Abstract
Community detection is an important task in the analysis of biological, social or technical networks. We survey different models of cohesive graphs, commonly referred to as clique relaxations, that are used in the detection of network communities. For each clique relaxation, we give an overview of basic model properties and of the complexity of the problem of finding large cohesive subgraphs under this model. Since this problem is usually NP-hard, we focus on combinatorial fixed-parameter algorithms exploiting typical structural properties of input networks.
Highlights
Networks, or in mathematical terms, graphs, are the standard tool to model interactions between entities
To gain an overview of the relationships between these tribes, Read [4] determined in a field study for pairs of tribes whether they have a friendly or hostile relationship and summarized these data in a social network
To better understand the society of the highland tribes, one might be interested in determining whether there are any communities of friendly tribes in this network
Summary
In mathematical terms, graphs, are the standard tool to model interactions between entities. We consider models relaxing the following clique-defining properties: Clique definition via vertex degrees: A size-k set S in a graph G is a clique if and only if every vertex v ∈ S has degree k − 1 in G [S], the subgraph induced by S. The degeneracy of a graph G is the smallest integer d such that every subgraph of G has at least one vertex whose degree is at most d This number is upper-bounded by the maximum degree, and in most real-world networks, it is much smaller than ∆. The value of the h-index is sandwiched between the degeneracy and the maximum degree of a graph The theme of this survey will be to inspect the computational problems involved in applying different types of clique relaxations from the viewpoint of multivariate algorithmics. We give a condensed overview of the considered clique relaxations and the corresponding complexity results
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