Emergent task differentiation on network filters

This paper aims to analyze the emergence of task differentiation in a model complex system, characterized by an absence of hierarchical control, yet able to exhibit coordinated behavior and collective function. The analysis focuses on linear network filters, i.e., networks of coupled linear oscillators with a differentiated steady-state response to exogenous harmonic excitation. It demonstrates how an optimal allocation of excitation sensitivities across the network nodes in a condition of resonance may be constructed either using global information about the network topology and spectral properties or through the iterated dynamics of a nonlinear, nonsmooth learning paradigm that only relies on local information within the network. Explicit conditions on the topology and desired resonant mode shape are derived to guarantee local asymptotic stability of fixed points of the learning dynamics. The analysis demonstrates the possibly semistable nature of the fixed point with all zero excitation sensitivities, a condition of system collapse that can be reached from an open set of initial conditions but that is unstable under the learning dynamics. Theoretical and numerical results also show the existence of periodic responses, as well as of connecting dynamics between fixed points, resulting in recurrent metastable behavior and noise-induced transitions along cycles of such connections. Structural additions to a core network that conserve desired spectral properties are proposed as a defensive mechanism for fault tolerance and shielding of the core against targeted harm.