of servers and beyond, it is crucial to equip them with a network that provides high performance. New topologies proposed to achieve such performance need to be evaluated under different traffic conditions, aiming to closely replicate real-world scenarios. While most optimizations should be guided by common traffic patterns, it is essential to ensure that no pathological traffic pattern can compromise the entire system. Determining synthetic adversarial traffic patterns for a network typically relies on a thorough understanding of its topology and routing. In this paper, we address the problem of identifying a generic adversarial traffic pattern for low-diameter direct interconnection networks. We first focus on Random Regular Graphs (RRGs), which represent a typical case for these networks. Moreover, RRGs have been proposed as topologies for interconnection networks due to their superior scalability and expandability, among other advantages. We introduce Ant Mill, an adversarial traffic pattern for RRGs when using routes of minimal length. Secondly, we demonstrate that the Ant Mill traffic pattern is also adversarial in other low-diameter direct interconnection networks such as Slimfly, Dragonfly, and Projective networks. Ant Mill is thoroughly motivated and evaluated, enabling future studies of low-diameter direct interconnection networks to leverage its findings.

Precise feature selection via non-convex regularized graph embedding and self-representation for unsupervised learning

In graph embedding learning with unsupervised feature selection fields, similarity matrices are usually obtained only from the initial noise-laden samples, and the probable association between various features is often ignored. To tackle the issues described above, this article proposes an effective unsupervised group feature selection approach via non-convex regularized graph embedding and self-representation (NLGMS). NLGMS integrates the similarity matrix and feature selection as a joint system on the basis of graph. It combines self-representation learning into the system to preserve the global structure and learns the nonconvex projection matrix to direct the procedure of feature selection. Specifically, first, NLGMS presents a unified framework for global structure learning, local structure learning, and feature selection process so the data’s intrinsic structure is excellently captured and learned adaptively. Second, nonconvex ℓ2,0-norm is enforced upon the projection matrix to perform the process of feature selection, ensuring that the optimal feature subset can be picked precisely. Nonconvex ℓ2,0-norm constraint considers the probable association between various features so that it can deliver optimal performance by considering the feature subset as an integral whole, which other convex regularized constraints often fail to achieve. Finally, the introduction to self-representation learning can facilitate the reconstruction of samples in low-rank subspaces, strengthening the model’s robustness. To tackle this challenging nonconvex model, an innovative alternative optimization technique is exploited. In addition, to validate the superiority of NLGMS, exhaustive experiments are performed on one synthetic dataset, six benchmark datasets, and one fish image dataset. The source code is available at: https://github.com/hrbai/NLGMS.

Weak External Bisections of Regular Graphs

Weak External Bisections of Regular Graphs

Optimal Layout of (Kp − Cp)n into Wheel-Like Networks

The problem of finding the induced subgraph with the maximum number of edges among all subsets of a fixed cardinality is known as the maximum subgraph problem (MSP). The lexicographic order has been proven optimal for Cartesian products of specific regular graphs [B. Sergei, B. Pavle and K. Nikola, New infinite family of regular edge isoperimetric graphs, Theoretical Computer Science 721 (2018) 42–53]. This paper is dedicated to the precise calculation of edge values within the resulting subgraphs, enhancing our comprehension of their structural properties and implications. In this paper, we focus on solving the MSP of [Formula: see text] for any [Formula: see text] and for odd [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is obtained by the Cartesian product of a complete graph with [Formula: see text] vertices with a removal of a cycle on [Formula: see text] vertices. It has been observed that the graph obtained by deleting a cycle of length [Formula: see text] from a complete graph [Formula: see text] is isomorphic to a circulant graph [Formula: see text], while the particular case [Formula: see text] has been previously studied in [A. Syeda and M. Rajesh, MinLA of [Formula: see text] and its optimal layout into certain trees, The Journal of Supercomputing (2023), https://doi.org/10.1007/s11227-023-05140-3]. Our work extends the understanding of the MSP to a broader class of graphs with similar properties and applications.

Full degree spanning trees in random regular graphs

We study the problem of maximizing the number of full degree vertices in a spanning tree T of a graph G; that is, the number of vertices whose degree in T equals its degree in G. In cubic graphs, this problem is equivalent to maximizing the number of leaves in T and minimizing the size of a connected dominating set of G. We provide an algorithm that produces (w.h.p.) a tree with at least 0.4591n vertices of full degree (and also, leaves) when run on a random cubic graph. This improves the previously best known lower bound of 0.4146n. We also provide lower bounds on the number of full degree vertices in the random regular graph G(n,r) for r≤10.

Semi-supervised Multi-view Clustering based on NMF with Fusion Regularization

Multi-view clustering has attracted significant attention and application. Nonnegative matrix factorization is one popular feature of learning technology in pattern recognition. In recent years, many semi-supervised nonnegative matrix factorization algorithms were proposed by considering label information, which has achieved outstanding performance for multi-view clustering. However, most of these existing methods have either failed to consider discriminative information effectively or included too much hyper-parameters. Addressing these issues, a semi-supervised multi-view nonnegative matrix factorization with a novel fusion regularization (FRSMNMF) is developed in this article. In this work, we uniformly constrain alignment of multiple views and discriminative information among clusters with designed fusion regularization. Meanwhile, to align the multiple views effectively, two kinds of compensating matrices are used to normalize the feature scales of different views. Additionally, we preserve the geometry structure information of labeled and unlabeled samples by introducing the graph regularization simultaneously. Due to the proposed methods, two effective optimization strategies based on multiplicative update rules are designed. Experiments implemented on six real-world datasets have demonstrated the effectiveness of our FRSMNMF comparing with several state-of-the-art unsupervised and semi-supervised approaches.

Unifying Graph Neural Networks with a Generalized Optimization Framework

Graph Neural Networks (GNNs) have received considerable attention on graph-structured data learning for a wide variety of tasks. The well-designed propagation mechanism, which has been demonstrated effective, is the most fundamental part of GNNs. Although most of the GNNs basically follow a message passing manner, little effort has been made to discover and analyze their essential relations. In this paper, we establish a surprising connection between different propagation mechanisms with an optimization problem. We show that despite the proliferation of various GNNs, in fact, their proposed propagation mechanisms are the optimal solutions of a generalized optimization framework with a flexible feature fitting function and a generalized graph regularization term. Actually, the optimization framework can not only help understand the propagation mechanisms of GNNs, but also open up opportunities for flexibly designing new GNNs. Through analyzing the general solutions of the optimization framework, we provide a more convenient way for deriving corresponding propagation results of GNNs. We further discover that existing works usually utilize naïve graph convolutional kernels for feature fitting function, or just utilize one-hop structural information (original topology graph) for graph regularization term. Correspondingly, we develop two novel objective functions considering adjustable graph kernels showing low-pass or high-pass filtering capabilities and one novel objective function considering high-order structural information during propagation respectively. Extensive experiments on benchmark datasets clearly show that the newly proposed GNNs not only outperform the state-of-the-art methods but also have good ability to alleviate over-smoothing, and further verify the feasibility for designing GNNs with the generalized unified optimization framework.

Robust sparse concept factorization with graph regularization for subspace learning

Concept factorization (CF) is a powerful tool in subspace learning. Recently graph-based CF and local coordinate CF have been proposed to exploit the intrinsic geometrical structure of data, and have been shown quite successful in improving performance. However, these methods have limited robustness and might be sensitive to noises and disturbances in practical applications. In this paper, we propose a novel robust sparse CF framework (RSCF) for subspace learning. Specifically, we present a robust loss function to effectively eliminate the impact of the large outliers. The local coordinate constraint and the graph regularization term are incorporated into RSCF to simultaneously guarantee the sparsity of the coefficient matrix and maintain the local structure of the data. We prove that the local coordinate constraint implies the orthogonality of the coefficient matrix. By using the half-quadratic optimization technique, we transform the objective function of RSCF into a quadratic form and provide the iterative updating rules and the convergence analysis. Extensive experiments demonstrate the robustness and superiority of the proposed RSCF in comparison to state-of-the-art CF methods.

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry.

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree d. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around W∼20 for a random regular graph d=4. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around W∼5, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.