# On the Radius of Nonsplit Graphs and Information Dissemination in Dynamic Networks

## TL;DR

This paper establishes that nonsplit graphs have a radius of O(log log n) and extends this to graph products, providing new bounds on information dissemination speed in dynamic distributed networks and implications for consensus.

## Contribution

The paper proves a tight bound on the radius of nonsplit graphs and their products, with applications to dynamic network communication and consensus lower bounds.

## Key findings

- Nonsplit graphs have radius in O(log log n).
- Product of nonsplit graphs also has bounded radius.
- Improved bounds on message dissemination in dynamic networks.

## Abstract

A nonsplit graph is a directed graph where each pair of nodes has a common incoming neighbor. We show that the radius of such graphs is in $O(\log \log n)$, where $n$ is the number of nodes. We then generalize the result to products of nonsplit graphs. The analysis of nonsplit graph products has direct implications in the context of distributed systems, where processes operate in rounds and communicate via message passing in each round: communication graphs in several distributed systems naturally relate to nonsplit graphs and the graph product concisely represents relaying messages in such networks. Applying our results, we obtain improved bounds on the dynamic radius of such networks, i.e., the maximum number of rounds until all processes have received a message from a common process, if all processes relay messages in each round. We finally connect the dynamic radius to lower bounds for achieving consensus in dynamic networks.