Grothendieck Graph Neural Network (GGNN): A Path-Based Framework For Network Modelling

We introduce a new framework for applying machine learning to graph data. Recently, graphs have emerged for modeling complex relations between components and giving a big picture of a system. The existence of paths, in addition to nodes and edges, helps represent some non-directed relations between nodes at different levels of importance. Understanding paths and interactions between them reveals more information about the entire graph. Most representation tools in graph theory are based on neighborhoods and lack special tools to represent paths. Therefore, it is very challenging to replace neighborhoods with paths in studying a graph. In this paper, we propose a matrix representation of paths and a binary operation to get a monoid. This algebraic point of view benefits the graph neural network (GNN) and can be seen as an alternative for neighborhoods in GNN. We apply this monoidal representation of graphs to introduce a new type of GNN called Grothendieck Graph Neural Network (GGNN), inspired by the Grothendieck Topology concept [1]. To evaluate our approach, we build a model to estimate path delays in networks based on GGNN. The results (MRE=0.0004) show the eligibility of applying GGNN in this kind of problem compared with RouteNet (MRE=0.022).