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Asymptotic Behavior of Coarse-grained Models for Opinion Dynamics on Large Networks

In this paper, we propose a general mathematical framework to represent many multi-agent signalling systems in recent works. Our goal is to apply previous results in monotonicity to this class of systems and study their asymptotic behavior. Hence we introduce a suitable partial order for these systems and prove nontrivial extensions of previous results on monotonicity. We also derive a convenient sufficient condition for a signalling system to be monotone and test our condition on the Naming Games, NG and K-NG on complete networks both with and without committed agents. We also give a counter example which fails to satisfy our condition. Next we further extend our conclusions to systems on sparse random networks. Finally we discuss several meaningful consequences of monotonicity which narrows down the possible asymptotic behavior of signalling systems in mathematical sociology and network science.