We study the multiparty communication model where players are the nodes of a network and each of these players knows his/her own identifier together with the identifiers of his/her neighbors. The players simultaneously send a unique message to a referee who must decide a graph property. The goal of this article is to separate, from the point of view of message size complexity, three different settings: deterministic protocols, randomized protocols with private coins and randomized protocols with public coins. For this purpose we introduce the boolean function Twins. This boolean function returns 1 if and only if there are two nodes with the same neighborhood.
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