Ambiguities in recurrence-based complex network representations of time series

Recently, different approaches have been proposed for studying basic properties of time series from a complex network perspective. In this work, the corresponding potentials and limitations of networks based on recurrences in phase space are investigated in some detail. We discuss the main requirements that permit a feasible system-theoretic interpretation of network topology in terms of dynamically invariant phase-space properties. Possible artifacts induced by disregarding these requirements are pointed out and systematically studied. Finally, a rigorous interpretation of the clustering coefficient and the betweenness centrality in terms of invariant objects is proposed.

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.

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 $N$.

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]].

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 $\left(\Delta t=0.05\right)$.