Random graph models are playing an increasingly important role in science and industry, and finds their applications in a variety of fields ranging from social and traffic networks, to recommendation systems and molecular genetics. In this paper, we perform an in-depth analysis of the random Kronecker graph model proposed in \cite{leskovec2010kronecker}, when the number of graph vertices $N$ is large. Built upon recent advances in random matrix theory, we show, in the dense regime, that the random Kronecker graph adjacency matrix follows approximately a signal-plus-noise model, with a small-rank (of order at most $\log N$) signal matrix that is linear in the graph parameters and a random noise matrix having a quarter-circle-form singular value distribution. This observation allows us to propose a ``denoise-and-solve'' meta algorithm to approximately infer the graph parameters, with reduced computational complexity and (asymptotic) performance guarantee. Numerical experiments of graph inference and graph classification on both synthetic and realistic graphs are provided to support the advantageous performance of the proposed approach.

翻译：随机图模型在科学和工业领域中扮演着日益重要的角色，其应用范围涵盖社交网络、交通网络、推荐系统以及分子遗传学等多个领域。本文针对文献\cite{leskovec2010kronecker}提出的随机Kronecker图模型，在图顶点数$N$较大时进行了深入分析。基于随机矩阵理论的最新进展，我们在稠密条件下证明了随机Kronecker图邻接矩阵近似服从信号加噪声模型：其中信号矩阵为低秩（阶数不超过$\log N$）且关于图参数线性，而噪声矩阵的奇异值分布呈现四分之一圆形式。这一发现使我们能够提出一种“去噪-求解”元算法用于图参数的近似推断，该算法在降低计算复杂度的同时具有渐近性能保证。在合成图与真实图上的图推断与图分类数值实验验证了所提方法的优越性能。