In this work we propose and unify classes of different models for information propagation over graphs. In a first class, propagation is modeled as a wave which emanates from a set of known nodes at an initial time, to all other unknown nodes at later times with an ordering determined by the time at which the information wave front reaches nodes. A second class of models is based on the notion of a travel time along paths between nodes. The time of information propagation from an initial known set of nodes to a node is defined as the minimum of a generalized travel time over subsets of all admissible paths. A final class is given by imposing a local equation of an eikonal form at each unknown node, with boundary conditions at the known nodes. The solution value of the local equation at a node is coupled the neighbouring nodes with smaller solution values. We provide precise formulations of the model classes in this graph setting, and prove equivalences between them. Motivated by the connection between first arrival time model and the eikonal equation in the continuum setting, we demonstrate that for graphs in the particular form of grids in Euclidean space mean field limits under grid refinement of certain graph models lead to Hamilton-Jacobi PDEs. For a specific parameter setting, we demonstrate that the solution on the grid approximates the Euclidean distance.
翻译:在这项工作中,我们提出并统一了不同模型的类别,用于在图形上传播信息。在第一类中,传播是先从一组已知节点的最初一组已知节点到后来所有其他未知节点的波,由信息波前端到达节点的时间决定。第二类模式基于在节点之间的路径上旅行时间的概念。从最初已知的节点到节点的信息传播时间被定义为在所有可接受路径子子上的通用旅行时间的最小值。最后一类是在每个未知节点上按已知节点的边界条件对每个网格的局部等式进行。节点的本地方程式的解决方案值与相邻节点与较小解点的节点结合。我们提供了该图表设置中模型班的精确配方,并证明了它们之间的等同性。根据首次到达时间模型和连续设置中的ekonal等值之间的关联,我们展示了Euclidea空间正轨表的本地等式等式,我们展示了Euclideidean平均阵列点模型中特定形式的图表。