Tight Bounds on the Minimum Size of a Dynamic Monopoly
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.
KeywordsDynamic monopoly Bootstrap percolation Threshold model Percolating set Target set selection
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