Lossy gossip and composition of metrics

TL;DR

This paper explores the algebraic structure of distance matrices under tropical multiplication, revealing a finite polyhedral fan structure and applying it to model gossip protocols over lossy communication lines.

Contribution

It introduces a novel geometric framework for the gossip monoid using tropical geometry and computes its structure for small dimensions, extending understanding of gossip dynamics.

Findings

The monoid forms a finite polyhedral fan of dimension n(n-1)/2.

The structure of the fan is explicitly computed for n up to 5.

A sharp bound on call chains where new information is learned is established.

Abstract

We study the monoid generated by n-by-n distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension n(n-1)/2, and we compute the structure of this fan for n up to 5. The monoid captures gossip among n gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid, as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.