For deriving maximal advantage from information on biomacromolecular flexibility and rigidity, results from rigidity analyses must be linked to biologically relevant characteristics of a structure. Here, we describe the Python-based software package Constraint Network Analysis (CNA) developed for this task. CNA functions as a front- and backend to the graph-based rigidity analysis software FIRST. CNA goes beyond the mere identification of flexible and rigid regions in a biomacromolecule in that it (I) provides a refined modeling of thermal unfolding simulations that also considers the temperature-dependence of hydrophobic tethers, (II) allows performing rigidity analyses on ensembles of network topologies, either generated from structural ensembles or by using the concept of fuzzy noncovalent constraints, and (III) computes a set of global and local indices for quantifying biomacromolecular stability. This leads to more robust results from rigidity analyses and extends the application domain of rigidity analyses in that phase transition points ("melting points") and unfolding nuclei ("structural weak spots") are determined automatically. Furthermore, CNA robustly handles small-molecule ligands in general. Such advancements are important for applying rigidity analysis to data-driven protein engineering and for estimating the influence of ligand molecules on biomacromolecular stability. CNA maintains the efficiency of FIRST such that the analysis of a single protein structure takes a few seconds for systems of several hundred residues on a single core. These features make CNA an interesting tool for linking biomacromolecular structure, flexibility, (thermo-)stability, and function. CNA is available from http://cpclab.uni-duesseldorf.de/software for nonprofit organizations.