Pathological Regimes of Closed-Loop Recommendation Systems over Social Networks

This paper addresses the problem of designing recommendation systems for social networks and e-commerce platforms from a control-theoretic perspective.

We formulate recommendation design as an infinite-horizon state-feedback optimal control problem whose performance index rewards alignment/engagement while penalizing polarization, large deviations from an uncontrolled baseline, recommendation mismatch, control effort, and exposure disagreement across neighboring users.

We derive explicit spectral conditions under which the reduced quadratic stage cost is strictly positive-definite, and we show that the failure of these conditions makes the resulting recommendation design exhibit pathological behaviors, such as unstable free modes, non-attainment of the infimum, or failure of the stationary affine synthesis.