Figure 1

(Color online) Effects of different metrics and embeddings on the $\rho \left(\epsilon \right)$ relationship, expressed in terms of the corresponding first derivative. [(A)–(D)] Manhattan distance; [(E)–(H)] Euclidean distance; [(I)–(L)] maximum distance. [(A), (E), and (I)] Lorenz system $\left[\dot{x}=10\left(y-x\right),\dot{y}=x\left(28-z\right),\dot{z}=xy-\frac{8}{3}z\right]$ with original components at three different randomly chosen initial conditions. [(B), (F), and (J)] Same Lorenz system embedded from the $x$ component with embedding delays ${\tau }_1=5,{\tau }_2=15$, and ${\tau }_3=20$. [(C), (G), and (K)] same as [(A), (E), and (I)] for the Rössler system $\left[\dot{x}=-y-z,\dot{y}=x+0.2y,\dot{z}=z\left(x-5.7\right)\right]$. [(D), (H), and (L)] same as (B), (F), and (J) for the Rössler system with ${\tau }_1=10,{\tau }_2=15$, and ${\tau }_3=20$. Circles indicate the respective maxima. In all cases, time series of $N=1000$ points with a sampling time of $\Delta t=0.05$ have been used, obtained with a 4th-order Runge-Kutta integrator with fixed step width $h=0.01$. The values of ${\tau }_2$ are guided by the first zeros of the corresponding autocorrelation functions.Reuse & Permissions

Figure 2

Mean values (squares) and range (shaded areas) of turning points ${\epsilon }_{crit}$ in 200 independent realizations of the Lorenz (a) and the Rössler system (b) in dependence on the network size $N$ (Euclidean distance, $\Delta t=0.05$). The insets show the corresponding link densities ${\rho }_{crit}$ for the same range of $N$.Reuse & Permissions

Figure 3

Examples for multiple turning points of $d\rho /d\epsilon$: (a) quasiperiodic trajectory of a continuous system (torus) and (b) a weakly chaotic orbit of the standard map [$x_{n+1}=x_n+5\text{sin}\left(y_n\right)\text{mod}1$, $y_{n+1}=y_n+x_{n+1}\text{mod}1$, see [15]].Reuse & Permissions

Figure 4

(Color online) Probability distribution function of betweenness centrality $b_v$ (in logarithmic scale) for different edge densities $\left({\rho }_1=0.005,{\rho }_2=0.01,{\rho }_3=0.015,{\rho }_4=0.2\right)$ for the Lorenz (A, $N=20000$) and Rössler system (B, $N=10000$), and corresponding relationships between local clustering coefficient ${\mathcal{C}}_v$ and betweenness centrality $b_v$ (C: Lorenz, D: Rössler, $\rho =0.01$) obtained from the original data using the Euclidean distance $\left(\Delta t=0.05\right)$.Reuse & Permissions