The number of available algorithms to infer a biological network from a dataset of high-throughput measurements is overwhelming and keeps growing. However, evaluating their performance is unfeasible unless a 'gold standard' is available to measure how close the reconstructed network is to the ground truth. One measure of this is the stability of these predictions to data resampling approaches. We introduce NetSI, a family of Network Stability Indicators, to assess quantitatively the stability of a reconstructed network in terms of inference variability due to data subsampling. In order to evaluate network stability, the main NetSI methods use a global/local network metric in combination with a resampling (bootstrap or cross-validation) procedure. In addition, we provide two normalized variability scores over data resampling to measure edge weight stability and node degree stability, and then introduce a stability ranking for edges and nodes. A complete implementation of the NetSI indicators, including the Hamming-Ipsen-Mikhailov (HIM) network distance adopted in this paper is available with the R package nettools. We demonstrate the use of the NetSI family by measuring network stability on four datasets against alternative network reconstruction methods. First, the effect of sample size on stability of inferred networks is studied in a gold standard framework on yeast-like data from the Gene Net Weaver simulator. We also consider the impact of varying modularity on a set of structurally different networks (50 nodes, from 2 to 10 modules), and then of complex feature covariance structure, showing the different behaviours of standard reconstruction methods based on Pearson correlation, Maximum Information Coefficient (MIC) and False Discovery Rate (FDR) strategy. Finally, we demonstrate a strong combined effect of different reconstruction methods and phenotype subgroups on a hepatocellular carcinoma miRNA microarray dataset (240 subjects), and we validate the analysis on a second dataset (166 subjects) with good reproducibility.