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The Densest Subgraph Problem requires to find, in a given graph, a subset of vertices whose induced subgraph maximizes a measure of density. The problem has received a great deal of attention in the algorithmic literature since the early 1970s, with many variants proposed and many applications built on top of this basic definition. Recent years have witnessed a revival of research interest in this problem with several important contributions, including some groundbreaking results, published in 2022 and 2023. This survey provides a deep overview of the fundamental results and an exhaustive coverage of the many variants proposed in the literature, with a special attention to the most recent results. The survey also presents a comprehensive overview of applications and discusses some interesting open problems for this evergreen research topic., Comment: Accepted to ACM Computing Surveys

Mining human-brain networks to discover patterns that can be used to discriminate between healthy individuals and patients affected by some neurological disorder, is a fundamental task in neuroscience. Learning simple and interpretable models is as important as mere classification accuracy. In this paper we introduce a novel approach for classifying brain networks based on extracting contrast subgraphs, i.e., a set of vertices whose induced subgraphs are dense in one class of graphs and sparse in the other. We formally define the problem and present an algorithmic solution for extracting contrast subgraphs. We then apply our method to a brain-network dataset consisting of children affected by Autism Spectrum Disorder and children Typically Developed. Our analysis confirms the interestingness of the discovered patterns, which match background knowledge in the neuroscience literature. Further analysis on other classification tasks confirm the simplicity, soundness, and high explainability of our proposal, which also exhibits superior classification accuracy, to more complex state-of-the-art methods., Comment: To be published at KDD 2020

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. In this work we study the problem of core decomposition of a multilayer network. The multilayer context is much 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 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. 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.