Arrival of information at a target set in a network
Karl Petersen, Ibrahim Salama
TL;DR
The paper studies when a nonnegative transition matrix $A$ on a finite alphabet yields the same arrivals on a target subtree $L_n$ regardless of the root symbol, formalized as $(\epsilon,L_n)$-fairness and completeness for a regular tree. It introduces a formal relation framework between starting symbols using $i \Rightarrow_n j$, and an explicit round-based algorithm that iteratively discovers all such relations via follower sets and subtree replacements/switches. The main result proves that if the tree dimension satisfies $k \ge s_A$ (with $s_A=\max_i |A_i|$), the algorithm terminates in finitely many rounds and finds all existing relations; the necessity of the bound is illustrated by counterexamples, and a matrix-product viewpoint clarifies the process. Overall, the work provides a constructive method to assess information-transfer fairness on trees, linking tree-shift dynamics to primitivity and topological pressure ideas in symbolic dynamics.
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.)