An energy function and its application to the periodic behavior of k-reversible processes

TL;DR

This paper introduces a new energy function for k-reversible graph processes, providing insights into their periodic behavior, maximum period length, and transient bounds, with applications to trees and general graphs.

Contribution

It presents a novel energy function for k-reversible processes, offering an alternative proof for maximum period length and improved bounds on transient length.

Findings

Established a monotonic energy function for k-reversible processes

Provided an alternative proof for maximum period length

Derived tighter bounds on transient length for trees and general graphs

Abstract

We consider the graph dynamical systems known as k-reversible processes. In such processes, each vertex in the graph has one of two possible states at each discrete time step. Each vertex changes its state between the current time and the next if and only if it currently has at least k neighbors in a state different than its own. For such processes, we present a monotonic function similar to the decreasing energy functions used to study threshold networks. Using this new function, we show an alternative proof for the maximum period length in a k-reversible process and provide better upper bounds on the transient length in both the general case and the case of trees.