Investigating the relationship between k-core and s-core network decompositions

Network decomposition methods, such as the much used k-core analysis, are able to identify globally central regions of networks. The decomposition approaches are hierarchical and identify nested sets of nodes with increasing centrality properties. While most studies have been concerned with unweighted networks, i.e. k-core analysis, recent works have introduced network decomposition methods that apply to weighted networks. Here, we investigate the relationship between k-core decomposition for unweighted networks and s-core decomposition for weighted networks by systematically employing a link-weight scheme that gradually discretizes the link weights. We applied this approach to the Erdős–Rényi model and the scale-free configuration model for five different weight distributions, and two empirical networks, the US air traffic network and a Facebook network. We find that (1) both uniformly random and positively correlated link-weight distributions give rise to highly stable s-core decompositions with respect to discretization levels. (2) For negatively correlated link-weight distributions, the resulting s-core decomposition has no similarity to the k-cores. Since several combinations of network topology and link-weight distributions give rise to a core-structure that is highly similar to the full s-core for a large range of link-discretization levels, it is possible to significantly speed up the numerical s-core analysis for these situations.