Oscillations of networks: the role of soft nodes

Abstract

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 which is naturally described using the eigenvectors as a basis. Assuming the graph is forced and damped at specific nodes, we derive the amplitude equations. These reveal the importance of a soft node where the eigenvector is zero. For example forcing the network at a resonant frequency reveals that damping can be ineffective if applied to such a soft node, leading to a disastrous resonance and destruction of the network. We give sufficient conditions for the existence of soft nodes and show that these exist for general graphs so that they can be expected for complex physical networks and engineering networks like power grids.

Inverse source problem in a forced network

“…. , N. This is the opposite of a "soft node" that was introduced in our previous work [12]. Also, for spatially continuous systems Jai and Pritchard stressed the importance of these strategic sets [25].…”

“…where A i,j m is the N × (N − 2) matrix obtained by removing the two columns i and j from the N × N matrix A m introduced in (13),X i,j m ∈ IR N −2 is the unknown vector defined by removing the two known componentsx i andx j from the vectorX m in (12) and…”

Localized solutions of nonlinear network wave equations

“…In the literature, for example [26] and references therein, the discrete Φ 4 model was studied only in the particular case of lattices where the graph Laplacian (∆u) i = u i+1 − 2u i + u i−1 (for a one-dimensional lattice) is a finite difference discretization of the continuous Laplacian. This formulation is natural since the linear graph wave equation arises from discrete conservation laws [24]. We formulate the discrete Φ 4 model using the graph Laplacian to describe general networks of arbitrary topology, like for example an electrical network.…”

“…The general model for these systems is the graph wave equation, where the usual continuum Laplacian is replaced by the graph Laplacian [23]. It arises naturally from discrete conservation laws [24]. The graph Laplacian, being a symmetric positive matrix, has real eigenvalues and we can choose a basis of orthogonal eigenvectors corresponding to the normal modes [24] giving rise to periodic solutions.…”

“…It arises naturally from discrete conservation laws [24]. The graph Laplacian, being a symmetric positive matrix, has real eigenvalues and we can choose a basis of orthogonal eigenvectors corresponding to the normal modes [24] giving rise to periodic solutions. When nonlinearity is present, these normal modes generally couple together.…”