Network connectivity during mergers and growth: optimizing the addition of a module

TL;DR

This paper investigates how adding a module to a network affects its principal eigenvalue, providing optimal strategies for connection to control network dynamics and robustness.

Contribution

It introduces methods to optimally connect modules to networks to control the principal eigenvalue, with broad applications in network growth and modification.

Findings

Optimal connection strategies for eigenvalue modification

Applications in network robustness and dynamics control

Framework applicable to various network growth scenarios

Abstract

The principal eigenvalue $\lambda$ of a network's adjacency matrix often determines dynamics on the network (e.g., in synchronization and spreading processes) and some of its structural properties (e.g., robustness against failure or attack) and is therefore a good indicator for how ``strongly'' a network is connected. We study how $\lambda$ is modified by the addition of a module, or community, which has broad applications, ranging from those involving a single modification (e.g., introduction of a drug into a biological process) to those involving repeated additions (e.g., power-grid and transit development). We describe how to optimally connect the module to the network to either maximize or minimize the shift in $\lambda$, noting several applications of directing dynamics on networks.