Regular Graphs Research Articles

Quantum-like states on complex synchronized networks

Recent work has exposed the idea that interesting quantum-like (QL) probability laws, including interference effects, can be manifest in classical systems. Here, we propose a model for QL states and QL bits. We suggest a way that huge, complex systems can host robust states that can process information in a QL fashion. Axioms that such states should satisfy are proposed. Specifically, it is shown that building blocks suited for QL states are networks, possibly very complex, that we defined based on k -regular random graphs. These networks can dynamically encode a lot of information that is distilled into the emergent states we can use for QL processing. Although the emergent states are classical, they have properties analogous to quantum states. Concrete examples of how QL functions are possible are given. The possibility of a ‘QL advantage’ for computing-type operations and the potential relevance for new kinds of function in the brain are discussed and left as open questions.

On the complexity of minimum maximal acyclic matchings

Low-Acy-Matching asks to find a maximal matching M in a given graph G of minimum cardinality such that the set of M-saturated vertices induces an acyclic subgraph in G. The decision version of Low-Acy-Matching is known to be NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{NP}}$$\\end{document}-complete. In this paper, we strengthen this result by proving that the decision version of Low-Acy-Matching remains NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{NP}}$$\\end{document}-complete for bipartite graphs with maximum degree 6 and planar perfect elimination bipartite graphs. We also show the hardness difference between Low-Acy-Matching and Max-Acy-Matching. Furthermore, we prove that, even for bipartite graphs, Low-Acy-Matching cannot be approximated within a ratio of n1-ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{1-\\epsilon }$$\\end{document} for any ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document} unless P=NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extsf{P}}={\ extsf{NP}}$$\\end{document}. Finally, we establish that Low-Acy-Matching exhibits APX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{APX}$$\\end{document}-hardness when restricted to 4-regular graphs.

Selective multiple kernel fuzzy clustering with locality preserved ensemble

Multiple kernel fuzzy clustering (MKFC) has demonstrated promising performance in capturing the non-linear relationships within data. However, its effectiveness relies heavily on the appropriate selection of the fuzzification coefficient and kernel functions. To address this challenge, this paper proposes a novel clustering ensemble approach for improving the robustness and accuracy of the MKFC algorithm. The proposed method employs a multi-objective evolutionary optimization approach to heuristically select the optimal fuzzification and kernel coefficients. By employing the selected coefficients, the MKFC algorithm generates a diverse set of accurate candidate clustering results. Subsequently, a locality preserved ensemble mechanism is introduced to derive the final partition matrix, ensuring that all candidate clusterings within the Pareto non-dominated set contribute to the final consensus matrix. Moreover, this mechanism incorporates the knowledge about the locality of the dataset via a graph regularization term, thereby further enhancing the clustering performance. Comprehensive experiments conducted on widely adopted benchmark datasets demonstrate the superiority of the proposed method over the state-of-the-art approaches.

ALGORITHM FOR CONSTRUCTING THE TRIPLE UNIT GRAPH OF TYPE II OF RING Z_n USING PYTHON

Let be a commutative ring with as the set of all unit elements in . This paper introduces a new graph associated with the ring , called the triple unit graph of type II, denoted by with the vertex set is − {0,1}. In TU2(R), two distinct vertices, and , are adjacent if there exists with and such that . This paper focuses on the algorithm for constructing using Python. This research uses the literature study research method. The Python programming language can be used to observe the characteristic result of the graph. From the patterns generated by the algorithm, some characteristics of are obtained. For example, if is a prime and , then is a connected graph, a complete graph, a regular graph, and a Hamiltonian graph

Identifying retinopathy in optical coherence tomography images with less labeled data via contrastive graph regularization.

Retinopathy detection using optical coherence tomography (OCT) images has greatly advanced with computer vision but traditionally requires extensive annotated data, which is time-consuming and expensive. To address this issue, we propose a novel contrastive graph regularization method for detecting retinopathies with less labeled OCT images. This method combines class prediction probabilities and embedded image representations for training, where the two representations interact and co-evolve within the same training framework. Specifically, we leverage memory smoothing constraints to improve pseudo-labels, which are aggregated by nearby samples in the embedding space, effectively reducing overfitting to incorrect pseudo-labels. Our method, using only 80 labeled OCT images, outperforms existing methods on two widely used OCT datasets, with classification accuracy exceeding 0.96 and an Area Under the Curve (AUC) value of 0.998. Additionally, compared to human experts, our method achieves expert-level performance with only 80 labeled images and surpasses most experts with just 160 labeled images.

Counting triangles in regular graphs

AbstractIn this paper, we investigate the minimum number of triangles, denoted by , in ‐vertex ‐regular graphs, where is an odd integer and is an even integer. The well‐known Andrásfai–Erdős–Sós Theorem has established that if . In a striking work, Lo has provided the exact value of for sufficiently large , given that . Here, we bridge the gap between the aforementioned results by determining the precise value of in the entire range . This confirms a conjecture of Cambie, de Joannis de Verclos, and Kang for sufficiently large .

Diverse joint nonnegative matrix tri-factorization for attributed graph clustering

Cluster analysis of attributed graphs is a demanding and challenging task in the analysis of network-structured data. It involves learning node representation by leveraging both node attributes and the topological structure of the graph, aiming to accomplish effective clustering. Typically, existing methods fuse the topological and non-topological information by learning a consensus representation, often resulting in redundancy and overlooking their inherent distinctions. To address this issue, this paper proposes the Diverse Joint Nonnegative Matrix Tri-Factorization (Div-JNMTF), an embedding based model to detect communities in attributed graphs. The novel JNMTF model attempts to extract two distinct node representations from topological and non-topological data. Simultaneously, a diversity regularization technique utilizing the Hilbert–Schmidt Independence Criterion (HSIC) is employed. Its objective is to reduce redundant information in the node representations while encouraging the distinct contributions of both types of information. In addition, two graph regularization terms are introduced to preserve the local structures in the topological and attribute representation spaces. The Div-JNMTF model is optimized by developing an iterative optimization approach. By conducting thorough experiments on four synthetic and eight real-world attributed graph datasets, it has been demonstrated that the proposed model excels in accurately detecting attributed communities and surpasses the performance of existing methods.

On regular triangle-distinct graphs

On regular triangle-distinct graphs

Beyond quantum annealing: optimal control solutions to maxcut problems

Quantum Annealing (QA) relies on mixing two Hamiltonian terms, a simple driver and a complex problem Hamiltonian, in a linear combination. The time-dependent schedule for this mixing is often taken to be linear in time: improving on this linear choice is known to be essential and has proven to be difficult. Here, we present different techniques for improving on the linear-schedule QA along two directions, conceptually distinct but leading to similar outcomes: 1) the first approach consists of constructing a Trotter-digitized QA (dQA) with schedules parameterized in terms of Fourier modes or Chebyshev polynomials, inspired by the Chopped Random Basis algorithm for optimal control in continuous time; 2) the second approach is technically a Quantum Approximate Optimization Algorithm (QAOA), whose solutions are found iteratively using linear interpolation or expansion in Fourier modes. Both approaches emphasize finding smooth optimal schedule parameters, ultimately leading to hybrid quantum–classical variational algorithms of the alternating Hamiltonian Ansatz type. We apply these techniques to MaxCut problems on weighted 3-regular graphs with N = 14 sites, focusing on hard instances that exhibit a small spectral gap, for which a standard linear-schedule QA performs poorly. We characterize the physics behind the optimal protocols for both the dQA and QAOA approaches, discovering shortcuts to adiabaticity-like dynamics. Furthermore, we study the transferability of such smooth solutions among hard instances of MaxCut at different circuit depths. Finally, we show that the smoothness pattern of these protocols obtained in a digital setting enables us to adapt them to continuous-time evolution, contrarily to generic non-smooth solutions. This procedure results in an optimized QA schedule that is implementable on analog devices.

Phases and Duality in the Fundamental Kazakov–Migdal Model on the Graph

Abstract We examine the fundamental Kazakov–Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on a graph similar to that in the small coupling region. We estimate the phase structure of the FKM model in both the small and large coupling regions by comparing it with the Gross–Witten–Wadia model. We further validate the theoretical analysis through detailed numerical simulations.

Near-term quantum algorithm for solving the MaxCut problem with fewer quantum resources

MaxCut is an NP-hard combinatorial optimization problem in graph theory. The quantum approximate optimization algorithms (QAOAs) offer new methods for solving such problems, which are potentially better than classical schemes. However, the requirement of N qubits for solving graphs with N-vertex in QAOA, coupled with the large-N trainability issue due to barren plateaus, poses a substantial challenge for noisy intermediate-scale quantum (NISQ) devices. Recently, a hybrid quantum–classical algorithm has been proposed for solving semidefinite programs (SDPs), named NISQ SDP Solver (NSS). In this paper, we study the performance of the NSS for solving MaxCut problem via the introduction of a near-term quantum algorithm and execution of experimental simulations. Since the MaxCut problem admits a relaxation into an SDP formulation, the SDP can be solved using NSS, with a subsequent classical post-processing step converting the hybrid density matrix into a MaxCut solution. After performing these steps, the near-term quantum algorithm will also inherit the advantages of NSS. We analyze the algorithm as compared to QAOAs and find that the depth of the quantum circuit is independent of the number of edges on the graph. Our algorithm requires logN qubits and has O(1) circuit depth. This implies that it uses exponentially fewer qubits compared to QAOAs while also requiring a significantly reduced circuit depth to solve the MaxCut problem. To demonstrate the effectiveness of the NSS, we focused on 3-regular graphs, 9-regular graphs, and ER graphs. Numerical results indicate that the quantum algorithm achieves a comparable approximation ratio with classical methods by solving the original high dimensional problem within a lower dimensional ansatz space. Analysis under various initial states shows that the algorithm exhibits a remarkably stable performance.

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Abstract Let $G$ be a vertex-weighted connected graph of $n$ vertices and let $T$ be a spanning tree of $G$. We call $T$ a maximum weighted internal spanning tree of $G$ if the sum of the weights of the internal vertices of $T$ is the maximum over all spanning trees of $G$. The maximum weighted internal spanning tree (MaxwIST) problem asks to find such a spanning tree $T$ of $G$. The problem is NP-hard. We give an $O(dn)$ time approximation algorithm for $d$-regular graphs of $n=|V|$ vertices that computes a spanning tree with total weight of the internal vertices is at least $\frac{\beta _{d}}{\beta _{d} +d-2} - \epsilon $ of the total weight of all the vertices of the graph for any $\epsilon>0$, where $\beta _{d} = (d-1)H_{d-1}$, and $H_{d-1} = \sum _{i=1}^{d-1} i^{-1}$ is the $(d-1)$th harmonic number. For every $d \geq 3$ and $n_{0} \geq 1$, we show the construction of a $d$-regular graph of at least $n_{0}$ vertices, such that for any of its spanning trees, $\frac{w(I)}{w(V)}\le \frac{d}{d+1}$ holds. We give an $O(dn)$ time approximation algorithm for subdivisions of $d$-regular graphs, where the ratio of the internal weight of the spanning tree with the total vertex weight of the graph is at least $\frac{d-1}{2d-3} - \epsilon $ for $\epsilon>0$. We extend our study to $x$-subdivisions of Hamiltonian and hypoHamiltonian graphs, where each edge of the original Hamiltonian or hypoHamiltonian graph has been subdivided at least $x$ times. For those two graph classes, we show that there exists a spanning tree with internal vertex weight at least $1-\frac{2}{x-1}$ of the total vertex weight of the graph. Furthermore, we give $O(n)$ time algorithm for $x$-subdivisions of biconnected outerplanar graphs and $4$-connected planar graphs to achieve the above bound.

The continuous-time magnetic quantum walk and its probability invariance on a class of graphs

Let [Formula: see text] be a nonnegative integer and [Formula: see text] the power set of the set [Formula: see text]. Then there is an adjacency relation in [Formula: see text] such that [Formula: see text] together with the relation forms a regular graph. In this paper, we propose a model of continuous-time magnetic quantum walk (MQW) on the graph [Formula: see text], and investigate its properties from a viewpoint of probability and quantum information. We first introduce a magnetic Laplacian [Formula: see text] on the graph [Formula: see text] and examine its spectrum. And then, with [Formula: see text] as the Hamiltonian, we construct our model of continuous-time MQW on the graph [Formula: see text]. We find that the model has probability distributions that are completely independent of the magnetic potential at all times. And we show that it has perfect state transfer at time [Formula: see text] when the magnetic potential satisfies some mild conditions. Some other interesting results are also obtained.

Renormalization group analysis of the Anderson model on random regular graphs

We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.

Cube is a good form: Hyperspectral band selection via multi-dimensional and high-order structure preserved clustering

As an effective strategy for reducing the noisy and redundant information for hyperspectral imagery (HSI), hyperspectral band selection intends to select a subset of original hyperspectral bands, which boosts the subsequent different tasks. In this paper, we introduce a multi-dimensional high-order structure preserved clustering method for hyperspectral band selection, referred to as MHSPC briefly. By regarding original hyperspectral images as a tensor cube, we apply the tensor CP (CANDECOMP/PARAFAC) decomposition on it to exploit the multi-dimensional structural information as well as generate a low-dimensional latent feature representation. In order to capture the local geometrical structure along the spectral dimension, a graph regularizer is imposed on the new feature representation in the lower dimensional space. In addition, since the low rankness of HSIs is an important global property, we utilize a nuclear norm constraint on the latent feature representation matrix to capture the global data structure information. Different to most of previous clustering based hyperspectral band selection methods which vectorize each band as a vector without considering the 2-D spatial information, the proposed MHSPC can effectively capture the spatial structure as well as the spectral correlation of original hyperspectral cube in both local and global perspectives. An efficient alternatively updating algorithm with theoretical convergence guarantee is designed to solve the resultant optimization problem, and extensive experimental results on four benchmark datasets validate the effectiveness of the proposed MHSPC over other state-of-the-arts.

Hippocampal hub failure is linked to long-term memory impairment in anti-NMDA-receptor encephalitis: insights from structural connectome graph theoretical network analysis

BackgroundAnti-N-methyl-d-aspartate receptor (NMDAR) encephalitis is characterized by distinct structural and functional brain alterations, predominantly affecting the medial temporal lobes and the hippocampus. Structural connectome analysis with graph-based investigations of network properties allows for an in-depth characterization of global and local network changes and their relationship with clinical deficits in NMDAR encephalitis.MethodsStructural networks from 61 NMDAR encephalitis patients in the post-acute stage (median time from acute hospital discharge: 18 months) and 61 age- and sex-matched healthy controls (HC) were analyzed using diffusion-weighted imaging (DWI)-based probabilistic anatomically constrained tractography and volumetry of a selection of subcortical and white matter brain volumes was performed. We calculated global, modular, and nodal graph measures with special focus on default-mode network, medial temporal lobe, and hippocampus. Pathologically altered metrics were investigated regarding their potential association with clinical course, disease severity, and cognitive outcome.ResultsPatients with NMDAR encephalitis showed regular global graph metrics, but bilateral reductions of hippocampal node strength (left: p = 0.049; right: p = 0.013) and increased node strength of right precuneus (p = 0.013) compared to HC. Betweenness centrality was decreased for left-sided entorhinal cortex (p = 0.042) and left caudal middle frontal gyrus (p = 0.037). Correlation analyses showed a significant association between reduced left hippocampal node strength and verbal long-term memory impairment (p = 0.021). We found decreased left (p = 0.013) and right (p = 0.001) hippocampal volumes that were associated with hippocampal node strength (left p = 0.009; right p < 0.001).ConclusionsFocal network property changes of the medial temporal lobes indicate hippocampal hub failure that is associated with memory impairment in NMDAR encephalitis at the post-acute stage, while global structural network properties remain unaltered. Graph theory analysis provides new pathophysiological insight into structural network changes and their association with persistent cognitive deficits in NMDAR encephalitis.

Self-representation with adaptive loss minimization via doubly stochastic graph regularization for robust unsupervised feature selection

Self-representation with adaptive loss minimization via doubly stochastic graph regularization for robust unsupervised feature selection

Some Two-Weight Codes Over Chain Rings and their Strongly Regular Graphs

Some Two-Weight Codes Over Chain Rings and their Strongly Regular Graphs

On an infinite family of integral Cayley graphs of Pauli groups

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph Cay(Pn,SPn) of the generalized Pauli group Pn on n-qubits; in fact, Pn may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set SPn allows us to recognize the structure of central product of graphs in Cay(Pn,SPn).Using this approach, we are able to recursively construct the adjacency matrix of Cay(Pn,SPn) for each n≥1, and to explicitly describe its spectrum and the associated eigenvectors. It turns out that Cay(Pn,SPn) is a (3n+2)-regular bipartite graph on 4n+1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral.

Sparse sampling photoacoustic reconstruction with a graph regularization group sparse dictionary

Photoacoustic tomography (PAT) has emerged as a promising biomedical imaging technique. The combination of optical contrast and ultrasound spatial resolution in photoacoustic tomography overcomes the limitations of optical scattering, enabling clear imaging of tissue structures. However, achieving high-resolution photoacoustic images typically requires a large number of sensor detection elements for sufficient angular coverage. This demand for extensive data acquisition and processing raises concerns about efficiency and system complexity. While sparse sampling strategies can improve efficiency, preserving detailed structural information becomes challenging with a minimal number of detectors. To address the challenges of sparse sampling, compressed sensing (CS) techniques have been successfully applied for image reconstructions in 2D and 3D photoacoustic embodiments. In this context, we propose a joint graph regularization group sparse dictionary and total variational regularization (GRGS-TV) algorithm based on our previous work of a group sparse dictionary. It preserves structured information and geometric relationships among dictionary atoms. Moreover, TV regularization effectively preserves edge structures while exhibiting a certain degree of robustness and flexibility. Numerical simulations and in vivo experiments on mice validate the effectiveness of this method in improving photoacoustic image quality and suppressing artifacts. Comparative evaluations against other algorithms show enhanced performance in terms of image reconstruction evaluation indices. This innovative approach holds promise for advancing photoacoustic imaging in biomedical research and clinical diagnostics.