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：我们引入了一种与离散对象（如图和超图）上的动力学相关联的新型信息几何结构。所提出的框架包含分别建立在顶点空间和边空间上的两个对偶平坦结构。前者是密度与势之间的传统对偶关系，例如，概率密度及其由凸热力学函数诱导的对数形式。后者是由凸且对称的耗散函数诱导的通量与力之间的对偶关系，该耗散函数驱动密度的动力学。这两个结构通过由底层离散对象诱导的同调代数关系在拓扑上相连。该双重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，包括马尔可夫跳跃过程和非线性化学反应动力学，以及自然梯度和镜像下降。该双重对偶平坦结构上的信息几何投影引出了亥姆霍兹-霍奇分解和$L^{2}$ Wasserstein几何中奥托结构的信息几何推广。该结构可扩展至非梯度非平衡流，由此我们还能得到循环空间上诱导的对偶平坦结构。这个抽象但通用的框架能将信息几何的适用性扩展至线性和非线性动力学中的各种问题。