Exact probabilities for the indeterminacy of complex networks as perceived through press perturbations

TL;DR

This paper derives exact formulas to quantify the likelihood of incorrect qualitative predictions in complex networks under uncertainty, enhancing probabilistic understanding of network responses to perturbations.

Contribution

It introduces exact formulas for predicting the probability of incorrect responses in complex networks considering uncertainties, reducing reliance on simulations.

Findings

Exact formulas for expected incorrect predictions

Tools for identifying sensitive network links

Methods to determine error bounds for qualitative predictions

Abstract

We consider the goal of predicting how complex networks respond to chronic (press) perturbations when characterizations of their network topology and interaction strengths are associated with uncertainty. Our primary result is the derivation of exact formulas for the expected number and probability of qualitatively incorrect predictions about a system's responses under uncertainties drawn form arbitrary distributions of error. These formulas obviate the current use of simulations, algorithms, and qualitative modeling techniques. Additional indices provide new tools for identifying which links in a network are most qualitatively and quantitatively sensitive to error, and for determining the volume of errors within which predictions will remain qualitatively determinate (i.e. sign insensitive). Together with recent advances in the empirical characterization of uncertainty in ecological networks, these tools bridge a way towards probabilistic predictions of network dynamics.