  Networks are often interconnected, with one system wielding greater influence
over another. However, the effects of such asymmetry on self-organized
phenomena (e.g., consensus and synchronization) are not well understood. Here,
we study collective dynamics using a generalized graph Laplacian for multiplex
networks containing layers that are asymmetrically coupled. We explore the
nonlinear effects of coupling asymmetry on the convergence rate toward a
collective state, finding that asymmetry induces one or more optima that
maximally accelerate convergence. When a faster and a slower system are
coupled, depending on their relative timescales, their optimal coupling is
either cooperative (network layers mutually depend on one another) or
non-cooperative (one network directs another without a reciprocated influence).
It is often optimal for the faster system to more-strongly influence the slower
one, yet counter-intuitively, the opposite can also be true. As an application,
we model collective decision-making for a human-AI system in which a social
network is supported by an AI-agent network, finding that a cooperative optimum
requires that these two networks operate on a sufficiently similar timescale.
More broadly, our work highlights the optimization of coupling asymmetry and
timescale balancing as fundamental concepts for the design of collective
behavior over interconnected systems.
