Sparse Graphs Research Articles

Thermal properties of structurally balanced systems on classical random graphs

The dynamics of social relations and the possibility of reaching the state of structural balance (Heider balance) under the influence of the temperature modeling the social noise level are discussed for interacting actors occupying nodes of classical random graphs. Depending on the graph density D, either a smooth crossover or a first-order phase transition from a balanced to an imbalanced state of the system is observed with an increase in the thermal noise level. The minimal graph density Dmin for which the first-order phase transition can be observed decreases with the system size N as Dmin∝N−0.58(1). For graph densities D>Dmin, the reduced critical temperature Tc⋆=Tc/Tc(D=1) increases with the graph density as Tc⋆∝D1.719(6) independently of the system size N.

Weakly interacting oscillators on dense random graphs

AbstractWe consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e. two particles are interacting if and only if they are connected in the underlying graph. We establish a law of large numbers for the empirical measure of the system that holds whenever the graph sequence is convergent to a graphon. The limit is the solution of a non-linear Fokker–Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on both deterministic and random graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the classical McKean–Vlasov mean-field limit.

Accurate regularized Tucker decomposition for image restoration

We propose a new accurate regularized Tucker decomposition (ARTD) method for image restoration (IR), which considers global low-rankness and local similarity of intrinsic image characteristics. Specifically, global low-rankness is represented by a sparse Tucker core tensor, whereas the local similarity is captured using nonnegative factor matrices and manifold regularization terms. Sparse nonnegative Tucker decomposition (SNTD) and graph nonnegative Tucker decomposition (GNTD) can be considered a special case of ARTD. We propose and implement an effective Alternating Proximal Gradient (APG) based algorithm to solve the ARTD model and deduce the closed-form updating rules. Notably, ARTD does not need to tune the Tucker rank and provides an initialization strategy and a first-order feedback control rule to accelerate its convergence. Experiments on real IR problems show that our method outperforms some existing state-of-the-art methods.

RelaGraph: Improving embedding on small-scale sparse knowledge graphs by neighborhood relations

Learning a continuous dense low-dimensional representation of knowledge graphs (KGs), known as knowledge graph embedding (KGE), has been viewed as the key to intelligent reasoning for deep learning (DL) and gained much attention in recent years. To address the problem that the current KGE models are generally ineffective on small-scale sparse datasets, we propose a novel method RelaGraph to improve the representation of entities and relations in KGs by introducing neighborhood relations. RelaGraph extends the neighborhood information during entity encoding, and adds the neighborhood relations to mine deeper level of graph structure information, so as to make up for the shortage of information in the generated subgraph. This method can well represent KG components in a vector space in a way that captures the structure of the graph, avoiding underlearning or overfitting. KGE based on RelaGraph is evaluated on a small-scale sparse graph KMHEO, and the MRR reached 0.49, which is 34 percentage points higher than that of the SOTA methods, as well as it does on several other datasets. Additionally, the vectors learned by RelaGraph is used to introduce DL into several KG-related downstream tasks, which achieved excellent results, verifying the superiority of KGE-based methods.

Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier–Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. Here the scaling of the average degree, as a function of the graph size, can range from nearly logarithmic to nearly linear.

Graph structure learning layer and its graph convolution clustering application

To learn the embedding representation of graph structure data corrupted by noise and outliers, existing graph structure learning networks usually follow the two-step paradigm, i.e., constructing a “good” graph structure and achieving the message passing for signals supported on the learned graph. However, the data corrupted by noise may make the learned graph structure unreliable. In this paper, we propose an adaptive graph convolutional clustering network that alternatively adjusts the graph structure and node representation layer-by-layer with back-propagation. Specifically, we design a Graph Structure Learning layer before each Graph Convolutional layer to learn the sparse graph structure from the node representations, where the graph structure is implicitly determined by the solution to the optimal self-expression problem. This is one of the first works that uses an optimization process as a Graph Network layer, which is obviously different from the function operation in traditional deep learning layers. An efficient iterative optimization algorithm is given to solve the optimal self-expression problem in the Graph Structure Learning layer. Experimental results show that the proposed method can effectively defend the negative effects of inaccurate graph structures. The code is available at https://github.com/HeXiax/SSGNN.

Exposing the Self-Supervised Space-Time Correspondence Learning via Graph Kernels

Self-supervised space-time correspondence learning is emerging as a promising way of leveraging unlabeled video. Currently, most methods adapt contrastive learning with mining negative samples or reconstruction adapted from the image domain, which requires dense affinity across multiple frames or optical flow constraints. Moreover, video correspondence predictive models require mining more inherent properties in videos, such as structural information. In this work, we propose the VideoHiGraph, a space-time correspondence framework based on a learnable graph kernel. Concerning the video as the spatial-temporal graph, the learning objectives of VideoHiGraph are emanated in a self-supervised manner for predicting unobserved hidden graphs via graph kernel manner. We learn a representation of the temporal coherence across frames in which pairwise similarity defines the structured hidden graph, such that a biased random walk graph kernel along the sub-graph can predict long-range correspondence. Then, we learn a refined representation across frames on the node-level via a dense graph kernel. The self-supervision of the model training is formed by the structural and temporal consistency of the graph. VideoHiGraph achieves superior performance and demonstrates its robustness across the benchmark of label propagation tasks involving objects, semantic parts, keypoints, and instances. Our algorithm implementations have been made publicly available at https://github.com/zyqin19/VideoHiGraph.

Detecting Anomalous Networks of Opioid Prescribers and Dispensers in Prescription Drug Data

The opioid overdose epidemic represents a serious public health crisis, with fatality rates rising considerably over the past several years. To help address the abuse of prescription opioids, state governments collect data on dispensed prescriptions, yet the use of these data is typically limited to manual searches. In this paper, we propose a novel graph-based framework for detecting anomalous opioid prescribing patterns in state Prescription Drug Monitoring Program (PDMP) data, which could aid governments in deterring opioid diversion and abuse. Specifically, we seek to identify connected networks of opioid prescribers and dispensers who engage in high-risk and possibly illicit activity. We develop and apply a novel extension of the Non-Parametric Heterogeneous Graph Scan (NPHGS) to two years of de-identified PDMP data from the state of Kansas, and find that NPHGS identifies subgraphs that are significantly more anomalous than those detected by other graph-based methods. NPHGS also reveals clusters of potentially illicit activity, which may strengthen state law enforcement and regulatory capabilities. Our paper is the first to demonstrate how prescription data can systematically identify anomalous opioid prescribers and dispensers, as well as illustrating the efficacy of a network-based approach. Additionally, our technical extensions to NPHGS offer both improved flexibility and graph density reduction, enabling the framework to be replicated across jurisdictions and extended to other problem domains.

Effect of plant-soil feedback on soil microbial co-occurrence network depends on the stage of secondary succession

Co-occurrence networks of soil microbial community are used to describe co-occurrence relationships in microbial communities, which can better understand the structure, function, and interaction patterns of microbial communities at different succession stages, but how co-occurrence networks within microbial communities respond to the effects exerted by plants come from different successional stages is unknown. Changes in the co-occurrence network of soil fungal and bacterial communities after plant species grown under monoculture were analyzed with a greenhouse experiment using the rhizosphere soils collected from early-, mid-, and late-successional species in the field. Results showed that compared with the soil microbial communities of plants grew in their own soil conditions, significant changes were observed in the Chao1 and ACE richness of the fungal community and Shannon and Simpson indices of the bacterial community after the growth of plants in other stages of succession; the relative abundance of the bacterial communities were mainly affected rather than the fungal communities at the phylum level. The late-successional species (or mid-successional species) mainly increased the graph density and average clustering coefficient (network complexity) of the fungal and bacterial communities in soil conditioned by the early- and mid-successional species (or early-successional species). Soil water-soluble nitrogen and phosphorus were the main soil factor affecting the relative abundance of the fungal and bacterial communities in species succession. The pre-successional species may be replaced directly by the subsequent plant species in a succession sequence given the network complexity of soil microbial communities.

Constructive–destructive heuristics for the Safe Set Problem

The Weighted Safe Set Problem aims to determine in an undirected graph a subset of vertices such that the weights of the connected components they induce exceed the weights of the adjacent components induced by the complementary subset. We tackle the problem with various approaches based on randomised constructive and destructive procedures, comparing them with each other and with the only other existing heuristic approach, that is a randomised destructive heuristic. The experiments concern all the available benchmark instances (random graphs up to 60 vertices), but also new benchmark instances with larger size and different topologies, namely random, small-world, regular, planar, grid and real-world graphs up to 300 vertices. Dense random graphs, in fact, appear to become progressively easier to solve as their size grows. On the other instances, the best performance is obtained by combining a randomised constructive procedure, a deterministic destructive procedure, and delayed termination conditions that allow a deeper exploration of the feasible region.

Intelligent fault diagnosis of rotating machinery under varying working conditions with global–local neighborhood and sparse graphs embedding deep regularized autoencoder

Nowadays, the health management of rotating machinery based on deep learning has achieved remarkable results. Nevertheless, in the presence of variable working conditions, the sparse nature of the valuable fault information makes the traditional deep models insufficient to achieve effective fault identification. Attributing to the aforementioned challenges, this research presented a new local–global neighborhood graph and sparse graph embedding deep-regularized autoencoder method (LGSDLRAE) framework for variable operating fault diagnosis (FD). More specifically, the FD scheme leverages manifold neighborhood graph embedding ability to mine fault information, combined with sparse theory ability in improving the generalization performance, to improve the performance of the original autoencoder (AE) algorithm. In the feature extraction (FE) phase, to enhance the compactness of homogeneous-classes and increase the separation between heterogeneous-classes by adopted global–local regularization terms; meanwhile, adding L1/2− sparse regularization terms gather the L1 regularization norm can make the data more sparse characteristics and L2 regularization norm prevents overfitting of data performance of the model and improve the generalization ability of the model. Finally, the identification accuracy of the datasets constructed variable operating conditions of double-span rotor test rig and planetary gearboxes are both above 98%, proving the superior performance of LGSDLRAE, respectively.

Understanding the class separation process of convolutional neural networks

Extensive research shows that deep neural networks (DNNs) capture better feature representations than the previous handcrafted feature engineering, which leads to a significant performance improvement. However, it raises the question of how does the neural network spontaneously learn the intermediate representation, during the training process, to correctly separate the test data set into different categories? In this paper, based on the existing research work, we continue to take a small step towards understanding the dynamics of convolutional neural networks (CNN). Specifically, we model the test data set as a relationship graph, and each intermediate layer representation learned by CNN will transform the relationship graph correspondingly. Through these continuous transformations, we investigate the class separation process of CNN from two perspectives: 1) show the evolution of the degree of correlation between samples through the histogram of similarity between vertices; 2) by visualizing and quantifying the degree of separation (i.e., modularity) of the important relationship subgraph, we can show the contribution of each convolution layer of CNN to class separation. In the preliminary experiment, it can be found that: 1) each convolution layer of CNN gradually reduced the similarity of most edges between the samples to a very low level, that is, the dense relationship graph gradually evolved into a sparse graph; 2) the remaining important relationship subgraph consist of edges with high similarity shows evident modularity; 3) both visualization and quantification results show that the modularity of intra-class subgraph increases when the layers go deeper. Moreover, the degradation and plateau in the modularity curve reveal the existence of redundant layers. In this paper, we reveal some more in-depth dynamics of CNN, and introduced another modularity, which is widely used as the theoretical guidance tool of layer pruning. It can save more model parameters without losing the accuracy of classification.

Predictive Quantitative Read-Across Structure-Property Relationship Modeling of the Retention Time (Log tR) of Pesticide Residues Present in Foods and Vegetables.

The retention time (log tR) of pesticidal compounds in a reverse-phase high-performance liquid chromatography (HPLC) analysis has a direct relationship with lipophilicity, which could be related to the ecotoxicity potential of the compounds. The novel quantitative read-across structure-property relationship (q-RASPR) modeling approach uses similarity-based descriptors for predictive model generation. These models have been shown to enhance external predictivity in previous studies for several end points. The current study describes the development of a q-RASPR model using experimental retention time data (log tR) in the HPLC experiments of 823 environmentally significant pesticide residues collected from a large compound database. To model the retention time (log tR) end point, 0D-2D descriptors have been used along with the read-across-derived similarity descriptors. The developed partial least squares (PLS) model was rigorously validated by various internal and external validation metrics as recommended by the Organization for Economic Co-operation and Development (OECD). The final q-RASPR model is proven to be a good fit, robust, and externally predictive (ntrain = 618, R2 = 0.82, Q2LOO = 0.81, ntest = 205, and Q2F1 = 0.84) that literally outperforms the external predictivity of the previously reported quantitative structure-property relationship (QSPR) model. From the insights of modeled descriptors, lipophilicity is found to be the most important chemical property, which positively correlates with the retention time (log tR). Various other characteristics, such as the number of multiple bonds (nBM), graph density (GD), etc., have a substantial and inversely proportionate relationship with the retention time end point. The software tools utilized in this study are user-friendly, and most of them are free, which makes our methodology quite cost-effective when compared to experimentation. In a nutshell, to obtain better external predictivity, interpretability, and transferability, q-RASPR is an efficient technique that has the potential to be employed as a good alternative approach for retention time prediction and ecotoxicity potential identification.

Sparsity of graphs that allow two distinct eigenvalues

The parameter q(G) of a graph G is the minimum number of distinct eigenvalues over the family of symmetric matrices described by G. It is shown that the minimum number of edges necessary for a connected graph G to have q(G)=2 is 2n−4 if n is even, and 2n−3 if n is odd. In addition, a characterization of graphs for which equality is achieved in either case is given.

A triplet graph convolutional network with attention and similarity-driven dictionary learning for remote sensing image retrieval

With the explosion in the volume of collected high-resolution aerial image data, the development of effective image retrieval methods for remote sensing (RS) has become a popular area of research. The problem of content-based image retrieval (CBIR) for high-resolution remote sensing (HRRS) requires robust feature extraction and representation to extract high-level semantics followed by a similarity measurement method to compute the similarity between images based on these semantics. In this paper, we propose a novel sparse graph learning-based triplet network for deep metric learning-based HRRS CBIR. First, the graph representations of the aerial images are constructed via local feature descriptors and unsupervised segmentation that provide an efficient graph-structured representation of the images. Then, a novel attention graph convolutional network is developed to extract high-level semantics from the spatial patterns of the images’ segmented regions by aggregating region-node features. The attention mechanism and the pyramid pooling structure of the graph network help the proposed model extract powerful spatial features by preserving the structural information of the images while reducing the data’s dimensionality. Finally, a novel task-driven dictionary learning (TDDL) method based on triplet loss constructs a sparse metric space for the similarity measurement of the images in which the computed sparse codes for the images promote intra-class similarity between same-class images and inter-class dissimilarity between images of different classes. The proposed sparse features reduce the dimensionality of model parameters which mitigates overfitting and improves the generalization of the overall framework for deep metric learning. Our TDDL method adopts a bi-level optimization strategy so that the dictionary can be trained alongside the parameters of the graph representation network in an end-to-end fashion. Finally, we conduct extensive evaluations of the performance of our proposed architecture on real-world datasets compared to state-of-the-art models and obtain superior results.

Lacon-, Shrub- and Parity-Decompositions: Characterizing Transductions of Bounded Expansion Classes

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts. In this paper, we show that one can encode graphs from a class with structurally bounded expansion via lacon-, shrub- and parity-decompositions from a class with bounded expansion. These decompositions are useful for lifting properties from sparse to structurally sparse graph classes.

Group Solidarity Based on Halal Network Labels: Moral Embeddedness of Agents of PT. HNI HPAI from an Economic Sociology Perspective

The number of consumers of halal products in the world has continued to increase since 2015 until now. For Muslim consumers, halal products are related to their moral and social values. Halal products meet the dietary standards of a devout Muslim. This research is intended to explore and present empirical evidence about group solidarity from the theory of moral embeddedness. It is important to provide empirical illustrations, especially those related to the halal product business network. At the same time, PT can also use empirical evidence. HNI HPAI in developing its business network. This study uses Beckert's perspective to analyze the foreign labor market. This research analyzes the formation of network label-based group solidarity in the moral-based halal product market. Researchers use digital research, namely SNA (Social Network Analysis) and TNA (Textual Network Analysis), as new methodologies. SNA to analyze the actor relation between HNI and Graph Density. TNA to analyze the narrative words of success stories of HNI agents in online media. The findings explain that morals, beliefs, and norms affect group solidarity based on the halal product business network label.

Induced Subgraphs and Path Decompositions

A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Similarly, we show that every graph excluding a fixed minor and with sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Finally, we present a characterisation for when a hereditary class defined by a finite set of forbidden induced subgraphs has bounded pathwidth.

Continuum Limits of Coupled Oscillator Networks Depending on Multiple Sparse Graphs

The continuum limit provides a useful tool for analyzing coupled oscillator networks. Recently, Medvedev (Commun Math Sci 17(4):883–898, 2019) gave a mathematical foundation for such an approach when the networks are defined on single graphs which may be dense or sparse, directed or undirected, and deterministic or random. In this paper, we consider coupled oscillator networks depending on multiple graphs, and extend his results to show that the continuum limit is also valid in this situation. Specifically, we prove that the initial value problem (IVP) of the corresponding continuum limit has a unique solution under general conditions and that the solution becomes the limit of those to the IVP of the networks in some adequate meaning. Moreover, we show that if solutions to the networks are stable or asymptotically stable when the node number is sufficiently large, then so are the corresponding solutions to the continuum limit, and that if solutions to the continuum limit are asymptotically stable, then so are the corresponding solutions to the networks in some weak meaning as the node number tends to infinity. These results can also be applied to coupled oscillator networks with multiple frequencies by regarding the frequencies as a weight matrix of another graph. We illustrate the theory for three variants of the Kuramoto model along with numerical simulations.

Graph Sparsification by Universal Greedy Algorithms

Graph sparsification is to approximate an arbitrary graph by a sparse graph and is useful in many applications, such as simplification of social networks, least squares problems, numerical solution of symmetric positive definite linear systems and etc. In this paper, inspired by the well-known sparse signal recovery algorithm called orthogonal matching pursuit (OMP), we introduce a deterministic, greedy edge selection algorithm called universal greedy approach (UGA) for graph sparsification. For a general spectral sparsification problem, e.g., positive subset selection problem from a set of $m$ vectors from $\mathbb{R}^n$, we propose a nonnegative UGA algorithm which needs $O(mn^2+ n^3/\epsilon^2)$ time to find a $\frac{1+\epsilon/\beta}{1-\epsilon/\beta}$-spectral sparsifier with positive coefficients with sparsity $\le\lceil\frac{n}{\epsilon^2}\rceil$, where $\beta$ is the ratio between the smallest length and largest length of the vectors. The convergence of the nonnegative UGA algorithm will be established. For the graph sparsification problem, another UGA algorithm will be proposed which can output a $\frac{1+O(\epsilon)}{1-O(\epsilon)}$-spectral sparsifier with $\lceil\frac{n}{\epsilon^2}\rceil$ edges in $O(m+n^2/\epsilon^2)$ time from a graph with $m$ edges and $n$ vertices under some mild assumptions. This is a linear time algorithm in terms of the number of edges that the community of graph sparsification is looking for. The best result in the literature to the knowledge of the authors is the existence of a deterministic algorithm which is almost linear, i.e. $O(m^{1+o(1)})$ for some $o(1)=O(\frac{(\log\log(m))^{2/3}}{\log^{1/3}(m)})$. Finally, extensive experimental results, including applications to graph clustering and least squares regression, show the effectiveness of proposed approaches.