Extracting connectivity from dynamics of networks with uniform bidirectional coupling

In the study of networked systems, a method that can extract information about how the individual nodes are connected with one another would be valuable. In this paper, we present a method that can yield such information of network connectivity using measurements of the dynamics of the nodes as the only input data. Our method is built upon a noise-induced relation between the Laplacian matrix of the network and the dynamical covariance matrix of the nodes, and applies to networked dynamical systems in which the coupling between nodes is uniform and bidirectional. Using examples of different networks and dynamics, we demonstrate that the method can give accurate connectivity information for a wide range of noise amplitude and coupling strength. Moreover, we can calculate a parameter $\Delta$ using again only the input of time-series data, and assess the accuracy of the extracted connectivity information based on the value of $\Delta$.

Figure 1

Values of $r_i\left(m\right)$ as a function of $m$ for a certain node $i$ of the random network with $\sigma =1$ and $g=1$.

Figure 2

$p_{\text{sen}}\left(i\right)$ (circles) and $p_{\text{spec}}\left(i\right)$ (triangles) for random network with (a) consensus dynamics with $\sigma =g=1$, (b) FHN1 dynamics with $\sigma =1$ and $g=10$, and (c) FHN2 dynamics with $\sigma =1$ and $g=10$.

Figure 3

Same as Fig. 2 for the C. elegans network.

Figure 4

Same as Fig. 2 for the BA scale-free network.

Figure 5

Dependence of $p_{\text{sen}}\left(i\right)$ on $k_i$ for the BA scale-free network and FHN1 dynamics with $\sigma =1$ and $g=10$. The dashed lines are plots of $100[1-\left(\text{number}\text{of}\text{errors}\right)/k_i]$ with the number of errors being 1, 2, 3, 4, 6, 8, and 13 from left to right.

Figure 6

Phase diagram showing the performance of the method in the parameter space for the random network with consensus dynamics with $T_{\text{av}}=1000$. Satisfactory results (both $P_{\text{SEN}}$ and $P_{\text{SPEC}}$ greater than 95) are represented by open circles, while unsatisfactory results are represented by closed circles.

Figure 7

Comparison of the degree distribution $P\left(k\right)$ calculated using $k_i^{\left(e\right)}$ (triangles) against the one calculated using the actual $k_i$ (circles) for the BA scale-free network with (a) consensus dynamics with $g=\sigma =1$, (b) FHN1 dynamics with $\sigma =1$ and $g=10$, and (c) FHN2 dynamics with $\sigma =1$ and $g=10$.

Figure 8

Comparison of the actual eigenvalue spectrum (circles) for the random network with $N=100$ with the extracted eigenvalue spectrum for consensus dynamics with $g=\sigma =1$ (squares), FHN1 dynamics with $\sigma =1$ and $g=10$ (triangles), and FHN2 dynamics with $\sigma =1$ and $g=10$ (diamonds), respectively. The analytical result for the actual spectrum for $N\to \infty$ is also shown as the solid line.

Figure 9

Same symbols as in Fig. 8 for the C. elegans network.

Figure 10

Same symbols as in Fig. 8 for the BA scale-free network. The solid line is the spectrum obtained numerically in Ref. [25]. As a comparison, we show also the extracted spectrum from consensus dynamics with a small $g=0.1$ and the same $\sigma =1$ (dashed line) for which our method does not give an accurate extraction.