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While the use of networks to understand how complex systems respond to perturbations is pervasive across scientific disciplines, the uncertainty associated with estimates of pairwise interaction strengths (edge weights) remains rarely considered. Mischaracterizations of interaction strength can lead to qualitatively incorrect predictions regarding system responses as perturbations propagate through often counteracting direct and indirect effects. Here, we introduce PressPurt , a computational package for identifying the interactions whose strengths must be estimated most accurately in order to produce robust predictions of a network's response to press perturbations. The package provides methods for calculating and visualizing these edge-specific sensitivities (tolerances) when uncertainty is associated to one or more edges according to a variety of different error distributions. The software requires the network to be represented as a numerical (quantitative or qualitative) Jacobian matrix evaluated at stable equilibrium. PressPurt is open source under the MIT license and is available as both a Python package and an R package hosted at https://github.com/dkoslicki/PressPurt and on the CRAN repository https://CRAN.R-project.org/package=PressPurt. 

We investigate the problem of recovering jointly [Formula: see text]-rank and [Formula: see text]-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that [Formula: see text] measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when [Formula: see text] for some exponent [Formula: see text]. We show that this is feasible for [Formula: see text], and that the proposed analysis cannot cover the case [Formula: see text]. The precise value of the optimal exponent [Formula: see text] is the object of a question, raised but unresolved in this paper, about head projections for the jointly low-rank and bisparse structure. Some related questions are partially answered in passing. For rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements.