Abstract
In a coupled system, predictive information flows from the causing to the caused variable. The amount of transferred predictive information can be quantified through the use of transfer entropy or, for Gaussian variables, equivalently via Granger causality. It is natural to expect and has been repeatedly observed that a tight coupling does not permit to reconstruct a causal connection between causing and caused variables. Here, we show that for a model of interacting social groups, carried from the master equation to the Fokker–Planck level, a residual predictive information flow can remain for a pair of uni-directionally coupled variables even in the limit of infinite coupling strength. We trace this phenomenon back to the question of how the synchronizing force and the noise strength scale with the coupling strength. A simplified model description allows us to derive analytic expressions that fully elucidate the interplay between deterministic and stochastic model parts.
Highlights
In many scientific problems, causal interactions of different processes or process components are clear from a mechanistic understanding of the system dynamics and reflected by model formulation.In other situations, the interaction structure of a large and/or complex system, e.g., the assembly dynamics of a chemical system [1,2], of an ecological [3,4] or social community [5,6] or of neuronal populations [7,8], may not be clear in advance
We find that a linearized dynamics with average-noise with state space averaged additive noise gives rise to the same values
The question whether the found residual Granger causality (GC) is a property of the specific socio-dynamic model or a generic property of a larger class of systems can be answered in the following way: our results follow from a minimalistic linear system (vector-Ornstein–Uhlenbeck process or VAR(1)), which means it is not a fancy effect of non-linear dynamics
Summary
Causal interactions of different processes or process components are clear from a mechanistic understanding of the system dynamics and reflected by model formulation.In other situations, the interaction structure of a large and/or complex system, e.g., the assembly dynamics of a chemical system [1,2], of an ecological [3,4] or social community [5,6] or of neuronal populations [7,8], may not be clear in advance. Causal interactions of different processes or process components are clear from a mechanistic understanding of the system dynamics and reflected by model formulation. Since the seminal works of Wiener [9] and Granger [10], this reconstruction is based on the rationale that a causal interaction can be inferred from an increased prediction error when excluding the causing process component from the prediction of the caused component. Operationalizing this idea leads to Granger causality (GC), a measure commonly used to compare causal interactions of different process components. As shown by Barnett et al [12], Granger causality and transfer entropy are equivalent for Gaussian variables
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