How does the graph change if we summarize it by merging nodes? How can we summarize the graph while preserving its spectral characteristics? Graph summarization aims to present a graph in a compact summary graph form while keeping its important structural information. Existing methods primarily focus on preserving the adjacency matrix. In contrast, spectral graph theory provides a powerful tool to describe the characteristics of a graph. In this paper, we propose a novel graph summarization method that preserves the spectral characteristics, including spectral moments and heat traces. We analyze the change of the spectral characteristics after summarization and design a simple yet effective summarization method based on agglomerative clustering. Our approach is extensively evaluated on real-world datasets. The experimental results show that our method excels in preserving the spectral characteristics and obtains better performance on the subsequent graph classification task.

Graph summarization is a useful tool for analyzing large-scale graphs. Some works tried to preserve original node embeddings encoding rich structural information of nodes on the summary graph. However, their algorithms are designed heuristically and not theoretically guaranteed. In this article, we theoretically study the problem of preserving node embeddings on summary graph. We prove that three matrix-factorization-based node embedding methods of the original graph can be approximated by that of the summary graph, and we propose a novel graph summarization method, named HCSumm , based on this analysis. Extensive experiments are performed on real-world datasets to evaluate the effectiveness of our proposed method. The experimental results show that our method outperforms the state-of-the-art methods in preserving node embeddings.

Given a large graph, can we learn its node embeddings from a smaller summary graph? What is the relationship between embeddings learned from original graphs and their summary graphs? Graph representation learning plays an important role in many graph mining applications, but learning em-beddings of large-scale graphs remains a challenge. Recent works try to alleviate it via graph summarization, which typ-ically includes the three steps: reducing the graph size by combining nodes and edges into supernodes and superedges,learning the supernode embedding on the summary graph and then restoring the embeddings of the original nodes. How-ever, the justification behind those steps is still unknown. In this work, we propose GELSUMM, a well-formulated graph embedding learning framework based on graph sum-marization, in which we show the theoretical ground of learn-ing from summary graphs and the restoration with the three well-known graph embedding approaches in a closed form.Through extensive experiments on real-world datasets, we demonstrate that our methods can learn graph embeddings with matching or better performance on downstream tasks.This work provides theoretical analysis for learning node em-beddings via summarization and helps explain and under-stand the mechanism of the existing works.

Given a large graph, how can we summarize it with fewer nodes and edges while maintaining its key properties, e.g.node degrees and graph spectrum?As a solution, graph summarization, which aims to find the compact representation for optimally describing and reconstructing a given graph, has received much attention, and numerous methods have been developed for it.However, many existing methods adopt the uniform reconstruction scheme, which is an unrealistic assumption as most real-world graphs have highly skewed node degrees, even within communities.Therefore we propose a degree-preserving graph summarization model, DPGS, with a novel reconstruction scheme based on the configuration model.To optimize the Minimum Description Length of our model, we deisgn a linearly scalable algorithm using hashing techniques.We theoretically show that the minimized reconstruction error bounds the perturbation of graph spectral information.Extensive experiments on realworld datasets show that DPGS yields more accurate summary graphs than several well-known baselines.Moreover, our reduced summary graphs can effectively train graph neural networks (GNNs) while saving computational cost.