No abstract

A measure of centrality is rank monotone if after adding an arc x → y, all nodes with a score smaller than (or equal to) y have still a score smaller than (or equal to) y. If, in particular, all nodes with a score smaller than or equal to y get a score smaller than y (i.e., all ties with y are broken in favor of y), the measure is called strictly rank monotone. We prove that harmonic centrality is strictly rank monotone, whereas closeness is just rank monotone on strongly connected graphs, and that some other measures, including betweenness, are not rank monotone at all (sometimes not even on strongly connected graphs). Among spectral measures, damped scores such as Katz's index and PageRank are strictly rank monotone on all graphs, whereas the dominant eigenvector is strictly monotone on strongly connected graphs only.

The expansion of coronavirus disease 2019 (COVID-19) prompted measures of disease containment by the Italian government with a national lockdown on March 9, 2020. The purpose of this study is to evaluate the rate of hospitalization and mode of in-hospital treatment of patients with chronic limb-threatening ischemia (CLTI) before and during lockdown in the Campania region of Italy. The study population includes all patients with CLTI hospitalized in Campania over a 10-week period: 5 weeks before and 5 weeks during lockdown ( n = 453). Patients were treated medically and/or underwent urgent revascularization and/or major amputation of the lower extremities. Mean age was 69.2 ± 10.6 years and 27.6% of the patients were women. During hospitalization, 21.9% of patients were treated medically, 78.1% underwent revascularization, and 17.4% required amputations. In the weeks during the lockdown, a reduced rate of hospitalization for CLTI was observed compared with the weeks before lockdown (25 vs 74/100,000 inhabitants/year; incidence rate ratio: 0.34, 95% CI 0.32–0.37). This effect persisted to the end of the study period. An increased amputation rate in the weeks during lockdown was observed (29.3% vs 13.4%; p < 0.001). This study reports a reduced rate of CLTI-related hospitalization and an increased in-hospital amputation rate during lockdown in Campania. Ensuring appropriate treatment for patients with CLTI should be prioritized, even during disease containment measures due to the COVID-19 pandemic or other similar conditions.