The combination of dynamical processes and networks has become the paradigmatic framework to model a wide variety of phenomena in complex systems and many significant advances have been achieved in the last decades towards a complete understanding on the interplay between structure and function. Furthermore, we have an increasing amount of empirical data that can be used to validate the theoretical models and build more sophisticated ones. However, the data is usually incomplete and show errors that can produce inaccurate theoretical predictions which may lead to dramatic consequences in real systems (as e.g. the underestimation of the epidemic threshold in the spreading of a disease or the abrupt synchronization threshold of neurones in epilepsy attacks).

Motivated by the unavoidable presence of errors in the measurements of real complex networks, in this work we tackle the problem of quantifying the uncertainty of the critical threshold when induced by microscopic noise in the weights of the network interactions. The obtained expressions, based on applying error propagation methods, are valid for a wide variety of dynamical processes and depend non-linearly on the explicit binary network structure and the features of the noise considered in the microscopic weights. We validate the theoretical predictions in synthetic and empirical networks with very different structural properties, obtaining good agreement against numerical simulations in most of the cases. Furthermore, we show that the heterogeneity in the degree distribution and other non-trivial connectivity patterns tend to amplify the fluctuations of the critical point, enhancing the macroscopic critical range of the system.

The proposed formalism also provides a quantitative and analytical tool to estimate confidence intervals and error bars in the prediction of the critical threshold, which is usually the most relevant quantity from both a theoretical point of veiw and for actuation and control purposes. The results can be applied in systems where the microscopic fluctuations can occur both due to the presence of intrinsic time-varying noise in the interactions among the units or due to a lack of precision in the measurement of the actual weights.