algorithms Article In Search of the Densest Subgraph András Faragó and Zohre R. Mojaveri * Department of Computer Science, Erik Jonsson School of Engineering and Computer Science, The University of Texas at Dallas, P.O.B. 830688, MS-EC31, Richardson, TX 75080, USA * Correspondence:
[email protected] Received: 23 June 2019; Accepted: 30 July 2019; Published: 2 August 2019 Abstract: In this survey paper, we review various concepts of graph density, as well as associated theorems and algorithms. Our goal is motivated by the fact that, in many applications, it is a key algorithmic task to extract a densest subgraph from an input graph, according to some appropriate definition of graph density. While this problem has been the subject of active research for over half of a century, with many proposed variants and solutions, new results still continuously emerge in the literature. This shows both the importance and the richness of the subject. We also identify some interesting open problems in the field. Keywords: dense subgraph; algorithms; graph density; clusters in graphs; big data 1. Introduction and Motivation In the era of big data, graph-based representations became highly popular to model various real-world systems, as well as diverse types of knowledge and data, due to the simplicity and visually appealing nature of graph models. The specific task of extracting a dense subgraph from a large graph has received a lot of attention, since it has direct applications in many fields. While the era of big data is still relatively young, the study and application of dense subgraphs started much earlier.