Arrival of information at a target set in a network

TL;DR

This paper introduces an algorithm to determine if the configurations at the last row of a regular tree are independent of the root symbol, extending the concept of matrix primitivity to tree structures with applications in topological pressure calculation.

Contribution

It proposes a novel algorithm for analyzing labelings on regular trees under transition constraints, with a proof of bounded execution time under certain conditions.

Findings

Algorithm succeeds in bounded steps when tree dimension exceeds max row sum of transition matrix

Extension of matrix primitivity concept to tree labelings

Application to topological pressure calculations on trees

Abstract

We consider labelings of a finite regular tree by a finite alphabet subject to restrictions specified by a nonnegative transition matrix, propose an algorithm for determining whether the set of possible configurations on the last row of the tree is independent of the symbol at the root, and prove that the algorithm succeeds in a bounded number of steps, provided that the dimension of the tree is greater than or equal to the maximum row sum of the transition matrix. (The question was motivated by calculation of topological pressure on trees and is an extension of the idea of primitivity for nonnegative matrices.)