Optimal Noise-Canceling Networks

Natural and artificial networks, from the cerebral cortex to large-scale power grids, face the challenge of converting noisy inputs into robust signals. The input fluctuations often exhibit complex yet statistically reproducible correlations that reflect underlying internal or environmental processes such as synaptic noise or atmospheric turbulence. This raises the practically and biophysically relevant question of whether and how noise filtering can be hard wired directly into a network’s architecture. By considering generic phase oscillator arrays under cost constraints, we explore here analytically and numerically the design, efficiency, and topology of noise-canceling networks. Specifically, we find that when the input fluctuations become more correlated in space or time, optimal network architectures become sparser and more hierarchically organized, resembling the vasculature in plants or animals. More broadly, our results provide concrete guiding principles for designing more robust and efficient power grids and sensor networks.

Figure 1

Even for spatially incoherent white noise input $\sigma$, $\tau \to 0$, optimal NCNs exhibit a nontrivial sparse topology independent of the nonlinear steady state. (a) The fraction of loops $f_{\ell }=N_{\ell }/N_{\mathrm{tri}}$, where $N_{\ell }$ is the number of loops in the optimized network and $N_{\mathrm{tri}}$ is the number of loops in a triangular grid, measures the topology of optimal networks. Each of the $30\times 30$ pixels in the cost-convexity phase diagram is an average over 15 optimal networks obtained for different uniformly random initial $B_{ij}$. In the white domain, no solutions to Eq. (2) were found. The NCN topology $f_{\ell }$ is effectively independent of $C$. Panels △, ☆, □ show examples of optimal NCNs with different sparsities, with edge thicknesses proportional to $B_{ij}^{\alpha }$. Backgrounds show one instance of the spatial feed-ins ${\xi }_i(t_0)$ normalized to $(-1,1)$. (b) Time-averaged variance $⟨|\mathit{ϵ}(t){|}^2⟩$ and instantaneous variances $|\mathit{ϵ}(t){|}^2$ (faint) obtained from numerical solutions of Eq. (1) on uniform and optimized network topologies for $\alpha =0.25$ (△) and $\alpha =0.5$ (☆) with edge cost $C=1$ and centered inputs. Analytically predicted variances (dashed) agree with the simulations.

Figure 2

Spatial and temporal input correlations lead to a similar hierarchical NCN organization despite acting through different mechanisms. (a),(b) Gaussian spatial correlations $\sigma >0$ with temporal white noise $\tau \to 0$. The loop fractions $f_{\ell }$ in (a) show that NCN topology depends largely on $\alpha$, although the transition between loopy and sparse phase shifts when the correlation scale $\sigma$ approaches the mean edge length $L_b$. For $\sigma \gg L_b$ networks become sparser when $\alpha \sim 1$. (b) The coupling variance ${\sigma }_B$, normalized by the mean ${\mu }_B$, indicates that nonuniform hierarchical patterns and sparsity are strongly correlated. (c),(d) Ornstein-Uhlenbeck colored noise $\tau >0$ with spatially incoherent feed-ins $\sigma \to 0$ shows hierarchical patterns similar to those in panels (a),(b). Examples of optimal networks at the positions marked by symbols in the phase diagrams illustrate the transitions from dense uniform networks to sparse hierarchical networks with increasing spatial or temporal correlation. Each of the $30\times 30$ pixels in (a)–(d) is an average over 15 optimal networks.

Figure 3

Combining spatial and temporal correlations leads to three qualitatively distinct NCN phases in the $(\tau ,\sigma )$ plane. (a) The loop density $f_{\ell }$ characterizes the three phases as follows: Short correlation times $\tau$ and short correlation lengths $\sigma$ favor highly reticulate redundant networks ($⊲$), large $\tau$ and small $\sigma$ lead to a moderate reticulation ($⊳$), whereas large $\tau$ and large $\sigma$ selects low reticulation ($△$). (b) The coupling spread ${\sigma }_B/{\mu }_B$ indicates a similar division of the $(\tau ,\sigma )$-phase plane: Low $\tau$, $\sigma$ lead to highly uniform networks ($⊲$), high $\tau$ and low $\sigma$ lead to networks with an intermediate coupling variability ($⊳$), and high $\tau$, $\sigma$ lead to strongly hierarchical networks with large spread in the couplings $B_{ij}$ ($△$). The three phases are separated approximately by the lines $\tau /{\gamma }^{-1}\sim 1$ and $\sigma /L_b\sim 1$. Each pixel in the $30\times 30$ plots (a),(b) is an average over 15 optimal networks; $\alpha =0.5$, $C=1$ in all panels.