We present a dynamically equivalent transformation between time series and complex networks based on coarse geometry theory. In terms of quasi-isometric maps, we characterize how the underlying geometrical characters of complex systems are preserved during transformations. Fractal dimensions are shown to be the same for time series (or complex network) and its transformed counterpart. Results from the Rössler system, fractional Brownian motion, synthetic networks, and real networks support our findings. This work gives theoretical evidences for an equivalent transformation between time series and networks.

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.

In Corollary 4.10 of [2], we have to assume that the group G is finitely presented. This is due to a gap in the proof of Lemma 2.2 when claiming that the real reduced group C * -algebra C * R (G) is G-dense. The proof only works for finitely generated free modules. We need to modify Definition 2.1 as follows.