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

2025-04-07

AI Summary

This paper investigates the mobile agent broadcasting problem in dynamic networks, examining whether edge density is a decisive graph-structural property governing the minimum number of agents required. Prior conjectures posited that sparse graphs require $o(n)$ agents while dense graphs necessitate $Omega(n)$. Method: Through explicit graph-family constructions, asymptotic analysis, and distributed algorithm design, the authors introduce a novel framework for modeling redundant edges and characterizing dynamic connectivity. Contribution/Results: The work refutes both conjectures: (i) it constructs an infinite family of graphs with edge density approaching 1 (i.e., arbitrarily dense) yet admitting broadcasting with $o(n)$ agents; (ii) for any $ ho > 1$ and $k > 0$, it exhibits an infinite family of graphs with edge density less than $ ho$ requiring exactly $k$ agents. Moreover, it presents the first deterministic broadcasting algorithm achieving $o(n)$ agent complexity on highly dense graphs. These results demonstrate only a weak correlation between edge density and agent requirement, challenging the conventional view of edge density as a critical discriminant.

Technology Category

• Multiagent Systems: Agent Communication
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we revisit the problem of extsc{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 $k = Omega(n)$ 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 $1.1overline{6}$ that require $Omega(n)$ agents, however, we construct graphs with edge density $>1.1overline{6}$ 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 $Omega(n/f(n))$ agents to solve, for any function $f(n) o infty$ as $n o infty$. We then construct an infinite family of graphs with edge density $< ho$ requiring exactly $k$ ignorant agents to solve extsc{Broadcast}, for any $k>0$ and $ ho>1$.

Problem

• Investigates agent-based broadcast in dynamic graphs
• Challenges edge density as key factor for agent count
• Explores graph properties affecting broadcast efficiency

Innovation

• Dynamic graph broadcast via mobile agents
• Algorithm for sparse graphs with o(n) agents
• Weak edge density and agent count relationship