Effects of hidden nodes on the reconstruction of bidirectional networks

Much research effort has been devoted to developing methods for reconstructing the links of a network from dynamics of its nodes. Many current methods require the measurements of the dynamics of all the nodes to be known. In real-world problems, it is common that either some nodes of a network of interest are unknown or the measurements of some nodes are unavailable. These nodes, either unknown or whose measurements are unavailable, are called hidden nodes. In this paper, we derive analytical results that explain the effects of hidden nodes on the reconstruction of bidirectional networks. These theoretical results and their implications are verified by numerical studies.

Figure 1

A path connecting the measured nodes $i$ and $j$ (open circles) via the hidden nodes ${\mu }_1,{\mu }_2,...,{\mu }_{m-1},{\mu }_m$ (closed circles).

Figure 2

Comparison of the distributions $P(X=x)$ of $X={\mathrm{\Sigma }}_{ij}^{-1}$ (circles) and $X={\left({\mathrm{\Sigma }}_m^{-1}\right)}_{ij}$ (triangles) for $i\ne j=1,2,...,n=50$ for network A.

Figure 3

Comparison of distributions $P(X=x)$ of $X={\mathrm{\Sigma }}_{ij}^{-1}$ (circles) and $X={\left({\mathrm{\Sigma }}_m^{-1}\right)}_{ij}$ (triangles) for network B with logistic dynamics and different number of hidden nodes: (a) $n_h=10$, (b) $n_h=30$, (c) $n_h=50$, and (d) $n_h=70$. The dashed lines are the theoretical results of $X={\mathrm{\Sigma }}_{ij}^{-1}-2C_{ij}/{\sigma }_n^2$ with $C_{ij}$ calculated using Eq. (20).

Figure 4

Same as described in the caption of Fig. 3 for network B with FHN dynamics but the dashed lines are now the theoretical estimates of $X={\mathrm{\Sigma }}_{ij}^{-1}-(2/{\sigma }_n^2){\left(E{L}_h^{-1}{E}^T\right)}_{ij}$. The dot-dashed lines are the improved theoretical estimates of $X={\mathrm{\Sigma }}_{ij}^{-1}-(2/{\sigma }_n^2)[{\left(E{L}_h^{-1}{E}^T\right)}_{ij}+b]$ with $b=1$ for (c) and $b=3$ for (d).

Figure 5

Same as described in the caption of Fig. 4 for network B with Rössler dynamics.

Figure 6

Similar to Fig. 3 for network C with both positive and negative $g_{ij}$'s for two different numbers of hidden nodes: (a) $n_h=20$ and (b) $n_h=40$.

Figure 7

Similar to Fig. 3 for scale-free network D with (a) $n_h=100$ and (b) $n_h=700$.

Figure 8

Comparison of distributions $P(X=x)$ of $X={\mathrm{\Sigma }}_{ij}^{-1}$ (circles) and $X={\left({\mathrm{\Sigma }}_m^{-1}\right)}_{ij}$ (triangles) for network E with $n_h=30$ hidden nodes that are preferentially linked to measured nodes that are unconnected. The dashed line is the theoretical result of $X={\mathrm{\Sigma }}_{ij}^{-1}-2C_{ij}/{\sigma }_n^2$ with $C_{ij}$ calculated using Eq. (20).