Asymptotic bounds for clustering problems in random graphs

Graph clustering is an important problem in network analysis. This problem can be approached by first finding a large cluster subgraph (i.e., a subgraph in which every connected component is a complete graph), perhaps in a relaxed form (connected components may have missing edges), and then assigning each of the remaining vertices to one of the connected components of the cluster subgraph according to some optimization criteria. The more vertices can be included in the initial cluster subgraph (also referred to as independent union of clusters), the more “clusterable” the graph is. This paper proposes a framework for establishing asymptotic bounds on the cardinality of independent unions of clusters in Erdős-Rényi random graphs $G(n,p)$ with constant $p$, referred to as uniform random graphs. In particular, sufficient conditions ensuring $O(\log n)$ (where $n$ is the number of nodes) upper bounds with probability 1 are developed and shown to be applicable for the maximum independent union of cliques as well as some clique relaxations. In addition, it is shown that every graph must have an independent union of cliques of cardinality at least $\mathrm{\Omega }(\log n)$. Since this bound is asymptotically tight on uniform random graphs, this suggests that these graphs can be viewed as a “least clusterable” class of graphs.