Distance-Based Clique Relaxations in Networks: s-Clique and s-Club

The concept of the clique, originally introduced as a model of a cohesive subgroup in the context of social network analysis, is a classical model of a cluster in networks. However, the ideal cohesiveness properties guaranteed by the clique definition put limitations on its applicability to situations where enforcing such properties is unnecessary or even prohibitive. Motivated by practical applications of diverse origins, numerous clique relaxation models, which are obtained by relaxing certain properties of a clique, have been introduced and studied by researchers representing different fields. Distance-based clique relaxations, which replace the requirement on pairwise distances to be equal to 1 in a clique with less restrictive distance bounds, are among the most important such models. This chapter surveys the up-to-date progress made in studying two common distance-based clique relaxation models called s-clique and s-club, as well as the corresponding optimization problems.