Incomplete Graph Research Articles

Tensor Completion-Based Incomplete Multiview Clustering.

Incomplete multiview clustering is a challenging problem in the domain of unsupervised learning. However, the existing incomplete multiview clustering methods only consider the similarity structure of intraview while neglecting the similarity structure of interview. Thus, they cannot take advantage of both the complementary information and spatial structure embedded in similarity matrices of different views. To this end, we complete the incomplete graph with missing data referring to tensor complete and present a novel and effective model to handel the incomplete multiview clustering task. To be specific, we consider the similarity of the interview graphs via the tensor Schatten p -norm-based completion technique to make use of both the complementary information and spatial structure. Meanwhile, we employ the connectivity constraint for similarity matrices of different views such that the connected components approximately represent clusters. Thus, the learned entire graph not only has the low-rank structure but also well characterizes the relationship between unmissing data. Extensive experiments show the promising performance of the proposed method comparing with several incomplete multiview approaches in the clustering tasks.

Consensus of Positive Networked Systems on Directed Graphs.

This article addresses the distributed consensus problem for identical continuous-time positive linear systems with state-feedback control. Existing works of such a problem mainly focus on the case where the networked communication topologies are of either undirected and incomplete graphs or strongly connected directed graphs. On the other hand, in this work, the communication topologies of the networked system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Specifically, the problem involves complex eigenvalues, the Hurwitzness of complex matrices, and positivity constraints, which make analysis difficult in the Laplacian matrix. First, a necessary and sufficient condition for the consensus analysis of directed networked systems with positivity constraints is given, by using positive systems theory and graph theory. Unlike the general Riccati design methods that involve solving an algebraic Riccati equation (ARE), a condition represented by an algebraic Riccati inequality (ARI) is obtained for the existence of a solution. Subsequently, an equivalent condition, which corresponds to the consensus design condition, is derived, and a semidefinite programming algorithm is developed. It is shown that, when a protocol is solved by the algorithm for the networked system on a specific communication graph, there exists a set of graphs such that the positive consensus problem can be solved as well.

Multi‐view subspace clustering with incomplete graph information

AbstractThe core of multi‐view clustering is how to exploit the shared and specific information of multi‐view data properly. The data missing and incompleteness bring great challenges to multi‐view clustering. In this paper, we propose an innovative multi‐view subspace clustering method with incomplete graph information, so‐called incomplete multiple graphs clustering. Specifically, we creatively separate one shared and multiple specific graphs from multiple raw graph data, and exploit the mask fusion strategy and block diagonal regulariser to obtain the inherent category information. To handle the incomplete multiple graph data, we utilise multiple indicator matrices to mark the missing elements existed in each raw graph. In addition, the weight of each raw graph is adaptively learnt according to the graph importance. The alternative direction optimization algorithm is employed to solve our proposed methods. Finally, we also analyse the algorithm convergence and the computation complexity in detail. The clustering results on six real‐world datasets show that our method obviously outperforms a serious of classic incomplete multi‐view clustering methods.

Positive Consensus of Directed Multiagent Systems

This technical note investigates the positive consensus problem of multiagent systems with directed communication topologies, where all the agents have identical continuous-time positive linear dynamics. Existing works of such a problem mainly focus on the case, where networked communication topologies are of either undirected and incomplete graphs, or strongly connected directed graphs. In contrast to them, we study this problem in which the communication topologies of the multiagent system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Based on the existing results in spectral graph theory and positive linear systems theory, several necessary, and sufficient conditions on positive consensus of the directed multiagent system are derived through using linear matrix inequality techniques. A primal-dual iterative algorithm is developed for the computation of solutions. Finally, several numerical simulations are provided to illustrate the effectiveness of the proposed theoretical results.

Inferring missing edges in a graph from observed collective patterns.

Many real-life networks are incomplete. Dynamical observations can allow estimating missing edges. Such procedures, often summarized under the term 'network inference', typically evaluate the statistical correlations among pairs of nodes to determine connectivity. Here, we offer an alternative approach: completing an incomplete network by observing its collective behavior. We illustrate this approach for the case of patterns emerging in reaction-diffusion systems on graphs, where collective behaviors can be associated with eigenvectors of the network's Laplacian matrix. Our method combines a partial spectral decomposition of the network's Laplacian matrix with eigenvalue assignment by matching the patterns to the eigenvectors of the incomplete graph. We show that knowledge of a few collective patterns can allow the prediction of missing edges and that this result holds across a range of network architectures. We present a numerical case study using activator-inhibitor dynamics and we illustrate that the main requirement for the observed patterns is that they are not confined to subsets of nodes, but involve the whole network.

Rough maximal cliques enumeration in incomplete graphs based on partially-known concept learning

The emerging massive noisy and incomplete data is transforming the conventional graph to the uncertain graph. In this paper, we study rough maximal cliques enumeration (RMCE) in incomplete graphs, which we define as the novel problem of enumerating all maximal cliques where some of edges are unknown to users. The hardness of RMCE is proved to NP-complete. To tackle this problem, an efficient framework for obtaining the rough maximal cliques based on Partially-Known Concept Learning (PKCL). With this framework, a given incomplete graph is initially represented as an incomplete formal context. Then, the partially-known SE-ISI concept lattice is generated through the constructed incomplete formal context. Based on the constructed SE-ISI concept lattice, an equivalence theorem between SE-ISI equiconcepts and rough maximal cliques is presented. The detailed topological structural analysis from the point of views of roughness and SE-ISI concept stability of rough maximal cliques are separately discussed. The evaluation results demonstrate that our proposed PKCL algorithm can better identify the rough maximal cliques under different probability distribution models of links compared to the existing baseline algorithms.

Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems

This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel matrix formulation for the HCP and TSP Hamiltonians, which enables the seamless and automatic integration of benchmark instances in quantum platforms. We also present a thorough mathematical analysis of the precise number of constraints required to express the HCP and TSP Hamiltonians. This analysis explains quantitatively why, almost always, running incomplete graph instances requires more qubits than complete instances. It turns out that QUBO models for incomplete graphs require more quadratic constraints than complete graphs, a fact that has been corroborated by a series of experiments. Moreover, we introduce a technique for the min-max normalization for the coefficients of the TSP Hamiltonian to address the problem of invalid solutions produced by the quantum annealer, a trend often observed. Our extensive experimental tests have demonstrated that the D-Wave Advantage_system4.1 is more efficient than the Advantage_system1.1, both in terms of qubit utilization and the quality of solutions. Finally, we experimentally establish that the D-Wave hybrid solvers always provide valid solutions, without violating the given constraints, even for arbitrarily big problems up to 120 nodes.

Real-Time Modulation of Physical Training Intensity Based on Wavelet Recursive Fuzzy Neural Networks.

In this study, a wavelet recurrent fuzzy neural network is used to conduct in-depth research and analysis on the real-time regulation of physical training intensity. Firstly, an inter-process control technique is proposed to solve the problem of incomplete control flow graph construction caused by the inability to effectively collect all program control flow information in the process of static analysis, in preparation for the research of fuzzy testing technique. Next, a wavelet recursive fuzzy neural network-guided fuzzy testing technique is proposed to solve the problem of fuzzy tests falling into invalid variation due to the lack of directionality in the fuzzy testing process. Each neuron in the feedforward network is divided into different groups according to the order of receiving information. Each group can be regarded as a neural layer. The neurons in each layer receive the output of the neurons in the previous layer and output to the neurons in the next layer. The empirical data show that injury-preventive fitness training can effectively improve all physical qualities in the first phase of preparation and can effectively maintain the physical state and effectively contribute to their abilities during the competition period, and its injury-preventive fitness training interventions were verified by statistical analysis to have a dangerous main effect on their pre and post-test performance. Therefore, it is still not possible to determine its correlation with the coordination and improvement of the athletes' physical fitness, and the integration of the basic physical training and rehabilitation physical training systems, making this theory a new special training theory.

Semisupervised Classification with High-Order Graph Learning Attention Neural Network

At present, the graph neural network has achieved good results in the semisupervised classification of graph structure data. However, the classification effect is greatly limited in those data without graph structure, incomplete graph structure, or noise. It has no high prediction accuracy and cannot solve the problem of the missing graph structure. Therefore, in this paper, we propose a high-order graph learning attention neural network (HGLAT) for semisupervised classification. First, a graph learning module based on the improved variational graph autoencoder is proposed, which can learn and optimize graph structures for data sets without topological graph structure and data sets with missing topological structure and perform regular constraints on the generated graph structure to make the optimized graph structure more reasonable. Then, in view of the shortcomings of graph attention neural network (GAT) that cannot make full use of the graph high-order topology structure for node classification and graph structure learning, we propose a graph classification module that extends the attention mechanism to high-order neighbors, in which attention decays according to the increase of neighbor order. HGLAT performs joint optimization on the two modules of graph learning and graph classification and performs semisupervised node classification while optimizing the graph structure, which improves the classification performance. On 5 real data sets, by comparing 8 classification methods, the experiment shows that HGLAT has achieved good classification results on both a data set with graph structure and a data set without graph structure.

Multistep Traffic Speed Prediction from Spatial–Temporal Dependencies Using Graph Neural Networks

Accurate traffic forecasting on citywide networks is one of the crucial urban data mining applications that accurately provide congestion warning and transportation scheduling. While previous work has made significant efforts to learn traffic temporal dynamics and spatial dependencies, two key limitations exist in current models: (1) most existing approaches solely capture spatial correlations among neighbors on predefined graph structure, and genuine relation may be missing due to the incomplete graph connections; and (2) existing methods are defective to capture the temporal trends because the recurrent and stacking structure employed in these methods suffers from the long-range temporal dependency vanquish problem. To overcome the difficulty in multistep prediction and further capture the dynamic spatial–temporal dependencies of traffic flows, we propose a new traffic speed prediction framework for multiscale graph attention networks (MS-GATNs). In particular, MS-GATNs is a hierarchically structured graph neural architecture that learns not only the local region-wise geographical dependencies but also the spatial semantics from a global perspective. Furthermore, a multiheads attention mechanism is introduced to empower our model with the capability of capturing complex nonstationary temporal dynamics. Experiments on real-world traffic data sets demonstrate that MS-GATNs outperforms the state-of-the-art baselines in long-term forecasting.

Incomplete multiview nonnegative representation learning with multiple graphs

Multiview clustering has become an important research topic during the past decade. However, partial views of many data instances are missing in some realistic multiview learning scenarios. To handle this problem, we develop an effective incomplete multiview nonnegative representation learning (IMNRL) framework, which is suitable for incomplete multiview clustering in various situations. The IMNRL framework performs matrix factorization on multiple incomplete graphs and decomposes these incomplete graphs into a consensus nonnegative representation and view-specific spectral representations, which integrates the advantages of multiview nonnegative representation learning and graph learning. The proposed framework has the following merits: (1) it learns a consensus nonnegative embedding and view-specific embeddings simultaneously; (2) the nonnegative embedding satisfies the neighbor constraint on each incomplete view, which directly reveals the multiview clustering results. Experimental results show that the proposed framework outperforms other state-of-the-art incomplete multiview clustering algorithms.

A complete restricted Boltzmann machine on an adiabatic quantum computer

Boltzmann Machines constitute a paramount class of neural networks for unsupervised learning and recommendation systems. Their bipartite version, called Restricted Boltzmann Machine (RBM), is the most developed because of its satisfactory trade-off between computability on classical computers and computational power. Though the diffusion of RBMs is quite limited as their training remains hard. Recently, a renewed interest has emerged as Adiabatic Quantum Computers (AQCs), which suggest a potential increase of the training speed with respect to conventional hardware. Due to the limited number of connections among the qubits forming the graph of existing hardware, associating one qubit per node of the neural network implies an incomplete graph. Thanks to embedding techniques, we developed a complete graph connecting nodes constituted by virtual qubits. The complete graph outperforms previous implementations based on incomplete graphs. Despite the fact that the learning rate per epoch is still slower with respect to a classical machine, the advantage is expected by the increase of number of nodes which impacts on the classical computational time but not on the quantum hardware based computation.

On the Convergence Properties of Social Hegselmann–Krause Dynamics

We study the convergence properties of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">social Hegselmann–Krause (HK) dynamics</i> , a variant of the HK model of opinion dynamics where a physical connectivity graph that accounts for the extrinsic factors that could prevent interaction between certain pairs of agents is incorporated. As opposed to the original HK dynamics (which terminates in finite time), we show that for any underlying connected and incomplete graph, under a certain mild assumption, the expected termination time of social HK dynamics is infinity. We then investigate the rate of convergence to the steady state, and provide bounds on the maximum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -convergence time in terms of the properties of the physical connectivity graph. We extend this discussion and observe that for almost all <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> , there exists an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -vertex physical connectivity graph on which social HK dynamics may not even <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -converge to the steady state within a bounded time frame. We then provide nearly tight necessary and sufficient conditions for arbitrarily slow merging (a phenomenon that is essential for arbitrarily slow <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -convergence to the steady state). Using the necessary conditions, we show that complete <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$r$</tex-math></inline-formula> -partite graphs have bounded <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -convergence times.

Incomplete Multi-view Clustering with Joint Partition and Graph Learning

Incomplete multi-view clustering (IMC) aims to integrate the complementary information from incomplete views to improve clustering performance. Most existing IMC methods try to fill the incomplete views or directly learn a common representation based on matrix factorization or subspace learning. The former may introduce useless even noisy information especially for data with a large missing ratio. The latter relies on the initialization and ignores the geometric structure of data. To address these issues, we propose a novel Joint Partition and Graph (JPG) learning method for IMC. Specifically, JPG jointly constructs local incomplete graph matrices, generates incomplete base partition matrices, stretches them to produce a unified partition matrix, and employs it to learn a consensus graph matrix. By this means, we transform incomplete multi-view data into a unified partition space and obtain the consensus graph in a mutual reinforcement manner. Moreover, a partition fusion strategy can allocate a large weight to the stretched base partition that is close to the unified matrix. The objective function is optimized in an alternating optimization fashion. Experimental results on several benchmark datasets demonstrate the effectiveness and superiority of JPG than the state-of-the-art baselines

Mixed Membership Graph Clustering via Systematic Edge Query

This work considers clustering nodes of a largely incomplete graph. Under the problem setting, only a small amount of queries about the edges can be made, but the entire graph is not observable. This problem finds applications in large-scale data clustering using limited annotations, community detection under restricted survey resources, and graph topology inference under hidden/removed node interactions. Prior works tackled this problem from various perspectives, e.g., convex programming-based low-rank matrix completion and active query-based clique finding. Nonetheless, many existing methods are designed for estimating the single-cluster membership of the nodes, but nodes may often have mixed (i.e., multi-cluster) membership in practice. Some query and computational paradigms, e.g., the random query patterns and nuclear norm-based optimization advocated in the convex approaches, may give rise to scalability and implementation challenges. This work aims at learning mixed membership of nodes using queried edges. The proposed method is developed together with a systematic query principle that can be controlled and adjusted by the system designers to accommodate implementation challenges -- e.g., to avoid querying edges that are physically hard to acquire. Our framework also features a lightweight and scalable algorithm with membership learning guarantees. Real-data experiments on crowdclustering and community detection are used to showcase the effectiveness of our method.

Network Topology Inference From Heterogeneous Incomplete Graph Signals

Inferring network topologies from observed graph-structured data (also known as graph signals) is a crucial task in many applications of network science. Existing papers on network topology inference typically assume that the observations at all nodes are available. However, there are many situations where only partial observations can be collected due to application-specific constraints. To handle the missing data problem, we propose a framework that relies on heterogeneous incomplete data from a collection of related networks to identify multiple network topologies simultaneously. This work advocates the Gaussian graphical model (GGM) and casts the topology inference problem in terms of estimating the precision matrix that has a form of graph Laplacian. Firstly, an unbiased estimator for the covariance matrix of incomplete data is established and then algorithms based on the alternating direction method of multipliers (ADMM) are developed to jointly estimate graph topologies, rather than estimate each graph topology separately. So that we can borrow information across the multiple related networks to eliminate the impact of missing data. Moreover, non-asymptotic statistical analysis is provided, which proves the consistency of the graph estimator and enables us to investigate the effect of several key factors on the graph estimation error bound. Furthermore, based on the consistent graph estimator, an adaptive algorithm that utilizes the reweighting scheme is proposed to improve the estimation accuracy of the graph-edge structure. Finally, we evaluate our method on both real and synthetic datasets, and the experimental results demonstrate the advantage of our method in comparison with benchmarking algorithms.

High performance GPU primitives for graph-tensor learning operations

Graph-tensor learning operations extend tensor operations by taking the graph structure into account, which have been applied to diverse domains such as image processing and machine learning. However, the running time of graph-tensor operations increases rapidly with the number of nodes and the dimension of data on nodes, making them impractical for real-time applications. In this paper, we propose a GPU library called cuGraph-Tensor for high-performance graph-tensor learning operations, which consists of eight key operations: graph shift (g-shift), graph Fourier transform (g-FT), inverse graph Fourier transform (inverse g-FT), graph filter (g-filter), graph convolution (g-convolution), graph-tensor product (g-product), graph-tensor SVD (g-SVD) and graph-tensor QR (g-QR). cuGraph-Tensor supports scalar, vector, and matrix data processing on each graph node. We propose optimization techniques on computing, memory accesses, and CPU–GPU communications that significantly improve the performance of the graph-tensor learning operations. Using the optimized operations, cuGraph-Tensor builds a graph data completion application for fast and accurate reconstruction of incomplete graph data. In the experiments, the proposed graph learning operations achieve up to 142.12× speedups versus CPU-based GSPBOX and CPU MATLAB implementations running on two Xeon CPUs. The graph data completion application achieves up to 174.38× speedups over the CPU MATLAB implementation, and up to 3.82× speedups with better accuracy over the GPU-based tensor completion in the cuTensor-tubal library.

Adaptive Weighted Graph Fusion Incomplete Multi-View Subspace Clustering.

With the enormous amount of multi-source data produced by various sensors and feature extraction approaches, multi-view clustering (MVC) has attracted developing research attention and is widely exploited in data analysis. Most of the existing multi-view clustering methods hold on the assumption that all of the views are complete. However, in many real scenarios, multi-view data are often incomplete for many reasons, e.g., hardware failure or incomplete data collection. In this paper, we propose an adaptive weighted graph fusion incomplete multi-view subspace clustering (AWGF-IMSC) method to solve the incomplete multi-view clustering problem. Firstly, to eliminate the noise existing in the original space, we transform complete original data into latent representations which contribute to better graph construction for each view. Then, we incorporate feature extraction and incomplete graph fusion into a unified framework, whereas two processes can negotiate with each other, serving for graph learning tasks. A sparse regularization is imposed on the complete graph to make it more robust to the view-inconsistency. Besides, the importance of different views is automatically learned, further guiding the construction of the complete graph. An effective iterative algorithm is proposed to solve the resulting optimization problem with convergence. Compared with the existing state-of-the-art methods, the experiment results on several real-world datasets demonstrate the effectiveness and advancement of our proposed method.

NOTE ON MINIMUM DEGREE AND PROPER CONNECTION NUMBER

AbstractAn edge-coloured graph $G$ is called properly connected if any two vertices are connected by a properly coloured path. The proper connection number, $pc(G)$, of a graph $G$, is the smallest number of colours that are needed to colour $G$ such that it is properly connected. Let $\unicode[STIX]{x1D6FF}(n)$ denote the minimum value such that $pc(G)=2$ for any 2-connected incomplete graph $G$ of order $n$ with minimum degree at least $\unicode[STIX]{x1D6FF}(n)$. Brause et al. [‘Minimum degree conditions for the proper connection number of graphs’, Graphs Combin.33 (2017), 833–843] showed that $\unicode[STIX]{x1D6FF}(n)&gt;n/42$. In this note, we show that $\unicode[STIX]{x1D6FF}(n)&gt;n/36$.

Multi-point shortest path planning based on an Improved Discrete Bat Algorithm

Multi-point shortest path planning problem is a typical problems in discrete optimization. The bat algorithm is a nature-inspired metaheuristic optimization algorithm that is used in a wide range of fields. However, there is one problem with the BA, which is easy to premature. To solve multi-point shortest path planning problem, an improved discrete bat algorithm (IDBA) is proposed in this paper. In this algorithm, the Floyd–Warshall algorithm is first used to transform an incomplete connected graph into a complete graph whose vertex set consists of a start point and necessary points. Then the algorithm simulates the bats’ foraging and obstacle avoidance process to find the shortest path in the complete graph to satisfy the constraints. Finally, the path is transferred to the original incomplete graph to get the solution. In order to overcome the premature phenomenon of a discrete bat algorithm, the modified neighborhood operator is proposed. To prove the effectiveness of our method, we compared its performance in 26 instances with the results obtained by three different algorithms: DBA, IBA and GSA-ACS-PSOT. We also performed a sensitivity analysis on the parameters. The results indicate that the improved bat algorithm outperforms all the other alternatives in most cases.