Must the Communication Graph of MPC Protocols be an Expander?

Secure multiparty computation (MPC) on incomplete communication networks has been studied within two primary models: (1) Where a partial network is fixed priori, and thus corruptions can occur dependent its structure, (2) edges in the graph are determined dynamically as part of protocol. Whereas rich literature succeeded mapping out feasibility limitations structures supporting secure fixed-graph model (including strong classical lower bounds), these bounds do not apply latter dynamic-graph setting, which recently seen exciting new results, but remains relatively unexplored. In this work, we initiate similar foundational study MPC model. As first step, investigate property expansion. All existing protocols (implicitly or explicitly) yield graphs expanders, it clear whether inherent. Our results consist types (for constant fraction corruptions): * Upper bounds: We demonstrate whose induced expander graphs, wide range settings (computational, information theoretic, with low locality, even locality adaptive security), each assuming some form input-independent setup. Lower plain (no setup) corruptions, that for certain functionalities, no protocol maintain non-expanding against all adversarial strategies. bound relies only correctness (not privacy), requires surprisingly delicate argument. More generally, provide formal framework analyzing evolving protocols, giving starting point studying relation between further, more general properties.