Reconstructing links in directed networks from noisy dynamics

We address the long-standing challenge of how to reconstruct links in directed networks from measurements, and present a general method that makes use of a noise-induced relation between network structure and both the time-lagged covariance of measurements taken at two different times and the covariance of measurements taken at the same time. When the coupling functions have certain additional properties, we can further reconstruct the weights of the links.

Figure 1

Verification of $M_{ij}=Q_{ij}=g_{ij}{A}_{ij}{h}_y(X_i,X_j)$ for $i\ne j$. (a) Case 2 with nonuniform noise and case 3 with inhomogeneous $f_i$ (inset), (b) case 5 with both excitatory and inhibitory links, (c) case 7, and (d) case 9 with $N_{\text{data}}=4\times {10}^6$ and $N_{\text{data}}=2\times {10}^6$ (inset). See Table 2 for detailed descriptions of the different cases. Dashed line is $y=x$.

Figure 2

Comparison of the reconstructed in degrees ${\widehat{k}}_{\mathrm{in}}$ (triangles) and out degrees ${\widehat{k}}_{\mathrm{out}}$ (circles) with the actual values for (a) case 5 and (b) case 9 with $N_{\text{data}}=4\times {10}^6$.

Figure 3

Comparison of reconstructed ${\widehat{G}}_{ij}$ and ${\widehat{G}}_j^{\mathrm{out}}\left(i\right)$ with the actual values for (a) case 5 and (b) case 9 with $N_{\text{data}}=4\times {10}^6$. Missed links (red) have $G_{ij}\ne 0$ $[G_j^{\text{out}}\left(i\right)\ne 0]$ but ${\widehat{G}}_{ij}=0$ $[{\widehat{G}}_j^{\mathrm{out}}\left(i\right)=0]$, while incorrectly inferred links (blue) have $G_{ij}=0$ $[G_j^{\mathrm{out}}\left(i\right)=0]$ but ${\widehat{G}}_{ij}\ne 0$ $[{\widehat{G}}_j^{\mathrm{out}}\left(i\right)\ne 0]$. Dashed line is $y=x$.