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Dynamic monopolies in tori

Dynamic monopolies in tori.
Authors: FLOCCHINI P; LODI E; LUCCIO, FABRIZIO; PAGLI, LINDA; SANTORO N.;

Dynamic monopolies in tori

Abstract

The authors study the following problem. Let \(G\) be a simple connected graph where every node is colored either black (a faulty element) or white (a non faulty element). Consider now the repetitive process where each node recolors itself, at each local time step, with the color held by the majority of its neighbors. Depending on the initial assignment of colors to the nodes and on the precise definition of majority, different dynamics can occur. \textit{Dynamos} are by definition initial assignments of colors which lead the system to a monochromatic configuration in a finite number of steps. The authors study two particular forms of dynamos (\textit{irreversible} and \textit{monotone}) in tori (in which each node has four neighbors), focusing on the minimum number of initial black elements needed to reach the fixed point. In \textit{reversible} coloring under \textit{simple} majority, each vertex becomes black if two of its neighbors are black, otherwise it becomes white. In \textit{reversible} coloring under \textit{strong} majority, if a vertex has three neighbors with the same color, it takes this color, otherwise it does not change its color. In \textit{irreversible} coloring under \textit{simple} majority a vertex becomes black if it has two black neighbors, otherwise it does not change its color. And in \textit{irreversible} coloring under \textit{strong} majority a vertex becomes black if it has three black neighbors, otherwise it does not change. An initial set \(S\) of black vertices is an irreversible dynamo under simple (respectively, strong) majority if an all black configuration is reached from \(S\) in a finite number of steps under the irreversible simple (respectively, strong) majority rule. For brevity, we call \(S\) a \textit{simple} (respectively, \textit{strong}) \textit{irreversible dynamo}. Similarly, a \textit{simple} (respectively, \textit{strong}) \textit{monotone dynamo} is an initial set of black vertices from which an all black configuration is reached in a finite number of steps under the reversible simple (respectively, strong) majority rule, and such that no black vertex ever becomes white during the process. Lower and upper bounds on the size of dynamos for three types of tori are derived, under simple and strong majority rule. These bounds are tight within an additive constant. In addition, the upper bounds are constructive: for each topology and each majority rule the authors exhibit a dynamo of the claimed size.

Keywords

distributed algorithm, Tori, majority rules, Applied Mathematics, Dynamic monopolies, Majority rules, dynamo, graph, Programming involving graphs or networks, dynamo; distributed algorithms; graph, dynamic monopolies, 004, tori, Graph algorithms (graph-theoretic aspects), Discrete Mathematics and Combinatorics

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
48
Top 10%
Top 10%
Average
hybrid