Fitting networks with a cancellation trick

TL;DR

This paper introduces the logit-DCBM model combining ideas from DCBM, LSM, and $eta$-model, and proposes R-SCORE, a recursive community detection algorithm that improves parameter fitting and community detection accuracy.

Contribution

The paper presents the logit-DCBM model and the R-SCORE algorithm, addressing the challenge of fitting nonlinear network models and enhancing community detection performance.

Findings

R-SCORE outperforms existing spectral methods in numerical experiments.

Theoretically, R-SCORE has a faster Hamming error rate than SCORE in certain sparse regimes.

R-SCORE effectively removes nonlinear factors, approximating a low-rank model for better community detection.

Abstract

The degree-corrected block model (DCBM), latent space model (LSM), and $\beta$-model are all popular network models. We combine their modeling ideas and propose the logit-DCBM as a new model. Similar as the $\beta$-model and LSM, the logit-DCBM contains nonlinear factors, where fitting the parameters is a challenging open problem. We resolve this problem by introducing a cancellation trick. We also propose R-SCORE as a recursive community detection algorithm, where in each iteration, we first use the idea above to update our parameter estimation, and then use the results to remove the nonlinear factors in the logit-DCBM so the renormalized model approximately satisfies a low-rank model, just like the DCBM. Our numerical study suggests that R-SCORE significantly improves over existing spectral approaches in many cases. Also, theoretically, we show that the Hamming error rate of R-SCORE is faster than that of SCORE in a specific sparse region, and is at least as fast outside this region.