Encoding Impact of Network Modification on Controllability via Edge Centrality Matrix

TL;DR

This paper introduces a new edge centrality matrix to quantify how network modifications affect controllability, providing analytical characterizations and bounds for directed stem-bud networks, validated through simulations.

Contribution

It proposes a novel edge centrality measure based on Gramian variations and characterizes its structure for specific network classes, enhancing understanding of network controllability impacts.

Findings

Edge centrality matrix captures the impact of edge modifications on controllability metrics.

Structural characterization of ECM for directed stem-bud networks.

Bounds on Gramian metrics ensuring stability after edge modifications.

Abstract

This paper develops tools to quantify the importance of agent interactions and its impact on global performance metrics for networks modeled as linear time-invariant systems. We consider Gramian-based performance metrics and propose a novel notion of edge centrality that encodes the first-order variation in the metric with respect to the modification of the corresponding edge weight, including for those edges not present in the network. The proposed edge centrality matrix (ECM) is additive over the set of inputs, i.e., it captures the specific contribution to each edge's centrality of the presence of any given actuator. We provide a full characterization of the ECM structure for the class of directed stem-bud networks, showing that non-zero entries are only possible at specific sub/super-diagonals determined by the network size and the length of its bud. We also provide bounds on the value of the trace, trace inverse, and log-det of the Gramian before and after single-edge modifications, and on the edge-modification weight to ensure the modified network retains stability. Simulations show the utility of the proposed edge centrality notion and validate our results.