The geometry of social networks

A key insight of general relativity is the view of the universe as a four-dimensional geometric space. Gravitation and other forces within physics are explained as properties of the geometry. Analogously, string theory includes many dimensions, as many as ten or more, in an effort to explain all the forces of the universe.

Given the geometry evident in nature, it is not surprising that sociologists suggested that social network inhabit a kind of multi-dimensional space. Agents in the network (i.e. you and I) are points or vectors in the space, and coordinates correspond to socio-demographic variables. For example, one coordinate could be the city you live in, another your income bracket, another your age, and so on. Agents are close in the space if they are similar; this is the principle of homophily: like is attracted to like.

Social space is commonly referred to as Blau space, after the sociologist Petr Blau who first proposed the idea. Blau space, therefore, can be viewed as the hidden geometry of social networks. Think of social networks as just dots and lines. Blau space embeds this network structure into a geometric space, complete with a metric which measures distance between agents.

Petr Blau

As is well-known to graph theorists, every network has an intrinsic geometry, determined by the geodesics (or shortest paths) separating nodes. However, this metric only reveals so much about the network. For example, by the small world property, most nodes are close. In Facebook for example, it was shown recently that on average, most nodes are distance four away in this network distance.  But people in the world are very different, and we shouldn’t expect them to be close in the corresponding Blau space of Facebook. A grandmother living in Topeka, Kansas who is republican and whose favorite pastime is knitting, would be light years away in Blau space from a Dutch ex-pat poet living in a beach house in Koh Samui, Thailand.

A recent paper of mine in PLOS ONE was the first to try and quantify properties of Blau space in networks like Facebook and LinkedIn. One of our models of social networks predicts a logarithmic dimension for the Blau space of social networks. For Facebook with its billion users, the dimension would be close to nine, the number of digits in a billion.  This means that only nine attributes would be needed to uniquely distinguish users among the millions of users of Facebook.

Using ideas from artificial intelligence (in particular, machine learning) along with some spectral analysis, we argued that this so-called Logarithmic Dimension Hypothesis is in fact an intrinsic property of social network.  We prefer to use more data sets to verify it if we had them, but we were buoyed up by the fact that some other prominent models posed independently by Leskovec and Kim, and Frieze and Tsourakakis also require the hypothesis to hold for their math models to work!

I go a step further. Social networks are just one kind of complex, real-world networks. There are many complex networks, such as the web graph or protein-interaction networks. My big idea here, which I call the Feature Space Thesis, is that every complex network lives in a feature space, with nodes with similar features closer in the space. For the web graph, the features would be topics; for protein networks, the features would be biochemical-properties. Note that as a thesis, this is not something that can be conclusively proved or refuted. Instead, it affords a view of complex networks; one that allows us to focus on aspects of the networks that might otherwise be ignored.

Protein-protein interaction network

My thinking is that the Feature Space Thesis is a critically important part of our understanding of complex networks. And the Logarithmic Dimension Hypothesis is just one piece of that larger puzzle. Once we know the dimension of a feature space, we should try and determine the underlying geometry of the space. Properties of the network should fall out naturally from the geometry. It will also allow us to better embed and cluster networks, as those close in the space are more likely to form communities.

Image result for community structure in networks

The great mathematician G.H. Hardy said: “The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics.” Whatever the geometric structure of social and other networks ultimately reveals, it has widened our view of these networks, and is likely to bring new ideas and approaches for many years to come.

Anthony Bonato



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