Imagine you, everyone you know, and everyone they know are represented as points floating in space. Once you’ve metaphorically assembled this enormous set of social connections, mentally draw lines connecting the points which represent their relationships to each other and to you. Soon enough you’ll have your entire social network represented as a spiderweb-looking graph of points and lines, or as network scientists call it- nodes and edges. Most of the relations that govern our daily lives can be represented by networks: the internet, our social behavior, the spread of information, and even gravity. Defining the geometries of these networks is vital to understanding the fabric of our most basic interactions.

[“”DK-Lab director Dima Krioukov tessellating his hyperbolic floor”/ DK Lab]

Dr. Dmitri Krioukov and his colleagues explore these concepts in the DK network theory lab. Visualized Hyperbolic Geometry: Their most recent publication in Springer Nature Physics, titled “Network Geometry,” reviews the fundamental properties of a network. Let’s revisit the social spiderweb example. Envision you are at your home-base node and are instructed to deliver a letter to an individual you’ve never met at the fringe of this giant network. Going randomly from node to node trying to find this individual would be inefficient and is not representative of how real-life social systems work. So, what are the parameters that make networks navigable and how do scientists define them?

In this publication, Dr. Krioukov’s work found that network geometries are actually defined by hidden, or latent properties. Our social networks are not made up of random connections between people in a geographic area, but rather by less obvious facts, or data, about us like our occupations,interests, and backgrounds. If you know the small details about how each person in your network behaves, you can now more easily find your way through the connections between people. With these latent properties, this network becomes more accessible.

In their next and most recent publication, “Ollivier-Ricci curvature convergence in random geometric graphs,” the team tackled one of the most fundamental and elusive aspects of defining a network: it’s curvature. The curvature of a network is the geometric road map that allows it to become easily navigable and thus, predictable. With over 20 different definitions of graph curvature, it is quite a challenge to tackle. Nevertheless, by focusing on the conclusions from the first study, Dr. Krioukov’s group determined that finding the curvatures of real-life networks is actually possible and predictable. “[Curvature] is perhaps the most basic characteristic of a space,” Dr. Krioukov explains, “[that] applies to systems we care about as humans.” This allows researchers to better understand phenomena characterized by spreading mechanisms such as information, viruses, and even fake news.

This finding is a major contribution to the field of network science. The team showed that, for networks living in latent space, curvature is definable and converges to a widely known geometry called Ollivier curvature: “If you know the curvature of the underlying graph then you know what kind properties to expect from your system, for example the spreading of the current pandemic, information online, and even the most fundamental behavior of gravity,” adds Dr. Krioukov.

The findings of these studies just scratch the surface of this exciting research. To learn more, visit the lab website: dk-lab.net.