Analysis on the Effects of Graph Perturbations on Centrality Metrics

On the sensitivity of centrality metrics

“…In this section, the experiments related to answer to RQ 1 are shown. Herein, indeed, we want to understand under which circumstances a perturbation on a graph G and, consequently, on its adjacency matrix A can be regarded as small [ 3 ]. Hence, we computed and evaluated the ψ variation (see Eq 3 ) as a function of p in the Uniform model (see Fig 1 ) and as a function of τ in the Best Connected Model (see Fig 2 ).…”

“…As we already discussed in [ 3 ], when we target a small fraction of nodes ( i.e ., τ = 0.1), then the ψ trend keeps constant up to p = 0.85; then, for p > 0.85, a steep decrease is noticed. This means that a small fraction of effectively failed nodes does not cause a relevant variation on the norm of the perturbation matrix.…”

“…This alteration compromises the connectivity of G and, consequently, the functionality of the system that G represents that, in the worst circumstances, such connectivity loss may lead to stopping the functioning of the whole system. For instance, it is well known that the failure of some routers on the Internet could cause the interruption of communications on a global scale [ 3 , 9 , 10 ]. In addition, in the work of Callaway et al .…”

“…Despite the extensive literature supporting the study of graph robustness upon node (and/or edge) failures, there is not enough knowledge about the consequences that the deactivation of some nodes along with their incident edges would have on node centrality [ 3 ]. What is more, the problem is usually addressed by the first assumption that all nodes in G can fail.…”

“…We target to answering the following two research questions through not only by an experimental evaluation, which was conducted in our previous work [ 3 ] but also and above all throughout a detailed and focused mathematical investigation on the reasons behind: RQ 1 Under which conditions can we classify a perturbation as “small” in the Uniform and in the Best Connected probabilistic failure models, respectively? RQ 2 How does the norm of the centrality vector vary in the two models?…”