Finding shortest and nearly shortest path nodes in large substantially incomplete networks by hyperbolic mapping

Abstract

Dynamic processes on networks, be it information transfer in the Internet, contagious spreading in a social network, or neural signaling, take place along shortest or nearly shortest paths. Computing shortest paths is a straightforward task when the network of interest is fully known, and there are a plethora of computational algorithms for this purpose. Unfortunately, our maps of most large networks are substantially incomplete due to either the highly dynamic nature of networks, or high cost of network measurements, or both, rendering traditional path finding methods inefficient. We find that shortest paths in large real networks, such as the network of protein-protein interactions and the Internet at the autonomous system level, are not random but are organized according to latent-geometric rules. If nodes of these networks are mapped to points in latent hyperbolic spaces, shortest paths in them align along geodesic curves connecting endpoint nodes. We find that this alignment is sufficiently strong to allow for the identification of shortest path nodes even in the case of substantially incomplete networks, where numbers of missing links exceed those of observable links. We demonstrate the utility of latent-geometric path finding in problems of cellular pathway reconstruction and communication security.

Zoo guide to network embedding

“…This algorithm can determine the angular coordinates of the hyperbolic embedding in H 2 and, more generally, extract other types of onedimensional embeddings. Hyperbolic latent space contains geometric information associated with the original network; in the context of H 2 , a recent study showed that shortest paths in the hyperbolic latent space are aligned along geodesic curves connecting endpoint nodes, and this alignment is sufficiently strong to allow the identification of shortest path nodes even in the case of substantially incomplete networks [179]. Finally, note that the hyperbolic embedding in the case of directed networks has been addressed in [180].…”