Core Decomposition in Multilayer Networks

Abstract

Multilayer networks are a powerful paradigm to model complex systems, where multiple relations occur between the same entities. Despite the keen interest in a variety of tasks, algorithms, and analyses in this type of network, the problem of extracting dense subgraphs has remained largely unexplored so far.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 study the problem of extracting the inner-most (also known as maximal) cores, i.e., the cores that are not dominated by any other core in terms of their core index in all the layers. Inner-most cores are typically orders of magnitude less than all the cores. Motivated by this, we devise an algorithm that effectively exploits the maximality property and extracts inner-most cores directly, without first computing a complete decomposition. This allows for a consistent speed up over a naïve method that simply filters out non-inner-most ones from all the cores.Finally, we showcase the multilayer core-decomposition tool in a variety of scenarios and problems. We start by considering the problem of densest-subgraph extraction in 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 exploit multilayer core decomposition to approximate this problem with quality guarantees. As further applications, we show how to utilize multilayer core decomposition to speed-up the extraction of frequent cross-graph quasi-cliques and to generalize the community-search problem to the multilayer setting.Dense structures in multilayer networks. Several recent works have dealt with the problem of extracting dense subgraphs from a set of multiple graphs sharing the same vertex set, which is a setting equivalent to the multilayer one we study in this work. Jethava and Beerenwinkel [49] define the densest common subgraph problem, i.e., find a subgraph maximizing the minimum average degree over all input graphs, and devise a linear-programming formulation and a greedy heuristic for it. Reinthal et al.[65] provide a Lagrangian relaxation of the Jethava and Beerenwinkel's linear program, which can be solved more efficiently. Semertzidis et al. [70] introduce three more variants of the problem, whose goal is to maximize the average average degree, the minimum minimum degree, and the average minimum degree, respectively. They show that the average-average varia...

Multilayer Network Science

“…12). For more detail on how to efficiently compute the complete core decomposition of a multilayer network, we refer to [59].…”

Hyper-cores promote localization and efficient seeding in higher-order processes

“…In fact, the coreness of a node, defined as the largest value of k such that the node belongs to the corresponding k-core, gives a centrality measure that largely determines the impact of a spreading process initiated (seeded) in that node 13 . This decomposition has also been extended to weighted networks 16 , via the s-core decomposition (where s represents the strength of a node, i.e., the sum of the weights of its adjacent links) 17 , to temporally evolving networks 18,19 , to multilayer networks 20 and to bipartite networks [21][22][23] .…”