Relationship between agents can be conveniently represented by graphs. When these relationships have different modalities, they are better modelled by multilayer graphs where each layer is associated with one modality. Such graphs arise naturally in many contexts including biological and social networks. Clustering is a fundamental problem in network analysis where the goal is to regroup nodes with similar connectivity profiles. In the past decade, various clustering methods have been extended from the unilayer setting to multilayer graphs in order to incorporate the information provided by each layer. While most existing works assume - rather restrictively - that all layers share the same set of nodes, we propose a new framework that allows for layers to be defined on different sets of nodes. In particular, the nodes not recorded in a layer are treated as missing. Within this paradigm, we investigate several generalizations of well-known clustering methods in the complete setting to the incomplete one and prove some consistency results under the Multi-Layer Stochastic Block Model assumption. Our theoretical results are complemented by thorough numerical comparisons between our proposed algorithms on synthetic data, and also on real datasets, thus highlighting the promising behaviour of our methods in various settings.

中文翻译：

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将缺少节点的多层图聚类

代理之间的关系可以方便地用图形表示。当这些关系具有不同的模态时，可以通过多层图对其进行更好的建模，其中每一层都与一个模态相关联。这样的图在包括生物和社会网络在内的许多情况下自然地出现。群集是网络分析中的一个基本问题，网络分析的目标是用相似的连接配置文件重新组合节点。在过去的十年中，各种聚类方法已从单层设置扩展到多层图，以便合并每个层提供的信息。尽管大多数现有工作都（相当限制性地）假设所有层共享同一组节点，但我们提出了一个新框架，该框架允许在不同节点集上定义层。特别是，没有记录在图层中的节点将被视为丢失。在这种范式下，我们研究了从完全设置到不完全设置的几种著名聚类方法的一般化，并在多层随机块模型假设下证明了一些一致性结果。通过对合成数据和真实数据集上的拟议算法进行全面的数值比较，我们的理论结果得到了补充，从而突显了我们的方法在各种环境下的有前途的行为。