Clean Graph Research Articles

Robust mixed-order graph learning for incomplete multi-view clustering

Incomplete multi-view clustering (IMVC) aims to address the clustering problem of multi-view data with partially missing samples and has received widespread attention in recent years. Most existing IMVC methods still have the following issues that require to be further addressed. They focus solely on the first-order correlation information among samples, neglecting the more intricate high-order connections. Additionally, these methods always overlook the noise or inaccuracies in the similarity matrix. To address above issues, a novel method named Robust Mixed-order Graph Learning (RMoGL) is proposed for IMVC. Specifically, to enhance the robustness to noise, the similarity matrices are separated into clean graphs and noise graphs. To capture complex high-order relationships among samples, the dynamic high-order similarity graphs are innovatively constructed from the recovered data. The clean graphs are endowed with mixed-order information and tend towards to obtain a consensus graph via a self-weighted manner. An efficient algorithm based on Alternating Direction Method of Multipliers (ADMM) is designed to solve the proposed RMoGL, and superior performance is demonstrated by compared with nine state-of-the-art methods across eight datasets. The source code of this work is available at https://github.com/guowei1314/RMoGL.

Classification of Rings Associated with the Genus of Clean Graphs

Classification of Rings Associated with the Genus of Clean Graphs

Profinite properties of algebraically clean graphs of free groups

We prove that for every prime p algebraically clean graphs of groups are virtually residually p-finite and cohomologically p-complete. We also prove that they are cohomologically good. We apply this to certain 2-dimensional Artin groups.

A Simple and Yet Fairly Effective Defense for Graph Neural Networks

Graph Neural Networks (GNNs) have emerged as the dominant approach for machine learning on graph-structured data. However, concerns have arisen regarding the vulnerability of GNNs to small adversarial perturbations. Existing defense methods against such perturbations suffer from high time complexity and can negatively impact the model's performance on clean graphs. To address these challenges, this paper introduces NoisyGNNs, a novel defense method that incorporates noise into the underlying model's architecture. We establish a theoretical connection between noise injection and the enhancement of GNN robustness, highlighting the effectiveness of our approach. We further conduct extensive empirical evaluations on the node classification task to validate our theoretical findings, focusing on two popular GNNs: the GCN and GIN. The results demonstrate that NoisyGNN achieves superior or comparable defense performance to existing methods while minimizing added time complexity. The NoisyGNN approach is model-agnostic, allowing it to be integrated with different GNN architectures. Successful combinations of our NoisyGNN approach with existing defense techniques demonstrate even further improved adversarial defense results. Our code is publicly available at: https://github.com/Sennadir/NoisyGNN.

CGDD: Multiview Graph Clustering via Cross-Graph Diversity Detection.

Multiview graph clustering has emerged as an important yet challenging technique due to the difficulty of exploiting the similarity relationships among multiple views. Typically, the similarity graph for each view learned by these methods is easily corrupted because of the unavoidable noise or diversity among views. To recover a clean graph, existing methods mainly focus on the diverse part within each graph yet overlook the diversity across multiple graphs. In this article, instead of merely considering the sparsity of diversity within a graph as previous methods do, we incline to a more suitable consideration that the diversity should be sparse across graphs. It is intuitive that the divergent parts are supposed to be inconsistent with each other, otherwise it would contradict the definition of diversity. By simultaneously and explicitly detecting the multiview consistency and cross-graph diversity, a pure graph for each view can be expected. The multiple pure graphs are further fused to the structured consensus graph with exactly r connected components where r is the number of clusters. Once the consensus graph is obtained, the cluster label to each instance can be directly allocated as each connected component precisely corresponds to an individual cluster. An alternating iterative algorithm is designed to optimize the subtasks of learning the similarity graphs adaptively, detecting the consistency as well as cross-graph diversity, fusing the multiple pure graphs, and assigning cluster label to each instance in a mutual reinforcement manner. Extensive experimental results on several benchmark multiview datasets demonstrate the effectiveness of our model, in comparison to several state-of-the-art algorithms.

Nil clean graphs associated with commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nilpotent elements of [Formula: see text]. The nil clean graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are all nonzero nil clean elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we focus on [Formula: see text], the subgraph of [Formula: see text] induced by the set [Formula: see text]. It is observed that [Formula: see text] has a crucial role in [Formula: see text]. We prove that [Formula: see text] is connected with [Formula: see text] and [Formula: see text]. We investigate also the interplay between the ring-theoretic properties of [Formula: see text] and the graph-theoretic properties of [Formula: see text]. Moreover, the clique number and the chromatic number of [Formula: see text] for some classes of rings are determined.

Multi-view Outlier Detection via Graphs Denoising

Recently, multi-view outlier detection attracts increasingly more attention. Although existing multi-view outlier detection methods have demonstrated promising performance, they still suffer from some problems. Firstly, many methods make the assumption that the data have a clear clustering structure and detect the outliers by using some off-the-shelf clustering methods. Therefore, the performance of these methods depends on the clustering methods they used, and thus these methods are hard to handle complicated data. Secondly, some methods ignore the complicated structure or distribution of class outliers and directly learn a consensus representation by simply combining the representation of different views linearly. To tackle these problems, we propose a novel method named Multi-view Outlier Detection with Graph Denoising (MODGD). We first construct a graph for each view, and then learn a consensus graph by ensembling the multiple graphs. When fusing the multiple graphs, we explicitly characterize and extract the structured outliers on each graph and recover the multiple clean graphs for the ensemble. During the process of multiple graph denoising and fusion, we carefully design an outlier measurement criterion based on the characteristics of attribute and class outliers. The extensive experiments on benchmark data sets demonstrate the effectiveness and superiority of the proposed method. The codes of this paper are released in http://Doctor-Nobody.github.io/codes/MODGD.zip.

Exploring interpretable graph convolutional networks for autism spectrum disorder diagnosis.

Finding the biomarkers associated with autism spectrum disorder (ASD) is helpful for understanding the underlying roots of the disorder and can lead to earlier diagnosis and more targeted treatments. In essence, we are faced with two challenges (i) how to learn a node representation and a clean graph structure from original graph data with high dimensionality and (ii) how to jointly model the procedure of node representation learning, structure learning and graph classification. We propose FSL-BrainNet, an interpretable graph convolution network (GCN) model for jointly Learning of node Features and clean Structures in brain networks for automatic brain network classification and interpretation. We formulate an end-to-end trainable and interpretable framework for graph classification and biomarkers (salient brain regions and potential subnetworks) identification. The experimental results on the ABIDE dataset show that our proposed methods not only achieve improved prediction performance compared with the state-of-the-art methods, but also find a compact set of highly suggestive biomarkers including relevant brain regions and subnetworks to ASD. Through node feature learning and structure learning, our model can simultaneously select important brain regions and identify subnetworks.

Scalable and Explainable 1-Bit Matrix Completion via Graph Signal Learning

One-bit matrix completion is an important class of positive-unlabeled (PU) learning problems where the observations consist of only positive examples, e.g., in top-N recommender systems. For the first time, we show that 1-bit matrix completion can be formulated as the problem of recovering clean graph signals from noise-corrupted signals in hypergraphs. This makes it possible to enjoy recent advances in graph signal learning. Then, we propose the spectral graph matrix completion (SGMC) method, which can recover the underlying matrix in distributed systems by filtering the noisy data in the graph frequency domain. Meanwhile, it can provide micro- and macro-level explanations by following vertex-frequency analysis. To tackle the computational and memory issue of performing graph signal operations on large graphs, we construct a scalable Nystrom algorithm which can efficiently compute orthonormal eigenvectors. Furthermore, we also develop polynomial and sparse frequency filters to remedy the accuracy loss caused by the approximations. We demonstrate the effectiveness of our algorithms on top-N recommendation tasks, and the results on three large-scale real-world datasets show that SGMC can outperform state-of-the-art top-N recommendation algorithms in accuracy while only requiring a small fraction of training time compared to the baselines.

Clean graph of a ring

Let [Formula: see text] be a ring (not necessarily commutative) with identity. The clean graph [Formula: see text] of a ring [Formula: see text] is a graph with vertices in form [Formula: see text], where [Formula: see text] is an idempotent and [Formula: see text] is a unit of [Formula: see text]; and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we focus on [Formula: see text], the subgraph of [Formula: see text] induced by the set [Formula: see text] is a nonzero idempotent element of [Formula: see text]. It is observed that [Formula: see text] has a crucial role in [Formula: see text]. The clique number, the chromatic number, the independence number and the domination number of the clean graph for some classes of rings are determined. Moreover, the connectedness and the diameter of [Formula: see text] are studied.

Mimicking disorder on a clean graph: Interference-induced inhibition of spread in a cyclic quantum random walk

We quantitatively differentiate between the spreads of discrete-time quantum and classical random walks on a cyclic graph. Due to the closed nature of any cyclic graph, there is additional “collision”-like interference in the quantum random walk along with the usual interference in any such walk on any graph, closed or otherwise. We find that the quantum walker exhibits inhibition of spread in comparison to the classical one, even in the absence of disorder, a phenomenon that is potentially attributable to the additional interference in the quantum case. This is to be contrasted with the situation on open graphs, where the quantum walker, being effectively denied the collision-like interference, garners a much higher spread than its classical counterpart. Inhibition of spread also occurs on open graphs, but with insertion of disorder. We use the Shannon entropy of the position probability distribution to quantify the spread of the walker in both quantum and classical cases. We find that for a given number of vertices on a cyclic graph, the entropy with respect to number of steps for the quantum walker saturates, on average, to a value lower than that for the corresponding classical one. We also analyze variations of the entropies with respect to system size, and look at the corresponding asymptotic growth rates.

Nil Clean Graphs of Rings

In this article, we define the nil clean graph of a ring R. The vertex set is the ring R, and two ring elements a and b are adjacent if and only if a + b is nil clean in R. Graph theoretic properties like the girth, dominating sets, diameter, etc., of the nil clean graph are studied for finite commutative rings.

Cleaning a network with brushes

Following the decontamination metaphor for searching a graph, we introduce a cleaning process, which is related to both the chip-firing game and edge searching. Brushes (instead of chips) are placed on some vertices and, initially, all the edges are dirty. When a vertex is ‘fired’, each dirty incident edge is traversed by only one brush, cleaning it, but a brush is not allowed to traverse an already cleaned edge; consequently, a vertex may not need degree-many brushes to fire. The model presented is one where the edges are continually recontaminated, say by algae, so that cleaning is regarded as an on-going process. Ideally, the final configuration of the brushes, after all the edges have been cleaned, should be a viable starting configuration to clean the graph again. We show that this is possible with the least number of brushes if the vertices are fired sequentially but not if fired in parallel. We also present bounds for the least number of brushes required to clean graphs in general and some specific families of graphs.