Core Decomposition and Densest Subgraph in Multilayer Networks

Multilayer networks are a powerful paradigm to model complex systems, where various relations might occur among the same set of entities. Despite the keen interest in a variety of problems, algorithms, and analysis methods in this type of network, the problem of extracting dense subgraphs has remained largely unexplored. As a first step in this direction, in this work we study the problem of core decomposition of a multilayer network. Unlike the single-layer counterpart in which cores are all nested into one another and can be computed in linear time, the multilayer context is much more challenging as no total order exists among multilayer cores; rather, they form a lattice whose size is exponential in the number of layers. In this setting we devise three algorithms which differ in the way they visit the core lattice and in their pruning techniques. We assess time and space efficiency of the three algorithms on a large variety of real-world multilayer networks. We then move a step forward and showcase an application of the multilayer core-decomposition tool to the problem of densest-subgraph extraction from multilayer networks. We introduce a definition of multilayer densest subgraph that trades-off between high density and number of layers in which the high density holds, and show how multilayer core decomposition can be exploited to approximate this problem with quality guarantees.