Communication complexity is generally discussed among many papers on secure multi-party computation (MPC). However, I do not still understand the importance of it.
What communication complexity affects is total computation time in a protocol depending on available bandwidth rate (10Mbps, 1Gbps, etc).
Is the communication complexity really important? When is it captured as primary result of some paper?
With a specific programming implementation detail, I can understand the total execution time including data transmission during the protocol. But from the communication complexity (even if some paper reduced it significantly compared with prior works), I wonder when and why it makes sense.
What is the advantage of reducing communication cost (in general at the expense of computation time on some nodes) in practice?