Information Geometry of Dynamics on Graphs and HypergraphsWe introduce a new information-geometric structure of dynamics on discrete
objects such as graphs and hypergraphs. The 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 on the manifold.
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 the information-geometric generalizations of Helmholtz-Hodge-Kodaira
decomposition and Otto structure in $L^{2}$ Wasserstein geometry. The structure
can be extended to non-gradient nonequilibrium flow, 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.
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