Oscillations of networks: the role of soft nodes

To describe the flow of a miscible quantity on a network, we consider the graph wave equation where the standard continuous Laplacian is replaced by the graph Laplacian. The structure of the graph influences strongly the dynamics. Assuming the graph is forced and damped at specific nodes, we derive the amplitude equations using a basis of eigenvectors of the graph Laplacian. These lead us to introduce the notion of soft nodes. We give sufficient conditions for their existence in general graphs. They can cause several effects as we show on small graphs, for example the ineffectiveness of damping applied to them. Soft nodes may be of critical importance for complex physical networks and engineering networks like power grids.