 
Summary: Graphs and Combinatorics 1,305310 (1985)
Graphsand
Combinatorics
9 SpringerVerlag 1985
Asynchronous Threshold Networks
Noga Alon
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel and
Bell Communications Research, Morristown, NJ 07960, USA
Abstract. 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)= 1ifthere is a tie).Gisin a stablestateifs(v)= s'(v)for all ve V.We show that for everygraph
G = (V,E) and every initialsigns, there is a sequence v~,v2..... v,of vertices of G,in which no vertex
appears more than once, such that ifvibecomes active at time i,(l < i _
reaches a stable state. This proves a conjecture of Miller. We also consider some generalizationsto
directed graphs with weighted edges.
1. Introduction
A threshold network N is a directed graph G = (V, E) with a fixed real weight w((u, v))
assigned to each (directed) edge (u, v)~ E and a variable sign s(v)e {+1} assigned
to each vertex ve V. For a vertex v~ V put N(v)= {u~ V: (u,v)eE}. Initially, the
vertices of N are not active and each sign has an initial value. When a node v e V
