Dynamical importance and network perturbations

TL;DR

This work investigates dynamical importance (FoEDI), a first-order measure of how edge perturbations alter the leading eigenvalue $\lambda$ of a network's adjacency matrix, and its connections to eigenvector changes and Kuramoto synchronization. The authors derive FoEDI, compare it to the true $\Delta\lambda$ across multiple graph models, and develop a first-order estimate $\delta v$ for the leading eigenvector change using the Moore–Penrose inverse, along with a refined FoEDI that incorporates this vector change. They also express the Kuramoto order parameter under edge perturbations in terms of FoEDI and demonstrate how edge additions can impact synchronization in networks with approximately homogeneous degree distributions. Overall, the results show that dynamical importance serves as a practical tool for linking structural perturbations to dynamical outcomes, while highlighting limitations and avenues for refinement in different network topologies.

Abstract

The leading eigenvalue $λ$ of the adjacency matrix of a graph exerts much influence on the behavior of dynamical processes on that graph. It is thus relevant to relate notions of the importance (specifically, centrality measures) of network structures to $λ$ and its associated eigenvector. We study a previously derived measure of edge importance known as ``dynamical importance'', which estimates how much $λ$ changes when one removes an edge from a graph or adds an edge to it. We examine the accuracy of this estimate for different network structures and compare it to the true change in $λ$ after an edge removal or edge addition. We then derive a first-order approximation of the change in the leading eigenvector. We also consider the effects of edge additions on Kuramoto dynamics on networks, and we express the Kuramoto order parameter in terms of dynamical importance. Through our analysis and computational experiments, we find that studying dynamical importance can improve understanding of the relationship between network perturbations and dynamical processes on networks.

Figures (7)

• Figure 1: We show the first-order edge dynamical importance (FoEDI) (dashed blue curve) and $\Delta \lambda / \lambda$ (solid red curve) for edge removals in various 200-node graphs. We order the curves for FoEDI and $\Delta\lambda$ by increasing value of FoEDI. The horizontal axis is the edge index. We plot results for graphs that we construct using (top left) the Erdős--Rényi (ER) model, (top right) the Barabási--Albert (BA) model, (bottom left) the Watts--Strogatz (WS) model, and (bottom right) a stochastic block model (SBM). We describe the parameters of each model in the main text. We use a single instantiation of each type of graph. In each panel, the number of data points equals the number of edges in the associated graph. The relative error is $\|x - y\|_2/\|x\|_2$, where $\| \cdot \|_2$ is the $\ell_2$ norm, $x$ is the vector of measured values, and $y$ is the vector of true values.
• Figure 2: We show the FoEDI (dashed blue curve) and $\Delta \lambda / \lambda$ (solid red curve) for edge additions. We divide $\Delta\lambda$ by $\lambda$ to normalize it. We order FoEDI and $\Delta\lambda$ by increasing FoEDI. We show results for (top left) an ER graph, (top right) a BA graph, (bottom left) a WS graph, and (bottom right) an SBM graph. The random-graph realization in each panel is the same as in the corresponding panel of Fig. \ref{['fig:edge_remove']}.
• Figure 3: The standard deviation $\sigma_d$ of the degree distribution of a graph $G$ as we add edges $e_{ij}$, one by one, that maximize $\iota_{ij}$. The horizontal axis is the number of edges that we add from the complement graph $G^C$. We show results for (top left) an ER graph, (top right) a BA graph, (bottom left) a WS graph, and (bottom right) an SBM graph. The random-graph realization in each panel is the same as in the corresponding panel of Fig. \ref{['fig:edge_remove']}.
• Figure 4: The relative error between $\delta v$ and $\Delta v$ for edge removals. The horizontal axis is the index of the edge, and the vertical axis is the relative error. We order the edge indices by increasing value of the relative error. We show results for (top left) an ER graph, (top right) a BA graph, (bottom left) a WS graph, and (bottom right) an SBM graph. The random-graph realization in each panel is the same in the corresponding panel of Fig. \ref{['fig:edge_remove']}.
• Figure 5: The relative error between $\delta v$ and $\Delta v$ for edge additions. The horizontal axis is the index of the edge, and the vertical axis is the relative error. We order the edge indices by increasing value of the relative error. We show results for (top left) an ER graph, (top right) a BA graph, (bottom left) a WS graph, and (bottom right) an SBM graph. The random-graph realization in each panel is the same as in the corresponding panel of Fig. \ref{['fig:edge_remove']}.