  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.
