# Tight Bounds on the Minimum Size of a Dynamic Monopoly

## TL;DR

This paper establishes precise bounds on the smallest initial set of nodes needed to eventually turn an entire graph black through specific bootstrap percolation processes, advancing understanding of influence spread in networks.

## Contribution

It provides tight upper and lower bounds on the minimum size of dynamic monopolies in two types of bootstrap percolation models, a novel theoretical result.

## Key findings

- Derived tight bounds for $r$-bootstrap percolation.
- Derived tight bounds for $eta$-bootstrap percolation.
- Enhanced understanding of influence dynamics in network models.

## Abstract

Assume that you are given a graph $G=(V,E)$ with an initial coloring, where each node is black or white. Then, in discrete-time rounds all nodes simultaneously update their color following a predefined deterministic rule. This process is called two-way $r$-bootstrap percolation, for some integer $r$, if a node with at least $r$ black neighbors gets black and white otherwise. Similarly, in two-way $\alpha$-bootstrap percolation, for some $0<\alpha<1$, a node gets black if at least $\alpha$ fraction of its neighbors are black, and white otherwise. The two aforementioned processes are called respectively $r$-bootstrap and $\alpha$-bootstrap percolation if we require that a black node stays black forever. For each of these processes, we say a node set $D$ is a dynamic monopoly whenever the following holds: If all nodes in $D$ are black then the graph gets fully black eventually. We provide tight upper and lower bounds on the minimum size of a dynamic monopoly.