Graph summarization has become integral for managing and analyzing large-scale graphs in diverse real-world applications, including social networks, biological networks, and communication networks. Existing methods for graph summarization often face challenges, being either computationally expensive, limiting their applicability to large graphs, or lacking the incorporation of node attributes. In response, we introduce SsAG, an efficient and scalable lossy graph summarization method designed to preserve the essential structure of the original graph.

SsAG computes a sparse representation (summary) of the input graph, accommodating graphs with node attributes. The summary is structured as a graph on supernodes (subsets of vertices of G), where weighted superedges connect pairs of supernodes. The methodology focuses on constructing a summary graph with k supernodes, aiming to minimize the reconstruction error (the difference between the original graph and the graph reconstructed from the summary) while maximizing homogeneity with respect to the node attributes. The construction process involves iteratively merging pairs of nodes.

To enhance computational efficiency, we derive a closed-form expression for efficiently computing the reconstruction error (RE) after merging a pair, enabling constant-time approximation of this score. We assign a weight to each supernode, quantifying their contribution to the score of pairs, and utilize a weighted sampling strategy to select the best pair for merging. Notably, a logarithmic-sized sample achieves a summary comparable in quality based on various measures. Additionally, we propose a sparsification step for the constructed summary, aiming to reduce storage costs to a specified target size with a marginal increase in RE.

Empirical evaluations across diverse real-world graphs demonstrate that SsAG exhibits superior speed, being up to 17 × faster, while generating summaries of comparable quality. This work represents a significant advancement in the field, addressing computational challenges and showcasing the effectiveness of SsAG in graph summarization.

图摘要已成为管理和分析各种现实世界应用（包括社交网络、生物网络和通信网络）中大规模图的不可或缺的一部分。现有的图汇总方法经常面临挑战，要么计算成本昂贵，限制其对大型图的适用性，要么缺乏节点属性的结合。为此，我们引入了SsAG，这是一种高效且可扩展的有损图摘要方法，旨在保留原始图的基本结构。

SsAG计算输入图的稀疏表示（摘要），以容纳具有节点属性的图。摘要的结构为超级节点（G的顶点子集）上的图，其中加权超级边连接超级节点对。该方法的重点是构建具有k 个超级节点的摘要图，旨在最小化重建误差（原始图与从摘要重建的图之间的差异），同时最大化节点属性的同质性。构建过程涉及迭代合并节点对。

为了提高计算效率，我们推导了一个封闭式表达式，用于在合并一对后有效计算重建误差（RE），从而实现该分数的恒定时间近似。我们为每个超级节点分配一个权重，量化它们对对得分的贡献，并利用加权采样策略来选择最佳的对进行合并。值得注意的是，对数大小的样本根据各种测量得出的质量可比的总结。此外，我们为构建的摘要提出了稀疏化步骤，旨在将存储成本降低到指定的目标大小，同时 RE 略有增加。

对不同现实世界图表的实证评估表明，SsAG表现出卓越的速度，速度提高了 17 倍，同时生成质量相当的摘要。这项工作代表了该领域的重大进步，解决了计算挑战并展示了SsAG在图摘要方面的有效性。