Asynchronous threshold networks

Let G = (V, E) be a graph with an initial sign s(v)e { _+ 1} for every vertex vs V. When a vertex v becomes active, it resets its sign to s'(v) which is the sign of the majority of its neighbors (s'(v) = 1 if there is a tie). G is in a stable state if s(v) = s'(v) for all v e V. We show that for every graph G = (V, E) and every initial signs, there is a sequence v~, v 2 ..... v, of vertices of G, in which no vertex appears more than once, such that ifv i becomes active at time i, (l < i _< r), then after these r steps G reaches a stable state. This proves a conjecture of Miller. We also consider some generalizations to directed graphs with weighted edges.