The appropriateness of ignorance in the inverse kinetic Ising model

Abstract

We develop efficient ways to consider and correct for the effects of hidden units for the paradigmatic case of the inverse kinetic Ising model with fully asymmetric couplings. We identify two sources of error in reconstructing the connectivity among the observed units while ignoring part of the network. One leads to a systematic bias in the inferred parameters, whereas the other involves correlations between the visible and hidden populations and has a magnitude that depends on the coupling strength. We estimate these two terms using a mean field approach and derive self-consistent equations for the couplings accounting for the systematic bias. Through application of these methods on simple networks of varying relative population size and connectivity strength, we assess how and under what conditions the hidden portion can influence inference and to what degree it can be crudely estimated. We find that for weak to moderately coupled systems, the effects of the hidden units is a simple rotation that can be easily corrected for. For strongly coupled systems, the non-systematic term becomes large and can no longer be safely ignored, further highlighting the importance of understanding the average strength of couplings for a given system of interest.