Regular Graphs Research Articles

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.

A complexity trichotomy for k-regular asymmetric spin systems with complex edge functions

We prove a complexity trichotomy theorem for a class of partition functions over k-regular graphs, where the signature is complex valued and not necessarily symmetric. In details, we establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: For every parameter setting in C for the spin system, the partition function is either (1) computable in polynomial time for every graph, or (2) #P-hard for general graphs but computable in polynomial time for planar graphs, or (3) #P-hard even for planar graphs.

Pink-noise dynamics in an evolutionary game on a regular graph.

We consider a multiplayer prisoner's dilemma game on a square lattice and regular graphs based on the pairwise-Fermi update rule, and we obtain heatmaps of the fraction of cooperators and the correlation of neighboring pairs. In the heatmap, we find a mixed region where cooperators and defectors coexist, and the correlation between neighbors is significantly enhanced. Moreover, we observe pink-noise behavior in the mixed region, where the power spectrum can be fitted by a power-law function of frequency. We also find that the pink-noise behavior can be reproduced in a simple random-walk model. In particular, we propose a modified random-walk model which can reproduce not only the pink-noise behavior but also the deviation from it observed in a low-frequency region.

The Lower Bound for Number of Hexagons in Strongly Regular Graphs with Parameters λ = 1 and μ = 2

The Lower Bound for Number of Hexagons in Strongly Regular Graphs with Parameters λ = 1 and μ = 2

An efficient iterative method for solving the graph regularization Q-weighted nonnegative matrix factorization problem in multi-view clustering

In this paper, we consider the graph regularization Q-weighted nonnegative matrix factorization problem in multi-view clustering. Based on the Q-weighted norm property, this problem is transformed into the minimization problem of the trace function. The necessary condition for the existence of a solution is given. The proximal alternating nonnegative least squares method and its acceleration method are designed to solve it. The convergence theorem is also given. The feasibility and effectiveness of the proposed methods are verified by numerical experiments.

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.