Oscillations of simple networks

TL;DR

This paper introduces a graph wave equation model to analyze oscillations in simple networks, revealing how graph structure influences dynamics, resonance phenomena, and the importance of strategic damping for network stability.

Contribution

It develops a novel graph wave equation framework and analyzes resonance effects in simple networks, highlighting the impact of graph topology on oscillatory behavior.

Findings

Eigenvalues correspond to specific network couplings.

Damping effectiveness depends on node placement.

Resonance can cause network destruction if damping is misapplied.

Abstract

To describe the flow of a miscible quantity on a network, we introduce the graph wave equation where the standard continuous Laplacian is replaced by the graph Laplacian. This is a natural description of an array of inductances and capacities, of fluid flow in a network of ducts and of a system of masses and springs. The structure of the graph influences strongly the dynamics which is naturally described using the basis of the eigenvectors. In particular, we show that if two outer nodes are connected to a common third node with the same coupling, then this coupling is an eigenvalue of the Laplacian. Assuming the graph is forced and damped at specific nodes, we derive the amplitude equations. These are analyzed for two simple non trivial networks: a tree and a graph with a cycle. Forcing the network at a resonant frequency reveals that damping can be ineffective if applied to the wrong node, leading to a disastrous resonance and destruction of the network. These results could be useful for complex physical networks and engineering networks like power grids.