Crossover and DPI-violation extent for transfer entropy under feedback

Determine, for three-variable stochastic processes with feedback of the form X1 ↔ X2 → X3, the location of the crossover between the feedforward and feedback-dominated regimes and quantify the extent to which the transfer entropy from X1 to X3 can exceed the transfer entropy from X1 to X2, thereby overcoming the data processing inequality in practice.

Background

In networks without feedback, the data processing inequality (DPI) ensures that mutual information and, equivalently, transfer entropy from an input to a downstream node does not increase along a feedforward chain. With feedback present, mutual information continues to obey the DPI, but transfer entropy does not necessarily do so, allowing potential DPI violations.

The paper studies the three-node motif X1 ↔ X2 → X3 and uses the TE-PWS algorithm to compute transfer entropy exactly, revealing empirical DPI violations as feedback increases. Despite these findings, the precise parameter regimes governing the crossover from feedforward to feedback-dominated information transfer and the quantitative extent of DPI violation remain explicitly unidentified.