Inferring directed networks using a rank-based connectivity measure

Inferring the topology of a network using the knowledge of the signals of each of the interacting units is key to understanding real-world systems. One way to address this problem is using data-driven methods like cross-correlation or mutual information. However, these measures lack the ability to distinguish the direction of coupling. Here, we use a rank-based nonlinear interdependence measure originally developed for pairs of signals. This measure not only allows one to measure the strength but also the direction of the coupling. Our results for a system of coupled Lorenz dynamics show that we are able to consistently infer the underlying network for a subrange of the coupling strength and link density. Furthermore, we report that the addition of dynamical noise can benefit the reconstruction. Finally, we show an application to multichannel electroencephalographic recordings from an epilepsy patient.

Figure 1

Different coupling strengths and link densities lead to different qualities of the network reconstruction. Sorted $\overline{W}$ values for four different exemplary cases (a–d) computed for the heterogeneous noise-free case. For cases (a–c), we fix $\rho =0.1$ with $\varepsilon =0$, 0.665, and 4.92, respectively. For case (d), we use $\rho =0.4$ and $\varepsilon =4.92$. Values corresponding to links (no links) are marked in green (black).

Figure 2

ROC curves allow us to quantify the accuracy of the reconstruction for a variable threshold. ROC curves for a heterogeneous noise-free system using different values of $\rho$ and $\varepsilon$. The curves A–D correspond to the cases (a–d) in Fig. 1. The blue line corresponds to the ROC curve of a random classification. In the legend we show the area under the ROC curve $\mathrm{\Omega }$.

Figure 3

Performance of the network reconstruction for the noise-free heterogeneous dynamics is highest for sparse networks with intermediate coupling between nodes. Area under the ROC curve $\mathrm{\Omega }$ averaged across $S=10$ independent realizations of the adjacency matrix $A$ in dependence on the link probability $\rho$ and the coupling strength $\varepsilon$. Values of $\varepsilon$ are placed equidistantly on a logarithmic scale. Only when $\mathrm{\Omega }=1$ across all the $S$ realizations we plot the corresponding field in green. Labels from A to D mark the set of parameters $(\rho ,\varepsilon )$ used in both Figs. 1 and 2.

Figure 4

Compared to the results for the heterogeneous noise-free system, the performance of the network reconstruction is degraded for the homogeneous noise-free system. Same as Fig. 3 but here for the homogeneous noise-free case.

Figure 5

For intermediate and high coupling strengths, heterogeneity helps the reconstruction. Difference in performance between the heterogeneous and the homogeneous systems $\mathrm{\Delta }\mathrm{\Omega }$ shown in Figs. 3 and 4. Blue and red correspond to higher performance obtained for the heterogeneous and homogeneous system, respectively.

Figure 6

Depending of the choice of $\varepsilon$ and $\rho$, there exist an ideal level of noise which improves the reconstruction as compared to the homogeneous noise-free case. Average performance of the reconstruction for $S=10$ adjacency matrix realizations as a function of the noise level for some characteristic values of $\varepsilon$ keeping $\rho =0.5$. The error bars represent the standard deviation across the ten different realizations.

Figure 7

Dynamical noise increases the performance for a wide range of parameters. Same as Fig. 3 but here for the homogeneous case with noise level of $\sqrt{2D}=0.5$.

Figure 8

Performance of the network reconstruction is decreased for increasing size $M$ when the link density $\rho$ is not adjusted. Area under the ROC curve $\mathrm{\Omega }$ averaged across $S=5$ independent realizations of the adjacency matrix $A$ in dependence on the network size $M$ and the coupling strength $\varepsilon$. Values of $\varepsilon$ are placed equidistantly on a logarithmic scale. We use a density $\rho =0.1$ for all the cases. Only when $\mathrm{\Omega }=1$ across all the $S$ realizations we plot the corresponding field in green.

Figure 9

Performance of the network reconstruction for the noise-free heterogeneous dynamics does not change if the system size grows. Same as described in the caption of Fig. 8 but using ${\rho }_M$.

Figure 10

High variability of connectivity structure is found across the different periods included in the EEG recording from an exemplary seizure. Temporal averages ${\langle W\rangle }_{\mathrm{period}}$ for the third seizure for the periods preictal (a), ictal (b), and postictal (c). Values below the mean plus one standard deviation taken separately across the elements of the matrix for three periods are displayed in white. Black lines delimit the different regions of the brain where the electrodes were placed. Contact names placed in the SOZ are marked with an asterisk.

Figure 11

A high overall connectivity is found during the seizure with a predominant direction from the SOZ to the remaining brain areas at the beginning of the seizure. (a) Spatially averaged connectivity matrix ${\overline{W}}^z$ in dependence on time for the seven seizures. (b) Spatially mean connectivity of SOZ driving the rest of the contacts (${\overline{W}}_{S\to R}$) minus SOZ being driven (${\overline{W}}_{R\to S}$) by the rest of the contacts for the same seven seizures. The gray frames depict windows that lay between two distinct periods. Left from the left frame is the preictal period, between the frames is the ictal period, and right from the right frame is the postictal period.