# Controlling percolation with limited resources

## TL;DR

This paper investigates how to control the emergence of large-scale connectivity in networks using limited resources, proposing an efficient intervention strategy that can induce a discontinuous percolation transition.

## Contribution

It introduces a novel control strategy for percolation with resource constraints and analyzes its impact on network connectivity transitions.

## Key findings

- Percolation can be effectively controlled with limited resources.
- The control strategy induces a discontinuous transition.
- Optimal control can be achieved even with scarce resources.

## Abstract

Connectivity - or the lack thereof - is crucial for the function of many man-made systems, from financial and economic networks over epidemic spreading in social networks to technical infrastructure. Often, connections are deliberately established or removed to induce, maintain, or destroy global connectivity. Thus, there has been a great interest in understanding how to control percolation, the transition to large-scale connectivity. Previous work, however, studied control strategies assuming unlimited resources. Here, we depart from this unrealistic assumption and consider the effect of limited resources on the effectiveness of control. We show that, even for scarce resources, percolation can be controlled with an efficient intervention strategy. We derive this strategy and study its implications, revealing a discontinuous transition as an unintended side-effect of optimal control.