Noise Bridges Dynamical Correlation and Topology in Coupled Oscillator Networks

We study the relationship between dynamical properties and interaction patterns in complex oscillator networks in the presence of noise. A striking finding is that noise leads to a general, one-to-one correspondence between the dynamical correlation and the connections among oscillators for a variety of node dynamics and network structures. The universal finding enables an accurate prediction of the full network topology based solely on measuring the dynamical correlation. The power of the method for network inference is demonstrated by the high success rate in identifying links for distinct dynamics on both model and real-life networks. The method can have potential applications in various fields due to its generality, high accuracy, and efficiency.

Figure 1

Distribution of the values of $[{\sigma }^2/(2c)]C_{ij}^{†}$, where $C_{ij}^{†}$ are the elements in the pseudo inverse matrix of the dynamical correlation matrix $\widehat{\mathbf{C}}$. Consensus dynamics [15] are used for (a) random [21], (b) small-world [22], (c) scale-free model networks [23] and three real-world networks: (d) friendship network of karate club [25], (e) network of American football games among colleges [26], and (f) the neural network of C. Elegans [22]. The theoretical threshold $[{\sigma }^2/(2c)]C_M^{†}$ is marked by red dashed lines. The sizes of model networks are all 500. For random networks, the connection probability among nodes is 0.024. For scale-free networks the minimum degree is $k_{\min }=6$. For small-world networks, $⟨k⟩=12$ and the rewiring probability is 0.1.

Figure 2

$C_{ii}$ as a function of node in-degree $k_{\mathrm{in}}$ for different node dynamics for directed networks where each link is assigned a random direction, and $\Lambda ={\sigma }^2(1+1/⟨k⟩)/2c$. Other network parameters are the same as in Fig. 1. The lines are predictions from Eq. (7).