Perfect Clustering in Very Sparse Diverse Multiplex Networks

TL;DR

This paper introduces a tensor-based method for perfect clustering of layers in very sparse, diverse multiplex networks, significantly improving over previous layer-by-layer approaches that required denser networks.

Contribution

The paper develops a novel tensor methodology that achieves perfect clustering in sparse multiplex networks with diverse subspace structures, extending beyond prior dense-network limitations.

Findings

Achieves perfect clustering in sparser networks than previous methods

Theoretical guarantees match computational lower bounds up to logarithmic factors

Applicable to a broad class of multiplex network models

Abstract

The paper studies the DIverse MultiPLEx Signed Generalized Random Dot Product Graph (DIMPLE-SGRDPG) network model (Pensky (2024)), where all layers of the network have the same collection of nodes. In addition, all layers can be partitioned into groups such that the layers in the same group are embedded in the same ambient subspace but otherwise matrices of connection probabilities can be all different. This setting includes majority of multilayer network models as its particular cases. The key task in this model is to recover the groups of layers with unique subspace structures, since the case where all layers of the network are embedded in the same subspace has been fairly well studied. Until now, clustering of layers in such networks was based on the layer-per-layer analysis, which required the multilayer network to be sufficiently dense. Nevertheless, in this paper we succeeded in pooling information in all layers together and providing a tensor-based methodology that ensures perfect clustering for a much sparser network. Our theoretical results, established under intuitive non-restrictive assumptions, assert that the new technique achieves perfect clustering under sparsity conditions that, up to logarithmic factors, coincide with the computational lower bound derived for a much simpler model.