A Geometric Principle for Path Selection in Directed Acyclic Graphs

Source abstract

There are many types of path between nodes of a Directed acyclic graph (DAG), including the shortest and longest paths. Here, we ask whether the choice of path used in any analysis reflects the geometry of the latent space in which the DAG is embedded. To do this we define L_p-cube spacetimes, a family of continuous ordered spaces in which a single parameter p determines whether the ordered distance satisfies the ordered triangle identity when p>1, or the reverse ordered triangle identity when 0 1, where the ordered triangle identity holds, the shortest paths provide the closest approximation to the continuum ordered geodesic while for 0<p<1 and p<0, where the reverse ordered triangle identity holds, this role is instead taken by the longest weighted path. We also test the robustness of this relationship in a geometric citation model in which the random geometric DAG is supplemented by a preferential-attachment-type mechanism allowing connections beyond the strict geometric cut-off. The same broad correspondence between latent geometry and ordered geodesic path type is recovered. Our results show that the preference for shortest or longest paths in a DAG is closely connected to the geometry assumed for its latent space, and suggest that path structure can provide information about latent geometry, even when that geometry is not directly observed.