提出新方法在极稀疏多层网络中实现层分组完美聚类。
Perfect Clustering in Very Sparse Diverse Multiplex Networks
- 基于张量融合全层信息，突破传统逐层分析限制。
- 在稀疏度接近计算下界时仍可实现完美聚类。
- 适用于低密度复杂多层网络，适合网络结构挖掘研究者。
本文研究了多层符号广义随机点积图（DIMPLE-SGRDPG）模型，该模型中所有层共享相同节点集，且各层可按嵌入子空间划分为组，同组层共享同一子空间但连接概率矩阵可不同。该设定涵盖多数多层网络模型。核心任务是识别具有独特子空间结构的层组，因所有层共用同一子空间的情形已研究较充分。以往方法依赖逐层分析，需网络足够稠密；而本文通过联合全层信息，提出基于张量的新方法，在更稀疏条件下实现完美聚类。理论分析在直观非限制性假设下成立，结果表明新方法的稀疏条件仅比简单模型的计算下界差对数因子。
原文摘要 · Abstract (English)
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.
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