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The principal eigenvalue $\lambda$ of a network's adjacency matrix oftendetermines dynamics on the network (e.g., in synchronization and spreadingprocesses) and some of its structural properties (e.g., robustness againstfailure or attack) and is therefore a good indicator for how ``strongly'' anetwork is connected. We study how $\lambda$ is modified by the addition of amodule, or community, which has broad applications, ranging from thoseinvolving a single modification (e.g., introduction of a drug into a biologicalprocess) to those involving repeated additions (e.g., power-grid and transitdevelopment). We describe how to optimally connect the module to the network toeither maximize or minimize the shift in $\lambda$, noting several applicationsof directing dynamics on networks.

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