Reconstruction of dynamic networks with time-delayed interactions in the presence of fast-varying noises

Most complex social, biological and technological systems can be described by dynamic networks. Reconstructing network structures from measurable data is a fundamental problem in almost all interdisciplinary fields. Network nodes interact with each other and those interactions often have diversely distributed time delays. Accurate reconstruction of any targeted interaction to a node requires measured data of all its neighboring nodes together with information on the time delays of interactions from these neighbors. When networks are large, these data are often not available and time-delay factors are deeply hidden. Here we show that fast-varying noise can be of great help in solving these challenging problems. By computing suitable correlations, we can infer the intensity and time delay of any targeted interaction with the data of two related nodes (driving and driven nodes) only while all other nodes in the network are hidden. This method is analytically derived and fully justified by extensive numerical simulations.

Figure 1

(a) Difficulties extensively appearing in the reconstruction of real-world networks such as complexity of network structures; strong nonlinearity of network dynamics; diverse and unknown impacts from the interiors of nodes and externals of networks, i.e., noises; many hidden nodes in networks; and different time delays of interactions. All the mentioned difficulties are present in our network reconstruction of Eq. (1). (b) Reconstruction of a local interaction from node $B$ to node $A$ based on minimal measured data of driving node $B$ and driven node $A$ only. All other nodes (black nodes) in the network are hidden and their data are unavailable. (c) Fast-varying noises $\mathbf{\Gamma }\left(t\right)$ [i.e., white noises in Eq. (2b)] can be of great help in correctly and accurately identifying directed interactions from node $B$ to node $A$, based on data of driving node $B$ and driven node $A$ only. With selected correlation terms of Eq. (8), fast-varying noise of node $B$ plays a key role in amplifying directed interactions from node $B$ to node $A$ only where correlation of these two nodes is dominant over those of all interactions from other nodes.

Figure 2

Reconstruction of the linear network (16) for $N=50$, ${\tau }_{ij}\in (0,0.25)$ with uniform distribution, $10\%$ active links randomly distributed in $M_{ij}\in (0.5,1.5)$, $10\%$ repressive links randomly distributed in $M_{ij}\in (-0.5,-1.5)$, and other links $M_{ij}=0$, $M_i=-8$, ${\tau }_C\in (10,20)$, $P_{ij}=0$ for $i\ne j$, $P_{ii}\in (0.5,1)$, and $Q_i\in (1,2)$. The numerical simulation is conducted with computation time step $dt=0.0005$ and measurement interval $\mathrm{\Delta }t=0.005$. In all the computations of our reconstruction algorithms of (12), (13), and (15), data of only a pair of related nodes $A$ and $B$ are used with all other nodes hidden [Fig. 1]. Different ${\tau }_{AB}$, $M_{AB}$, and $Q_B$ are computed by searching all different pairs. (a) A clear discontinuity in the correlation function of $V_B\left(\mathrm{\Delta }k\right)$ exists at $\mathrm{\Delta }k=0$. (b) Plot of $Q_B^{\prime }$ computed by Eq. (12) against the actual $Q_B$. The correlation function $R_{AB}\left(\mathrm{\Delta }k\right)$ is computed for (c) active, (d) null, and (e) repressive links. There is no discontinuity in (d) for a null link and discontinuity gaps in (c) and (e) at different $\mathrm{\Delta }k$ with different directions for active and repressive links, respectively. Also shown are (f) $n_{AB}^{\prime }$ and (g) $J_{AB}$, computed by Eqs. (13) and (15), plotted against the actual $n_{AB}$ and $M_{AB}$, respectively. The results of reconstruction are excellent, in particular, in (f), where the elements of the time-delay matrix $\widehat{\tau }$ are reconstructed with extremely high accuracy. (h) Error $E$ of Eq. (17) plotted against the computation time $T=L\mathrm{\Delta }t$ with different $\mathrm{\Delta }t$. Here $E$ decays as $\sqrt{T}$. The results of Eqs. (13) and (15) are exact for white noises in the limits of $\mathrm{\Delta }t\to 0$ and $T\to \infty$.

Figure 3

(a)–(c) Reconstruction of the neural network (18a) for $N=50$, ${\tau }_{ij}\in (0,0.25)$, $a=0.84$, $b=0.07$, $\varepsilon =0.04$, and $Q_i=Q=0.10$. The network is randomly constructed with link probability $p=0.20$. (a) Example of a pair of time series of $u_A\left(t\right)$ and $u_B\left(t\right)$ recorded, used in our reconstruction algorithms of Eqs. (13) and (15). (b) Time delay $n_{AB}$ plotted against those $n_{AB}^{\prime }$ computed by the algorithm (13). All dots are strictly at the diagonal line, showing that actual time delays in the interactions are accurately estimated. (c) Different noise intensities $Q$ are applied for reconstruction of the network (18a). In a wide range of noise intensities, the reconstruction algorithm (15) works robustly. It is remarkable that without noise $Q=0$ reconstruction fails completely. (d)–(f) Same as (a)–(c), respectively, with coupled Rössler oscillators (19) considered for $N=50$, ${\tau }_{ij}\in (0,0.25)$, ${\omega }_i\in (0.7,1.3)$, $a_i=0.2$, $b_i=0.2$, $c_i=9$, and $Q=5.0$. All predictions in Eqs. (13) and (15) are convincingly verified in the network reconstruction of Eq. (19) as well.

Figure 4

Reconstruction of gene regulatory network. (a) Network structure of the repressilator in Eq. (20) for the activation coefficient $\kappa =0.30$, Hill exponent $h=3.0$, ${\tau }_{ij}\in (0,0.25)$, and $Q=0.02$. (b) Two time series of $x_A\left(t\right)$ and $x_B\left(t\right)$ are available which are used in our reconstruction algorithms of Eqs. (13) and (15). It is demonstrated that all (c) time delays $n_{AB}$ and interaction signs and (d) intensities $M_{AB}$ can be accurately inferred by the algorithms of Eqs. (13) and (15), respectively, based on the minimal measured data of driving and driven nodes only.