| Summary: | Granger causality (GC) is a powerful method for causal inference for time series. In general, the GC value is computed using discrete time series sampled from continuous-time processes with a certain sampling interval length $tau$, emph{i.e.}, the GC value is a function of $tau$. Using the GC analysis for the topology extraction of the simplest integrate-and-fire neuronal network of two neurons, we discuss behaviors of the GC value as a function of $tau$, which exhibits (i) oscillations, often vanishing at certain finite sampling interval lengths, (ii) the GC vanishes linearly as one uses finer and finer sampling. We show that these sampling effects can occur in both linear and nonlinear dynamics: the GC value may vanish in the presence of true causal influence or become nonzero in the absence of causal influence. Without properly taking this issue into account, GC analysis may produce unreliable conclusions about causal influence when applied to empirical data. These sampling artifacts on the GC value greatly complicate the reliability of causal inference using the GC analysis, in general, and the validity of topology reconstruction for networks, in particular. We use idealized linear models to illustrate possible mechanisms underlying these phenomena and to gain insight into the general spectral structures that give rise to these sampling effects. Finally, we present an approach to circumvent these sampling artifacts to obtain reliable GC values.
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