# Tight Bounds on the Minimum Size of a Dynamic Monopoly

• Ahad N. Zehmakan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11417)

## 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.

## Keywords

Dynamic monopoly Bootstrap percolation Threshold model Percolating set Target set selection

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