A control analysis perspective on Katz centrality

TL;DR

This paper offers a new interpretation of Katz centrality as a steady-state solution to continuous-time network dynamics, enabling sensitivity analysis akin to metabolic control analysis, and enhancing understanding of node influence in directed networks.

Contribution

It introduces a novel perspective on Katz centrality, linking it to dynamic system steady states and sensitivity analysis, which improves network influence assessment.

Findings

Katz centrality can be viewed as a steady-state of continuous-time dynamics.

The new interpretation allows for sensitivity analysis of nodes.

High-centrality nodes are crucial for propagating network dynamics.

Abstract

Methods for efficiently controlling dynamics propagated on networks are usually based on identifying the most influential nodes. Knowledge of these nodes can be used for the targeted control of dynamics such as epidemics, or for modifying biochemical pathways relating to diseases. Similarly they are valuable for identifying points of failure to increase network resilience in, for example, social support networks and logistics networks. Many measures, often termed `centrality', have been constructed to achieve these aims. Here we consider Katz centrality and provide a new interpretation as a steady-state solution to continuous-time dynamics. This enables us to implement a sensitivity analysis which is similar to metabolic control analysis used in the analysis of biochemical pathways. The results yield a centrality which quantifies, for each node, the net impact of its absence from the network. It also has the desirable property of requiring a node with a high centrality to play a central role in propagating the dynamics of the system by having the capacity to both receive flux from others and then to pass it on. This new perspective on Katz centrality is important for a more comprehensive analysis of directed networks.