Random majority percolation

Abstract

ABSTRACT:We shall consider the discrete time synchronous random majority-vote cellular automata on the n by n torus, in which every vertex is in one of two states and, at each time step t, every vertex goes into the state the majority of its neighbors had at time t − 1 with a small chance p of error independently of all other events. We shall show that, if n is fixed and p is sufficiently small, then the process spends almost half of its time in each of two configurations. Further more, we show that the expected time for it to reach one of these configurations from the other is (1/p

Pattern-based computing via sequential phase transitions in hierarchical mean field neuropercolation

“…Neuropercolation operates with emergent spatio-temporal oscillation patterns in the style of brains. Since the early 2000's, there have been several theoretical advances in the mathematical, graph-theoretical aspects of neuropercolation, see [48,49]; but the main progress has been achieved through Monte Carlo simulations [34,45,62,39].…”

“…There are mathematically rigorous results for homogeneous two-dimensional lattice graphs with random majority update rule [48,49], when the lattice dynamics is bimodal. The system stays in one mode for a long time, then rapidly transits (switches) to the other mode.…”

Tight Bounds on the Minimum Size of a Dynamic Monopoly

“…We first present the construction of graph G step by step. For 1 ≤ i ≤ K := n/(r + 1) − 1, let G i be the clique on the node set V i := {v (j) i : 1 ≤ j ≤ r + 1} minus the edge {v (1) i , v (2) i }. To create the first part of graph G, we connect G i s with a path.…”

“…To create the first part of graph G, we connect G i s with a path. More precisely, we add the edge set {{v (2) i , v (1) i+1 } : 1 ≤ i ≤ K − 1}. So far the generated graph is r-regular except the nodes v and v (2) K which are of degree r − 1 and we have := n − K(r + 1) nodes left.…”

“…Thus, eventually the graph is going to be fully white. Now, assume that v is a node in V i \{v (1) i , v (2) i }. Again, in the next round v and v (4) i get white and the same argument follows.…”