Revealing Network Connectivity From Dynamics

TL;DR

This work considers networks of coupled phase oscillators and explicitly study their long-term stationary response to temporally constant driving, finding good predictions of the actual connectivity even for formally underdetermined problems.

Abstract

We present a method to infer network connectivity from collective dynamics in networks of synchronizing phase oscillators. We study the long-term stationary response to temporally constant driving. For a given driving condition, measuring the phase differences and the collective frequency reveals information about how the oscillators are interconnected. Sufficiently many repetitions for different driving conditions yield the entire network connectivity from measuring the dynamics only. For sparsely connected networks we obtain good predictions of the actual connectivity even for formally under-determined problems.

Figures (4)

• Figure 1: (color) Driving induces phase patterns, implicitly defined by (\ref{['eq:PhaseLockedm']}). The network has $N=16$ oscillators, each connected with a constant coupling strength $J_{ij}=1/k_{i}$ to $k_{i}\equiv8$ randomly selected others ($J_{ij}=0$ otherwise). (a) Homogeneous frequencies, $\omega_{i}\equiv1$; (b) inhomogeneous random frequencies $\omega_{i}\in[1,1+\Delta\omega]$, $\Delta\omega=0.1$. Both panels display the phase differences $\Delta\phi_{i}:=\max_{j}\{\phi_{j}\}-\phi_{i}$ in the phase-locked states versus $i$. The responses to three different driving conditions, (blue $\hbox{\boldmath ${\bigcirc}$}$) one oscillator $i=5$ driven, $I_{5,1}=0.3$; (red $\hbox{\boldmath ${\bigcirc}$}$) two oscillators $i\in\{2,8\}$ driven, $I_{2,2}=I_{8,2}=0.3$; (grey $\hbox{\boldmath ${\bullet}$}$) all oscillators driven by a signal of random strength $I_{i,3}\in[0,0.3]$ are shown along with the undriven dynamics ($\hbox{\boldmath ${\times}$}$).
• Figure 2: Inferring connectivity from measuring response dynamics. $M=N=16$ experiments DrivingSignals. (a) Connectivity of the network of Fig 1a as obtained using Eqs. (\ref{['eq:DiffMatrix']}--\ref{['eq:MatrixEquation']}). The matrix of connection strengths $J_{ij}$ is gray-coded from light gray ($J_{ij}=0$) to black ($J_{ij}=\max_{i',j'}\{ J_{i'j'}\}$). (b) Element-wise absolute difference $|J_{ij}^{\textrm{original}}-J_{ij}^{\textrm{derived }}|$, plotted on the same scale as (a). Inset shows magnified difference $100\times|J_{ij}^{\textrm{original}}-J_{ij}^{\textrm{derived }}|$ with a cutoff at unity (black). Panels (c) and (d) are analogous to (a) and (b) for the network with inhomogeneous frequencies of Fig. 1b.
• Figure 3: Revealing connectivity with $M<N$ measurements. Network ($N=64$, $k=10$, $\Delta\omega=0$) reconstructed by minimizing the 1-norm, (a) $M=38$, (b) $M=24$. The insets show the element-wise absolute differences to the original network.
• Figure 4: Quality of reconstruction and required number of experiments. (a) Quality of reconstruction ($\alpha=0.95$) displayed for $k=10$ and $N=24$ ($\diamond$), $N=36$ ($\triangle$), $N=66$ ($\circ$), $N=96$ ($\bigcirc$) . (b) Minimum number of experiments required ($q=0.90$, $\alpha=0.95$) versus network size $N$ with best linear and logarithmic fits (gray and black solid lines). Inset shows same data with $N$ on logarithmic scale.