Analysis and Approximate Inference of Large Random Kronecker Graphs
This provides an efficient solution for analyzing large networks in fields like social networks and biology, though it is incremental as it builds on existing random matrix theory.
The paper tackles the problem of inferring parameters in large random Kronecker graphs by proving a spectral decomposition of the adjacency matrix into a low-rank signal and noise, enabling a 'denoise-and-solve' method that achieves comparable or better performance than baselines like KronFit and graph neural networks with linear time scaling in graph size.
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 $N$.