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Abstract:Finding important edges in a graph is a crucial problem for various research fields, such as network epidemics, signal processing, machine learning, and sensor networks. In this paper, we tackle the problem based on sampling theory on graphs. We convert the original graph to a line graph where its nodes and edges, respectively, represent the original edges and the connections between the edges. We then perform node sampling of the line graph based on the edge smoothness assumption: This process selects the most critical edges in the original graph. We present a general framework of edge sampling based on graph sampling theory and reveal a theoretical relationship between the degree of the original graph and the line graph. We also propose an acceleration method for edge sampling in the proposed framework by using the relationship between two types of Laplacian of the node and edge domains. Experimental results in synthetic and real-world graphs validate the effectiveness of our approach against some alternative edge selection methods.

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Abstract:This paper proposes a compression framework for adjacency matrices of weighted graphs based on graph filter banks. Adjacency matrices are widely used mathematical representations of graphs and are used in various applications in signal processing, machine learning, and data mining. In many problems of interest, these adjacency matrices can be large, so efficient compression methods are crucial. In this paper, we propose a lossy compression of weighted adjacency matrices, where the binary adjacency information is encoded losslessly (so the topological information of the graph is preserved) while the edge weights are compressed lossily. For the edge weight compression, the target graph is converted into a line graph, whose nodes correspond to the edges of the original graph, and where the original edge weights are regarded as a graph signal on the line graph. We then transform the edge weights on the line graph with a graph filter bank for sparse representation. Experiments on synthetic data validate the effectiveness of the proposed method by comparing it with existing lossy matrix compression methods.

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