Analysis and Approximate Inference of Large Random Kronecker Graphs
Zhenyu Liao, Yuanqian Xia, Chengmei Niu, Yong Xiao
TL;DR
The paper addresses scalable parameter inference for large random Kronecker graphs by proving a spectral signal-plus-noise decomposition: the adjacency ${\mathbf A}$ is close to a low-rank signal plus zero-mean noise after appropriate permutation, with a signal ${\mathbf S}_K$ of rank at most $(m-1)K+1$ that is linear in the initiator perturbations. Building on this, the authors propose a denoise-and-solve meta algorithm that first denoises the adjacency via a shrinkage-based estimator to recover the low-rank signal, then solves a permuted linear regression to recover the initiator parameters ${\mathbf P}_1$ (or ${\mathbf X}$), achieving near-linear time complexity in the number of nodes $N$ and offering RNLA-assisted speedups. The approach is validated through synthetic experiments against KronFit and through realistic graph classification benchmarks, showing competitive or superior performance in many regimes and substantial scalability improvements. The work advances scalable graph inference and representation learning by combining high-dimensional spectral analysis with practical, scalable optimization for Kronecker graph models.
Abstract
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 $N$.