  This paper is concerned with structure-preserving numerical approximations
for a class of nonlinear nonlocal Fokker-Planck equations, which admit a
gradient flow structure and find application in diverse contexts. The
solutions, representing density distributions, must be non-negative and satisfy
a specific energy dissipation law. We design an arbitrary high-order
discontinuous Galerkin (DG) method tailored for these model problems. Both
semi-discrete and fully discrete schemes are shown to admit the energy
dissipation law for non-negative numerical solutions. To ensure the
preservation of positivity in cell averages at all time steps, we introduce a
local flux correction applied to the DDG diffusive flux. Subsequently, a hybrid
algorithm is presented, utilizing a positivity-preserving limiter, to generate
positive and energy-dissipating solutions. Numerical examples are provided to
showcase the high resolution of the numerical solutions and the verified
properties of the DG schemes.
