How Robust Is the Core of a Network?

The k-core is commonly used as a measure of importance and well connectedness for nodes in diverse applications in social networks and bioinformatics. Since network data is commonly noisy and incomplete, a fundamental issue is to understand how robust the core decomposition is to noise. Further, in many settings, such as online social media networks, usually only a sample of the network is available. Therefore, a related question is: How robust is the top core set under such sampling? We find that, in general, the top core is quite sensitive to both noise and sampling; we quantify this in terms of the Jaccard similarity of the set of top core nodes between the original and perturbed/sampled graphs. Most importantly, we find that the overlap with the top core set varies non-monotonically with the extent of perturbations/sampling. We explain some of these empirical observations by rigorous analysis in simple network models. Our work has important implications for the use of the core decomposition and nodes in the top cores in network analysis applications, and suggests the need for a more careful characterization of the missing data and sensitivity to it.