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.

翻译：本文引入了一种与图、超图等离散对象动力学相关的全新信息几何结构。所提出的框架由分别构建于顶点空间和边空间上的两个对偶平坦结构组成。前者是密度与势之间的常规对偶关系，例如由凸热力学函数诱导的概率密度及其对数形式。后者是由驱动密度动力学的凸对称耗散函数诱导的通量与力之间的对偶关系。这两个结构通过底层离散对象诱导的同调代数关系在拓扑上相互连接。该双对偶平坦结构中的广义梯度流是黎曼流形上梯度流的推广，涵盖了马尔可夫跳变过程、非线性化学反应动力学，以及自然梯度和镜像下降。该双对偶平坦结构上的信息几何投影将导致亥姆霍兹-霍奇分解和$L^{2}$ Wasserstein几何中奥托结构的信息几何扩展。该结构可进一步推广至非梯度非平衡流，由此我们还获得了循环空间上诱导的对偶平坦结构。这一抽象但通用的框架能够将信息几何的适用性扩展到线性和非线性动力学中的各类问题。