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Goles, E., & Montealegre, P. (2020). The complexity of the asynchronous prediction of the majority automata. Inf. Comput., 274(SI).
Abstract: We consider the asynchronous prediction problem for some automaton as the one consisting in determining, given an initial configuration, if there exists a non-zero probability that some selected site changes its state, when the network is updated picking one site at a time uniformly at random. We show that for the majority automaton, the asynchronous prediction problem is in NC in the two-dimensional lattice with von Neumann neighborhood. Later, we show that in three or more dimensions the problem is NP-Complete.
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Kiwi, M., de Espanes, P. M., Rapaport, I., Rica, S., & Theyssier, G. (2014). Strict Majority Bootstrap Percolation in the r-wheel. Inf. Process. Lett., 114(6), 277–281.
Abstract: In the strict Majority Bootstrap Percolation process each passive vertex v becomes active if at least [deg(v)+1/2] of its neighbors are active (and thereafter never changes its state). We address the problem of finding graphs for which a small proportion of initial active vertices is likely to eventually make all vertices active. We study the problem on a ring of n vertices augmented with a “central” vertex u. Each vertex in the ring, besides being connected to u, is connected to its r closest neighbors to the left and to the right. We prove that if vertices are initially active with probability p > 1/4 then, for large values of r, percolation occurs with probability arbitrarily close to I as n -> infinity. Also, if p < 1/4, then the probability of percolation is bounded away from 1. (c) 2014 Elsevier B.V. All rights reserved.
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Rapaport, I., Suchan, K., Todinca, I., & Verstraete, J. (2011). On Dissemination Thresholds in Regular and Irregular Graph Classes. Algorithmica, 59(1), 16–34.
Abstract: We investigate the natural situation of the dissemination of information on various graph classes starting with a random set of informed vertices called active. Initially active vertices are chosen independently with probability p, and at any stage in the process, a vertex becomes active if the majority of its neighbours are active, and thereafter never changes its state. This process is a particular case of bootstrap percolation. We show that in any cubic graph, with high probability, the information will not spread to all vertices in the graph if p < 1/2. We give families of graphs in which information spreads to all vertices with high probability for relatively small values of p.
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