We propose and unify classes of different models for information propagation over graphs. In a first class, propagation is modelled 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 arrival time of the information wave front. 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 generalised 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 to those of neighbouring nodes with lower values. We provide precise formulations of the model classes and prove equivalences between them. Motivated by the connection between first arrival time model and the eikonal equation in the continuum setting, we derive formal limits for graphs based on uniform grids in Euclidean space under grid refinement. For a specific parameter setting, we demonstrate that the solution on the grid approximates the Euclidean distance, and illustrate the use of front propagation on graphs to trust networks and semi-supervised learning.

翻译：本文提出并统一了图上的不同信息传播模型类别。第一类模型中，传播被建模为从初始时刻已知节点集合出发的波，在后续时刻按信息波前到达时间确定的顺序传播至其他未知节点。第二类模型基于节点间路径的传播时间概念：从初始已知节点集合到某节点的信息传播时间定义为所有可行路径子集上的广义传播时间最小值。最终类别通过在每个未知节点上施加程函形式的局部方程，并在已知节点处设置边界条件。节点局部方程的解值与其邻域中具有较低值的节点耦合。我们给出了各类模型的精确表述，并证明了它们之间的等价性。受连续介质中首达时间模型与程函方程关联的启发，我们基于欧氏空间中的均匀网格，推导了网格细化下图的正式极限。针对特定参数设置，我们证明网格上的解近似欧氏距离，并展示了图上前向传播在信任网络与半监督学习中的应用。