Broadcast via Mobile Agents in a Dynamic Network: Interplay of Graph Properties & Agents

TL;DR

The paper addresses Broadcast via mobile agents on dynamic graphs, questioning whether edge density alone can determine the minimum number of ignorant agents $k$ required. It develops a generalized-theta graph framework and a 5-phase algorithm that solves Broadcast with $k\ge\ell$ on $\theta_{d_1,\dots,d_{\ell}}$, while also constructing diverse graph families to show that edge density is not the sole predictor of solvability. By presenting low-density graphs needing many agents and arbitrary-density constructions (including lollipop graphs), the authors demonstrate that no fixed density threshold separates $k=o(n)$ from $k=\Omega(n)$ in general. They further explore connectivity-based bounds (edge/vertex connectivity and bonds) to derive partial limitations and progress toward a more nuanced understanding of which graph properties govern Broadcast in dynamic networks.

Abstract

In this paper, we revisit the problem of \textsc{Broadcast}, introduced by Das, Giachoudis, Luccio, and Markou [OPODIS, 2020], where $k+1$ agents are initially placed on an $n$ node dynamic graph, where $1$ agent has a message that must be broadcast to the remaining $k$ ignorant agents. The original paper studied the relationship between the number of agents needed to solve the problem and the edge density of the graph. The paper presented strong evidence that edge density of a graph, or the number of redundant edges within the graph, may be the correct graph property to accurately differentiate whether $k= o(n)$ agents (low edge density) or $k = Ω(n)$ agents (high edge density) are needed to solve the problem. In this paper, we show that surprisingly, edge density may not in fact be the correct differentiating property. The original paper presents graphs with edge density $1.1\overline{6}$ that require $Ω(n)$ agents, however, we construct graphs with edge density $> 1.1\overline{6}$ and develop an algorithm to solve the problem on those graphs using only $o(n)$ agents. We subsequently show that the relationship between edge density and number of agents is fairly weak by first constructing graphs with edge density tending to $1$ from above that require $Ω(n/f(n))$ agents to solve, for any function $f(n) \to \infty$ as $n \to \infty$. We then construct an infinite family of graphs with edge density $< ρ$ requiring exactly $k$ ignorant agents to solve \textsc{Broadcast}, for any $k>0$ and $ρ>1$.

Figures (2)

• Figure 1: An example theta graph $\theta_{7,4,2,3,4,3,1,7}$
• Figure 2: Potential operation of the algorithm presented in Theorem \ref{['the:alg-works']}, blue nodes are ignorant agents, and red are source. This figure assumes each phase ends by transitioning to the next rather than returning to a previous phase.

Theorems & Definitions (14)

• Definition 1: The Broadcast Problem
• Lemma 2
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 6
• Corollary 7
• Theorem 8
• Theorem 9
• Corollary 10