Adjacency Matrices
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Xun Liu ◽  
Fangyuan Lei ◽  
Guoqing Xia ◽  
Yikuan Zhang ◽  
Wenguo Wei

AbstractSimple graph convolution (SGC) achieves competitive classification accuracy to graph convolutional networks (GCNs) in various tasks while being computationally more efficient and fitting fewer parameters. However, the width of SGC is narrow due to the over-smoothing of SGC with higher power, which limits the learning ability of graph representations. Here, we propose AdjMix, a simple and attentional graph convolutional model, that is scalable to wider structure and captures more nodes features information, by simultaneously mixing the adjacency matrices of different powers. We point out that the key factor of over-smoothing is the mismatched weights of adjacency matrices, and design AdjMix to address the over-smoothing of SGC and GCNs by adjusting the weights to matching values. Experiments on citation networks including Pubmed, Citeseer, and Cora show that our AdjMix improves over SGC by 2.4%, 2.2%, and 3.2%, respectively, while achieving same performance in terms of parameters and complexity, and obtains better performance in terms of classification accuracy, parameters, and complexity, compared to other baselines.

2021 ◽  
Vol 1 (1) ◽  
pp. 1-38
Alicia Kollár ◽  
Peter Sarnak

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals ( 2 2 , 3 ) (2 \sqrt {2},3) and [ − 3 , − 2 ) [-3,-2) achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in [ − 3 , 3 ] [-3,3] which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in [ − 3 , 3 ) [-3,3) can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1522
Anna Concas ◽  
Lothar Reichel ◽  
Giuseppe Rodriguez ◽  
Yunzi Zhang

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

Willem H. Haemers ◽  
Leila Parsaei Majd

AbstractA conference matrix of order n is an $$n\times n$$ n × n matrix C with diagonal entries 0 and off-diagonal entries $$\pm 1$$ ± 1 satisfying $$CC^\top =(n-1)I$$ C C ⊤ = ( n - 1 ) I . If C is symmetric, then C has a symmetric spectrum $$\Sigma $$ Σ (that is, $$\Sigma =-\Sigma $$ Σ = - Σ ) and eigenvalues $$\pm \sqrt{n-1}$$ ± n - 1 . We show that many principal submatrices of C also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.

2021 ◽  
pp. 147592172098866
Shunlong Li ◽  
Jin Niu ◽  
Zhonglong Li

The novelty detection of bridges using monitoring data is an effective technique for diagnosing structural changes and possible damages, providing a critical basis for assessing the structural states of bridges. As cable forces describe the state of cable-stayed bridges, a novelty detection method was developed in this study using spatiotemporal graph convolutional networks by analysing spatiotemporal correlations among cable forces determined from different cable dynamometers. The spatial dependency of the sensor network was represented as a directed graph with cable dynamometers as vertices, and a graph convolutional network with learnable adjacency matrices was used to capture the spatial dependency of the locally connected vertices. A one-dimensional convolutional neural network was operated along the time axis to capture the temporal dependency. Sensor faults and structural variations could be distinguished based on the local or global anomalies of the spatiotemporal model parameters. Faulty sensors were detected and isolated using weighted adjacency matrices along with diagnostic indicators of the model residuals. After eliminating the effect of the sensor fault, the underlying variations in the state of the cable-stayed bridge could be determined based on the changing data patterns of the spatiotemporal model. The application of the proposed method to a long-span cable-stayed bridge demonstrates its effectiveness in sensor fault localization and structural variation detection.

Hilal A. Ganie

Let [Formula: see text] be a digraph of order [Formula: see text] and let [Formula: see text] be the adjacency matrix of [Formula: see text] Let Deg[Formula: see text] be the diagonal matrix of vertex out-degrees of [Formula: see text] For any real [Formula: see text] the generalized adjacency matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text] This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find [Formula: see text]-spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the [Formula: see text]-spectrum of various families of unsymmetric digraphs.

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