Tempological Control of Network Dynamics

TL;DR

This work shows how it is possible to drive a system to a desired state or even stabilize otherwise unstable states, all in the absence of external forcing, by strategically switching network topology on-the-fly based on the current states of the nodes.

Abstract

Feedback control is an effective strategy for stabilizing a desired state and has been widely adopted in maintaining the stability of systems such as flying birds and power grids. By default, this framework requires continuous control input to offset deviations from the desired state, which can be invasive and cost a considerable amount of energy. Here, we introduce tempological (temporal + topological) control -- a novel, noninvasive strategy that harnesses the inherent flexibility of time-varying networks to control the dynamics of a general nonlinear system. By strategically switching network topology on-the-fly based on the current states of the nodes, we show how it is possible to drive a system to a desired state or even stabilize otherwise unstable states, all in the absence of external forcing. We demonstrate the utility of our approach using networks of Kuramoto and Stuart-Landau oscillators, achieving synchronization out of sets of unsynchronizable networks. Finally, we develop a statistical theory that explains why tempological control will almost always succeed in the limit of large and diverse temporal networks, with diversity of network configurations overcoming the deficiencies of any snapshot in isolation.

Figures (4)

• Figure 1: Illustrative example of tempological control. (a, top) Two snapshots of 3-node networks coupling Kuramoto oscillators. The number beside each directed link $j \rightarrow i$ denotes the corresponding weight $A_{ij}$, and the links are colored blue (red) if $A_{ij}$ is positive (negative). (a, bottom) Vector fields of the corresponding dynamics [\ref{['eq:kuramoto']}], which we present in 2D by measuring the phases of the first two nodes ($\theta_1$ and $\theta_2$) relative to the third. In both snapshots, the phase-locked state at the origin ($\theta_1 = \theta_2 = \theta_3$) is unstable. However, by strategically switching between these two unstable systems, one can drive the trajectory to the synchronous state from far away, as shown in (b), where we color the controlled trajectory according to which snapshot is active. In (c), we show the corresponding energy ($V$, top), the normalized works of both snapshots ($W/V$, middle), and the switching signal ($\sigma$, bottom) as functions of time.
• Figure 2: Tempological control almost always succeeds with enough network snapshots. The success rate $R$ measures the probability of reaching the target synchronization state from random initial conditions by switching between $m$ unstable networks using our tempological control strategy. Each data point is averaged over $500$ independent trials. Here, a trial is considered successful if the energy function $V(\bm{\theta})$ goes below $10^{-6}$ within $300$ time units. The parameter $p$ is the percentage of repulsive links in the networks. Larger $p$ makes the snapshots more unstable, increasing the difficulty of the control task.
• Figure 3: Successful control depends strongly on the available network snapshots, not the initial conditions. We attempt to control an ensemble of 1,000 initial conditions towards synchrony, generated to have energies distributed log-uniformly over the range $0.001 \le V_0 \le 0.99$. For each value of $p$ (percentage of repulsive links) and each initial condition, we test 1,000 realizations of temporal networks on $n = 20$ nodes, each with $m = 10$ undirected snapshots and a density $q$ corresponding to $\langle k \rangle = 2$. (a) The success rate of tempological control is largely independent of $V_0$; states initially far from synchrony ($V_0 \approx 1$) are on average no harder to control than those close to it ($V_0 \approx 0$). (b) In contrast, the success rates of different network realizations are strongly bimodal at $0\%$ and $100\%$. That is, a given set of snapshots will either succeed for almost all initial conditions, or fail for almost all. Changing $p$ merely changes the proportion of temporal networks at either of these extremes.
• Figure 4: Predicting the distribution and scaling of the minimal work, $W_\text{min}$. We compare the distribution of $W_\text{min}$ for undirected 100-node temporal networks with: (a) $m = 5$ snaphshots vs. (b) $m = 500$ snapshots, generated with varying $p$ and with $\langle k \rangle = 3$. Solid lines show the density corresponding to our theoretically-predicted Gumbel distribution of \ref{['eq:gumbel']}, while each set of bars is a histogram over 1,000 random temporal networks with the given parameters. Vertical axes represent density, with the same scale across panels to facilitate comparison. (c) Tail probability of $W_\textrm{min}$ being negative vs. the number of snapshots, $m$. Solid lines denote the theoretical predictions of \ref{['eq:gumbel']} while each marker represents an average over 1,000 random temporal networks. All simulations consider $W$ evaluated at the initial condition $\bm{\theta}_0$ defined by $\theta_{i + 1} - \theta_i = 0.05$ for $i = 1,\ldots,99$, with energy $V_0 \approx 0.943$. Other states show similar qualitative behavior, and a similarly excellent fit with theory.