Sparse Graphs Research Articles

Analysis and comparison of two classical greedy algorithms for minimum spanning tree

The total weight of the minimum spanning is the smallest in the connected graph. It can be used to solve many practical problems in urban life. Prims algorithm and Kruskals algorithm are greedy algorithms for solving the minimum spanning tree problem. But they make greedy choices in different ways. The paper focuses on two greedy algorithms for solving the minimum spanning tree problem. The author will evaluate each algorithms complexity and determine their most suitable condition as well. The author compares the running process of the two algorithms and analyzes the relationship between their algorithm complexity and the number of edges, which can describe the sparsity of the graphs. The result shows that Kruskals complexity is related to the number of edges, so it is better for sparse graphs. Prims complexity is related to the number of vertices, so it is better at analyzing connected graphs with lots of vertices, that is, dense graphs. As a consequence, Prims algorithm is better for dense graphs.

Collaborative bi-aggregation for directed graph embedding

Directed graph is able to model asymmetric relationships between nodes and research on directed graph embedding is of great significance in downstream graph analysis and inference. Learning source and target embeddings of nodes separately to preserve edge asymmetry has become the dominant approach, but also poses challenge for learning representations of low or even zero in/out degree nodes that are ubiquitous in sparse graphs. In this paper, a collaborative bi-directional aggregation method (COBA) for directed graph embedding is proposed. Firstly, the source and target embeddings of the central node are learned by aggregating from the counterparts of the source and target neighbors, respectively; Secondly, the source/target embeddings of the zero in/out degree central nodes are enhanced by aggregating the counterparts of opposite-directional neighbors (i.e. target/source neighbors); Finally, source and target embeddings of the same node are correlated to achieve collaborative aggregation. Both the feasibility and rationality of the model are theoretically analyzed. Extensive experiments on real-world datasets demonstrate that COBA comprehensively outperforms state-of-the-art methods on multiple tasks and meanwhile validates the effectiveness of proposed aggregation strategies.

Singularity of the k-core of a random graph

Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix) due to the presence of “low-degree dependencies” such as isolated vertices and pairs of degree 1 vertices with the same neighborhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants k≥3 and λ>0, an Erdős–Rényi random graph G∼G(n,λ∕n) with n vertices and edge probability λ∕n typically has the property that its k-core (its largest subgraph with minimum degree at least k) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for “extremely sparse” random matrices with density O(1∕n). A key aspect of our proof is a technique to extract high-degree vertices and use them to “boost” the rank, starting from approximate rank bounds obtainable from (nonquantitative) spectral convergence machinery due to Bordenave, Lelarge, and Salez.

Application of graph neural network and feature information enhancement in relation inference of sparse knowledge graph

At present, knowledge embedding methods are widely used in the field of knowledge graph (KG) reasoning, and have been successfully applied to those with large entities and relationships. However, in research and production environments, there are a large number of KGs with a small number of entities and relations, which are called sparse KGs. Limited by the performance of knowledge extraction methods or some other reasons (some common-sense information does not appear in the natural corpus), the relation between entities is often incomplete. To solve this problem, a method of the graph neural network and information enhancement is proposed. The improved method increases the mean reciprocal rank (MRR) and Hit@3 by 1.6% and 1.7%, respectively, when the sparsity of the FB15K-237 dataset is 10%. When the sparsity is 50%, the evaluation indexes MRR and Hit@10 are increased by 0.8% and 1.8%, respectively.

An Efficient Compressive Sensing Event-Detection Scheme for Internet of Things System Based on Sparse-Graph Codes

This work studied the event-detection problem in an Internet of Things (IoT) system, where a group of sensor nodes are placed in the region of interest to capture sparse active event sources. Using compressive sensing (CS), the event-detection problem is modeled as recovering the high-dimensional integer-valued sparse signal from incomplete linear measurements. We show that the sensing process in IoT system produces an equivalent integer CS using sparse graph codes at the sink node, for which one can devise a simple deterministic construction of a sparse measurement matrix and an efficient integer-valued signal recovery algorithm. We validated the determined measurement matrix, uniquely determined the signal coefficients, and performed an asymptotic analysis to examine the performance of the proposed approach, namely event detection with integer sum peeling (ISP), with the density evolution method. Simulation results show that the proposed ISP approach achieves a significantly higher performance compared to existing literature at various simulation scenario and match that of the theoretical results.

Order-disorder transition in the zero-temperature Ising model on random graphs.

The zero-temperature Ising model is known to reach a fully ordered ground state in sufficiently dense random graphs. In sparse random graphs, the dynamics gets absorbed in disordered local minima at magnetization close to zero. Here, we find that the nonequilibrium transition between the ordered and the disordered regime occurs at an average degree that slowly grows with the graph size. The system shows bistability: The distribution of the absolute magnetization in the reached absorbing state is bimodal, with peaks only at zero and unity. For a fixed system size, the average time to absorption behaves nonmonotonically as a function of average degree. The peak value of the average absorption time grows as a power law of the system size. These findings have relevance for community detection, opinion dynamics, and games on networks.

Weighted non-negative matrix factorization based on adaptive robust local sparse graph

Weighted non-negative matrix factorization based on adaptive robust local sparse graph

Temporal graph patterns by timed automata

Temporal graphs represent graph evolution over time, and have been receiving considerable research attention. Work on expressing temporal graph patterns or discovering temporal motifs typically assumes relatively simple temporal constraints, such as journeys or, more generally, existential constraints, possibly with finite delays. In this paper we propose to use timed automata to express temporal constraints, leading to a general and powerful notion of temporal basic graph pattern (BGP). The new difficulty is the evaluation of the temporal constraint on a large set of matchings. An important benefit of timed automata is that they support an iterative state assignment, which can be useful for early detection of matches and pruning of non-matches. We introduce algorithms to retrieve all instances of a temporal BGP match in a graph, and present results of an extensive experimental evaluation, demonstrating interesting performance trade-offs. We show that an on-demand algorithm that processes total matchings incrementally over time is preferable when dealing with cyclic patterns on sparse graphs. On acyclic patterns or dense graphs, and when connectivity of partial matchings can be guaranteed, the best performance is achieved by maintaining partial matchings over time and allowing automaton evaluation to be fully incremental. The code and datasets used in our analysis are available at http://github.com/amirpouya/TABGP.

Continuum limit ofp-Laplacian evolution problems on graphs:Lqgraphons and sparse graphs

In this paper we study continuum limits of the discretizedp-Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This goes far beyond known results by handling much more general class of kernels, possibly singular, and graph sequences whose limit are the so-calledLq-graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e., with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of thep-Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.

WITHDRAWN: Application of graph neural network and feature information enhancement in relation inference of sparse knowledge graph

WITHDRAWN: Application of graph neural network and feature information enhancement in relation inference of sparse knowledge graph

FMRI-Based Brain Disease Diagnosis: A Graph Network Approach

As a non-invasive brain imaging technology, functional magnetic resonance imaging (fMRI) provides a basic tool for brain functional network modeling and brain disease diagnosis. Problems, such as large number of parameters, low training efficiency, and poor interpretability, are encountered in mainstream models because of the high complexity of fMRI and brain networks. To solve these problems, a novel structure feature combined graph neural network (SFC-GNN) with a low number of parameters is proposed. In particular, SFC-GNN is composed of 1) the graph convolution layer of the brain region perception and 2) the node pooling layer of the graph structure feature (GSF). It also receives the sparse brain graph modeled by each subject’s fMRI as input. Especially, the GSF layer can select brain regions that are important for classification, thereby localizing all active regions related to brain disease. Moreover, a group network is constructed according to the correlation among subjects, and SFC-GNN can be extended further to a node classification model to achieve better diagnosis performance. The proposed method has been validated on the ABIDE and ADNI datasets, thereby showing the effectiveness of our proposed method in various experiments.

Internet of Vehicles Data-Oriented Arterial Travel Time Estimation Framework With Dynamic Multigraph Model

The rapid development of Internet of Vehicles (IoV) data powers various online intelligent transportation applications, such as network travel time reporting. However, the accuracy might be severely compromised due to limited probe vehicle sampling frequency. On that account, this article ­proposes a dynamic multigraph model-enabled framework to estimate reliable network travel time, even in low-IoV-frequency arterial corridors. The proposed framework first develops an improved sparse IoV travel time decomposition method. The segment travel time is further divided into the free-flow running time and static and dynamic delays. Second, a dynamic multigraph traffic network model (DMGTN) is developed to aid the proposed decomposition method. The model analyzes complicated spatiotemporal relevance between segments from multiple perspectives: the real-time travel time, congestion level, signal control (which is frequently neglected in previous research), and segment properties. Additionally, two distinct enhanced modules are designed for handing dense and sparse network graphs, respectively. This allows for a more efficient inspection over large-scale intricate arterial networks while maintaining precision. Field implementation is conducted in the downtown area of Zhangzhou, China. Compared to other high-performance baseline models, the designed DMGTN model as well as the proposed decomposition method demonstrate state-of-the-art accuracy and successfully capture travel time variability. The proposed framework better utilizes available IoV data to provide valuable traffic information for commuters and traffic management agencies.

Sparse-Representation-Network-Based Feature Learning of Vibration Signal for Machinery Fault Diagnosis

Although deep neural networks (DNNs) have been widely applied in machinery fault diagnosis, the key problem about the impulsive component extraction and noise filtering in the learning procedure are not addressed very well. Thus, sparse representation network (SRNet) is developed to extract impulses from collected signals and then used for machinery fault recognition. For purpose of improving the feature extraction capacity of SRNet, convolutional sparse graph is developed in sparse representation (SR) layer to suppress noise and reserve impulsive characteristics of signals. A selective residual learning is developed to effectively optimize gradient propagation and further enhance the feature learning performance of SRNet. Finally, the feature learning and fault classification capacity of SRNet is evaluated on two gearbox cases. The recognition accuracy of SRNet are 98.72% and 99.68% on two cases, respectively, demonstrating the effectiveness of SRNet compared with other DNNs.

BP-Based Sparse Graph List Decoding of Polar Codes

How to construct an effective polar decoding scheme has attracted researchers in the field of communication. The belief propagation list (BPL) decoder has performance improvement over the traditional BP decoder but comes with much higher complexity. To solve the issue of high complexity & latency, a low-density parity-check (LDPC) like BP decoder was proposed but it suffered from performance degradation over the original BP decoder. In this paper, a BP-based sparse graph list (BP-SGL) decoder is proposed by leveraging both list decoding scheme and LDPC-like BP decoding algorithm to achieve performance improvement while maintaining low complexity & latency. The key idea of the proposed list generation method is the similarity comparison of decoding graphs. Testing results verify that selecting graphs with large structural differences helps to construct a list with good overall performance. Simulation results show that the proposed scheme is superior to LDPC-like BP, and even outperforms the original BPL and some state-of-the-art (SOTA) BP-based decoding algorithms with significant reduction in complexity & latency.

Feature preserving 3D mesh denoising with a Dense Local Graph Neural Network

Graph neural networks (GNNs) are ideally suited for mesh denoising. However, existing solutions such as those based on graph convolutional networks (GCNs) are built for a fixed graph thus making them not naturally generalizable to unseen meshes. Furthermore, their graph Laplacian based global node embedding algorithms can cause excessive smoothing while achieving feature preserving mesh denoising requires a GNN to possess local processing capability. This paper presents a mesh denoising method via a new Dense lOcal Graph neural NETwork (DOGNET). DOGNET implements a local node embedding algorithm that generates node embeddings through aggregating information from a node’s connected local neighbours which automatically make DOGNET inductive as well as effective for feature preserving mesh denoising. We present extensive experimental results to demonstrate quantitatively and qualitatively that DOGNET is superior to SOTA meshing denoising techniques.

Stirling Numbers of Uniform Trees and Related Computational Experiments

The Stirling numbers for graphs provide a combinatorial interpretation of the number of cycle covers in a given graph. The problem of generating all cycle covers or enumerating these quantities on general graphs is computationally intractable, but recent work has shown that there exist infinite families of sparse or structured graphs for which it is possible to derive efficient enumerative formulas. In this paper, we consider the case of trees and forests of a fixed size, proposing an efficient algorithm based on matrix algebra to approximate the distribution of Stirling numbers. We also present a model application of machine learning to enumeration problems in this setting, demonstrating that standard regression techniques can be applied to this type of combinatorial structure.

Graph Traversal (GT) on Linear and Quadratic Graphic Processing Units with CPU

Nowadays, Graph Traversal (GT) is a fundamental procedure for developing a measure in computing systems and algorithms. Graph traversal algorithms usually iterate input datasets of a graph to provide a logical convergence in every iteration. Graphic Processing Units (GPUs) have been utilized broadly as accelerators of GT due to their massive parallelism and accessibility of high-transfer speed memory. Furthermore, graph-based computation is pervasive and challenging for memory systems implementation in a specific time period. This paper provides quadratic and single GPUs, allowing developers to extend graph-based GPU computation effortlessly. The paper concentrates on the creditability and validation of sparse graphs in GPUs. Since the benchmark characteristic of Breadth-First Search (BFS) plays a critical role in graphs, this paper considers a primary characteristic benchmark for BFS computation usage. This paper explores the effectiveness and applicability of BFS for analyzing graph systems by utilizing the mentioned characteristics in BFS computation. This paper also utilizes the penalization of GPU BFS during the analysis and simulation results. In addition, using a comparative simulation result, this paper concludes that improved traversal rates speed up the computation and reduce memory usage for the single and quadratic GPU structures.

Ramsey upper density of infinite graphs

AbstractFor a fixed infinite graph $H$ , we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-colouring of the edges of $K_{\mathbb{N}}$ . This is called the Ramsey upper density of $H$ and was introduced by Erdős and Galvin in a restricted setting, and by DeBiasio and McKenney in general. Recently [4], the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs $H$ up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approach relates this problem to the solution of an optimisation problem for continuous functions. We show that, under certain conditions, the density depends only on the chromatic number of $H$ , the number of components of $H$ and the expansion ratio $|N(I)|/|I|$ of the independent sets of $H$ .

Mediation analysis for high-dimensional mediators and outcomes with an application to multimodal imaging data

Multimodal neuroimaging data have attracted increasing attention for brain research. An integrated analysis of multimodal neuroimaging data and behavioral or clinical measurements provides a promising approach for comprehensively and systematically investigating the underlying neural mechanisms of different phenotypes. However, such an integrated data analysis is intrinsically challenging due to the complex interactive relationships between the multimodal multivariate imaging variables. To address this challenge, a novel multivariate-mediator and multivariate-outcome mediation model (MMO) is proposed to simultaneously extract the latent systematic mediation patterns and estimate the mediation effects based on a dense bi-cluster graph approach. A computationally efficient algorithm is developed for dense bi-cluster structure estimation and inference to identify the mediation patterns with multiple testing correction. The performance of the proposed method is evaluated by an extensive simulation analysis with comparison to the existing methods. The results show that MMO performs better in terms of both the false discovery rate and sensitivity compared to existing models. The MMO is applied to a multimodal imaging dataset from the Human Connectome Project to investigate the effect of systolic blood pressure on whole-brain imaging measures for the regional homogeneity of the blood oxygenation level-dependent signal through the cerebral blood flow.

Strong bounds and exact solutions to the minimum broadcast time problem

AbstractGiven a graph and a subset of its nodes, referred to as source nodes, the minimum broadcast time problem asks for the minimum number of steps in which a signal can be transmitted from the sources to all other nodes in the graph. In each step, the sources and the nodes that already have received the signal can forward it to at most one of their neighbour nodes. The problem has previously been proved to be NP‐hard. In the current work, we develop a compact integer programming model for the problem. We also devise procedures for computing lower bounds on the minimum number of steps required, along with methods for constructing near‐optimal solutions. Computational experiments demonstrate that in a wide range of instances, in particular instances with sufficiently dense graphs, the lower and upper bounds under study collapse. In instances where this is not the case, the integer programming model proves strong capabilities in closing the remaining gap and proves to be considerably more efficient than previously studied models.