Universal Graph Compression: Stochastic Block Models
Alankrita Bhatt, Ziao Wang, Chi Wang, Lele Wang
TL;DR
This work develops a universal, polynomial-time compressor for labeled graphs generated by SBMs, achieving the asymptotically optimal first-order rate without knowing SBM parameters. The core idea is to decompose the adjacency matrix into many small blocks that become almost i.i.d., enabling KT/Laplace coding and adaptive arithmetic coding to approach the graph entropy $H(A_n)$. Theoretical results establish universality across broad SBM regimes and provide minimax redundancy bounds, while local weak convergence analysis shows BC-entropy-level performance in the sparse regime. Empirically, the method substantially reduces storage compared to existing universal and domain-specific compressors on four real graphs. This advances scalable, parameter-free graph compression with practical coding efficiency.
Abstract
Motivated by the prevalent data science applications of processing large-scale graph data such as social networks and biological networks, this paper investigates lossless compression of data in the form of a labeled graph. Particularly, we consider a widely used random graph model, stochastic block model (SBM), which captures the clustering effects in social networks. An information-theoretic universal compression framework is applied, in which one aims to design a single compressor that achieves the asymptotically optimal compression rate, for every SBM distribution, without knowing the parameters of the SBM. Such a graph compressor is proposed in this paper, which universally achieves the optimal compression rate with polynomial time complexity for a wide class of SBMs. Existing universal compression techniques are developed mostly for stationary ergodic one-dimensional sequences. However, the adjacency matrix of SBM has complex two-dimensional correlations. The challenge is alleviated through a carefully designed transform that converts two-dimensional correlated data into almost i.i.d. submatrices. The sequence of submatrices is then compressed by a Krichevsky--Trofimov compressor, whose length analysis is generalized to identically distributed but arbitrarily correlated sequences. In four benchmark graph datasets, the compressed files from competing algorithms take 2.4 to 27 times the space needed by the proposed scheme.