Random graph models are playing an increasingly important role in various fields ranging from social networks, telecommunication systems, to physiologic and biological networks. Within this landscape, the random Kronecker graph model, emerges as a prominent framework for scrutinizing intricate real-world networks. In this paper, we investigate large random Kronecker graphs, i.e., the number of graph vertices $N$ is large. Built upon recent advances in random matrix theory (RMT) and high-dimensional statistics, we prove that the adjacency of a large random Kronecker graph can be decomposed, in a spectral norm sense, into two parts: a small-rank (of rank $O(\log N)$) signal matrix that is linear in the graph parameters and a zero-mean random noise matrix. Based on this result, we propose a ``denoise-and-solve'' approach to infer the key graph parameters, with significantly reduced computational complexity. Experiments on both graph inference and classification are presented to evaluate the our proposed method. In both tasks, the proposed approach yields comparable or advantageous performance, than widely-used graph inference (e.g., KronFit) and graph neural net baselines, at a time cost that scales linearly as the graph size $N$.

翻译：随机图模型在社会科学网络、通信系统、生理与生物网络等多个领域发挥着日益重要的作用。在此背景下，随机Kronecker图模型作为分析复杂真实网络的重要框架崭露头角。本文研究大型随机Kronecker图（即图顶点数$N$较大时的情形）。基于随机矩阵理论和高维统计学的最新进展，我们证明大型随机Kronecker图的邻接矩阵在谱范数意义下可分解为两部分：一个秩为$O(\log N)$的小秩信号矩阵（其元素与图参数呈线性关系）和一个零均值随机噪声矩阵。基于此结论，我们提出一种"去噪-求解"方法以推断关键图参数，该方法显著降低了计算复杂度。通过图推断与分类实验对所提方法进行验证，结果表明：在两项任务中，我们的方法在时间成本随图规模$N$线性增长的前提下，取得了与广泛使用的图推断方法（如KronFit）及图神经网络基线相当或更优的性能。