  We introduce a new information-geometric structure associated with the
dynamics on discrete objects such as graphs and hypergraphs. The presented
setup consists of two dually flat structures built on the vertex and edge
spaces, respectively. The former is the conventional duality between density
and potential, e.g., the probability density and its logarithmic form induced
by a convex thermodynamic function. The latter is the duality between flux and
force induced by a convex and symmetric dissipation function, which drives the
dynamics of the density. These two are connected topologically by the
homological algebraic relation induced by the underlying discrete objects. The
generalized gradient flow in this doubly dual flat structure is an extension of
the gradient flows on Riemannian manifolds, which include Markov jump processes
and nonlinear chemical reaction dynamics as well as the natural gradient and
mirror descent. The information-geometric projections on this doubly dual flat
structure lead to information-geometric extensions of the Helmholtz-Hodge
decomposition and the Otto structure in $L^{2}$ Wasserstein geometry. The
structure can be extended to non-gradient nonequilibrium flows, from which we
also obtain the induced dually flat structure on cycle spaces. This abstract
but general framework can extend the applicability of information geometry to
various problems of linear and nonlinear dynamics.
