Interacting Particle Systems on Networks: joint inference of the network
  and the interaction kernel
Quanjun Lang Xiong Wang Fei Lu Mauro Maggioni
Modeling multi-agent systems on networks is a fundamental challenge in a wide
variety of disciplines. We jointly infer the weight matrix of the network and
the interaction kernel, which determine respectively which agents interact with
which others and the rules of such interactions from data consisting of
multiple trajectories. The estimator we propose leads naturally to a non-convex
optimization problem, and we investigate two approaches for its solution: one
is based on the alternating least squares (ALS) algorithm; another is based on
a new algorithm named operator regression with alternating least squares
(ORALS). Both algorithms are scalable to large ensembles of data trajectories.
We establish coercivity conditions guaranteeing identifiability and
well-posedness. The ALS algorithm appears statistically efficient and robust
even in the small data regime but lacks performance and convergence guarantees.
The ORALS estimator is consistent and asymptotically normal under a coercivity
condition. We conduct several numerical experiments ranging from Kuramoto
particle systems on networks to opinion dynamics in leader-follower models.