SIMPLE-RC: Group Network Inference with Non-Sharp Nulls and Weak Signals

TL;DR

This paper introduces SIMPLE-RC, a novel network inference method that tests for similar membership profiles among groups of nodes under weaker signals, improving robustness and applicability in large-scale networks.

Contribution

It extends the SIMPLE framework to handle non-sharp null hypotheses with weaker signals using random coupling, along with new theoretical analysis for asymptotic distributions.

Findings

SIMPLE-RC effectively detects shared profiles under weak signals.

The method maintains high power while reducing test correlation.

Theoretical analysis confirms the asymptotic properties of the test.

Abstract

Large-scale network inference with uncertainty quantification has important applications in natural, social, and medical sciences. The recent work of Fan, Fan, Han and Lv (2022) introduced a general framework of statistical inference on membership profiles in large networks (SIMPLE) for testing the sharp null hypothesis that a pair of given nodes share the same membership profiles. In real applications, there are often groups of nodes under investigation that may share similar membership profiles at the presence of relatively weaker signals than the setting considered in SIMPLE. To address these practical challenges, in this paper we propose a SIMPLE method with random coupling (SIMPLE-RC) for testing the non-sharp null hypothesis that a group of given nodes share similar (not necessarily identical) membership profiles under weaker signals. Utilizing the idea of random coupling, we construct our test as the maximum of the SIMPLE tests for subsampled node pairs from the group. Such technique reduces significantly the correlation among individual SIMPLE tests while largely maintaining the power, enabling delicate analysis on the asymptotic distributions of the SIMPLE-RC test. Our method and theory cover both the cases with and without node degree heterogeneity. These new theoretical developments are empowered by a second-order expansion of spiked eigenvectors under the $\ell_\infty$-norm, built upon our work for random matrices with weak spikes. Our theoretical results and the practical advantages of the newly suggested method are demonstrated through several simulation and real data examples.