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

Abstract

In this paper, we revisit the problem of 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 is not in fact the differentiating property. The original paper presents graphs with edge density 1.1 ̅{6} that require Ω(n) agents, however, we construct graphs with edge density > 1.1 ̅{6} and develop an algorithm to solve the problem on those graphs using only o(n) agents. We further 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) → ∞ as n → ∞. We then construct an infinite family of graphs with edge density < ρ requiring exactly k ignorant agents to solve Broadcast, for any k > 0 and ρ > 1.
Finally, we investigate different versions of connectivity as possible properties determining the number of agents. We show that edge connectivity is not related to the number of agents: it is possible for a graph to have low (constant) edge connectivity but require a high (linear) number of agents to solve Broadcast. More generally, for any arbitrary λ, we construct a family of graphs on n nodes with edge connectivity (n-1)/λ requiring exactly k = n - 2 λ + 1 agents. On the other hand, we show vertex connectivity can impact the required number of agents: for any graph with vertex connectivity ≥ 3 and minimum degree δ, k ≥ δ - 1 agents are required to solve Broadcast. Finally, we show that for any graph containing a certain type of bond with m edges, Broadcast is unsolvable when k ≤ m-2.