Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Pinning Control for Synchronization over Temporal Networks

Aandrew Baggio S, Rachel Kalpana Kalaimani

TL;DR

The paper tackles finite-time synchronization of nonlinear dynamical networks with time-varying interactions modeled as temporal networks. It introduces a pinning-control approach where a subset of nodes is externally driven to a trajectory and the rest synchronize through network interactions, leveraging a fused graph $\mathcal{G}$ to capture cross-snapshot influence. A linear-programming formulation minimizes the number of pinning nodes needed for full synchronization, while a greedy heuristic addresses maximizing synchronized nodes under pinning constraints, with NP-hardness established for the latter. Numerical simulations on Van der Pol oscillators and random temporal networks demonstrate near-optimal performance of the proposed greedy method and validate the finite-time synchronization framework. The results provide a scalable strategy for robust synchronization in temporally evolving networks relevant to power grids, biological systems, and distributed control tasks.

Abstract

In this paper, we address the finite time synchronization of a network of dynamical systems with time-varying interactions modeled using temporal networks. We synchronize a few nodes initially using external control inputs. These nodes are termed as pinning nodes. The other nodes are synchronized by interacting with the pinning nodes and with each other. We first provide sufficient conditions for the network to be synchronized. Then we formulate an optimization problem to minimize the number of pinning nodes for synchronizing the entire network. Finally, we address the problem of maximizing the number of synchronized nodes when there are constraints on the number of nodes that could be pinned. We show that this problem belongs to the class of NP-hard problems and propose a greedy heuristic. We illustrate the results using numerical simulations.

Optimal Pinning Control for Synchronization over Temporal Networks

TL;DR

The paper tackles finite-time synchronization of nonlinear dynamical networks with time-varying interactions modeled as temporal networks. It introduces a pinning-control approach where a subset of nodes is externally driven to a trajectory and the rest synchronize through network interactions, leveraging a fused graph to capture cross-snapshot influence. A linear-programming formulation minimizes the number of pinning nodes needed for full synchronization, while a greedy heuristic addresses maximizing synchronized nodes under pinning constraints, with NP-hardness established for the latter. Numerical simulations on Van der Pol oscillators and random temporal networks demonstrate near-optimal performance of the proposed greedy method and validate the finite-time synchronization framework. The results provide a scalable strategy for robust synchronization in temporally evolving networks relevant to power grids, biological systems, and distributed control tasks.

Abstract

In this paper, we address the finite time synchronization of a network of dynamical systems with time-varying interactions modeled using temporal networks. We synchronize a few nodes initially using external control inputs. These nodes are termed as pinning nodes. The other nodes are synchronized by interacting with the pinning nodes and with each other. We first provide sufficient conditions for the network to be synchronized. Then we formulate an optimization problem to minimize the number of pinning nodes for synchronizing the entire network. Finally, we address the problem of maximizing the number of synchronized nodes when there are constraints on the number of nodes that could be pinned. We show that this problem belongs to the class of NP-hard problems and propose a greedy heuristic. We illustrate the results using numerical simulations.
Paper Structure (11 sections, 3 theorems, 30 equations, 3 figures, 1 algorithm)

This paper contains 11 sections, 3 theorems, 30 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Preliminaries
  2. Problem Formulation
  3. Main Results
  4. Graph Conditions for Synchronizing the temporal network
  5. Optimal pinning nodes
  6. Maximum synchronizing nodes with constraints on pinning nodes
  7. Simulation results
  8. Conclusion
  9. Appendix
  10. Finite time synchronization for pinning nodes
  11. Finite time synchronization in each snapshot of temporal network

Key Result

Theorem 1

Consider a network of dynamical systems each following the dynamics as given in eq:sysmodel - unpinned, with interactions depicted by a temporal network $\{G_k\}$, $k=1,\dots, T$. Let $\mathcal{G}$ denote the fused graph obtained from all $G_k$'s. Given an $S_0$, if the set of all root nodes in $G_{

Figures (3)

  • Figure 1: Fused graph $\mathcal{G}$ of the snapshots $G_{1}$, $G_{2}$ and $G_{3}$. The solid nodes are root nodes.
  • Figure 2: Snapshot representation of the Van der Pol Network
  • Figure 3: Synchronization of a 5 Node Temporal Network

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • ...and 1 more