An inclusion hierarchy of irreversible dynamos

Abstract

In this paper we extend the setting of irreversible dynamic monopolies (or shortly, dynamos) from two to more than two colors. The distributed protocol can be described as follows: let G be a simple connected graph where every node has a color from a finite ordered set C={1,…,k} of colors. At each round, each node can increase its color by 1 according to the colors held by its neighbors. We are interested in the so called SCIM-dynamos, that is, subsets of nodes initially having color k leading all the nodes to recolor by k under our protocol. We show that there is a tradeoff between the size of a SCIM-dynamo and the size of C: if C is limited to two colors, our protocol coincides with the irreversible strong majority. Differently, increasing the number of colors employed permits to reduce the size of a SCIM-dynamo to that of a minimum simple majority dynamo. This gives rise to an inclusion hierarchy of dynamos from simple majority dynamos to strong majority dynamos. Our goal is to minimize the size of a SCIM-dynamo for a given G, and then to determine the minimum number of colors accordingly. In particular, given an m×n torus, we show that there exists a special class of minimum size SCIM-dynamos for k≈min(m,n)/2.