This post is based on a paper that appeared a couple of weeks ago on the Computer Science section of arXiv. You can find it here. I’m going to write a few things about the ideas behind the paper, and avoid pretty much entirely the convincing data the authors present, as well as questions of implementing the algorithms discussed.

The setting is a social network, which we can describe as a graph. Nodes stand for people, and an edge represents that the two associated people have some social connection. This paper focuses on edges corresponding to friendship in the Facebook graph.

A key empirical feature of the graph topology of such social networks as compared to most mathematical models of random graphs is the prevalence of short cycles, and so-called *clustering*. Loosely speaking, in an Erdos-Renyi random graph, any potential edges appear in the graph independently of the rest of the configuration. But this does not accord well with our experience of our own Facebook friend circle. In general, knowledge that A is friends with both B and C increases the likelihood that B and C are themselves friends. This effect appears to be more present in other models, such as Preferential Attachment and the Configuration Model, but that is really more a consequence of the degree sequence being less concentrated.

The reason for this phenomenon appearing in social networks is clear. People meet other people generally by sharing common activities, whether that be choice of school, job or hobbies. The question of how readily people choose to add others on Facebook is a worthwhile one, but not something I have the time or the sociological credibility to consider! In any case, it is not a controversial idea that for some typical activity, it is entirely possible that almost all the participants will end up as friends, leading to a large (almost-) ‘clique’ in the graph. Recall a clique is a copy of a complete graph embedded in a larger graph – that is, a set of nodes all of which are pairwise connected.

We could think of much of the structure of this sort of network as being generated in the following way. Suppose we were able to perform the very unlikely-sounding task of listing every conceivable activity or shared attribute that might engender a friendship. Each activity corresponds to a set of people. We then construct a graph on the set of people by declaring that a pair of nodes are connected by an edge precisely if the people corresponding to these nodes can both be found in some activity set.

Another way of thinking about this setup is to consider a bipartite graph, with people as one class of vertices, and activities as the other. Predictably, we join a person to an activity if they engage in that activity. The edges within the class of people are then induced by the bipartite edges. Obviously, under this interpretation, we could equally well construct a graph on the set of activities. Here, two activities would be joined if there is a person who does them both. Graphs formed in this way can be called *Intersection Graphs*, and there is lots of interest in investigating various models of *Random Intersection Graphs*.

The question addressed by the authors of the paper can be summarised as follows. A social network graph tells us whether two people are ‘friends’, but it does not directly tell us how close their relationship is. It is certainly an interesting question to ask whether the (local) network topology can give us a more quantitative measure of the strength of a friendship.

As the authors explain, a first approach might be to consider how many mutual friends two people have. (We consider only pairs of people who are actually friends. It seems reasonable to take this as a pre-requisite for a strong relationship among people who do actually use Facebook.) However, this can fail because of the way these social networks organise themselves around shared attributes and activities. The size of one of these cliques (which are termed *social foci *in parts of the literature) is not especially likely to be well correlated to the strengths of the friendships within the clique. In particular, the clique corresponding to someone’s workplace is likely to grow in size over time, especially when people grow towards an age where, on average, they move job much less. So it seems likely that, according to a naive examination of the number of mutual friends, we would predict that a person’s strongest friend is likely to be someone they work with, who perhaps by chance also does some other activity with that person.

The authors phrase this problem slightly differently. They examine algorithms for establishing a person’s spouse or long-term partner with good accuracy from only the local network structure.

Heuristically we might expect that a husband knows many of his wife’s work colleagues, and vice versa. Not all of these ties might be so strong that they actually lead to friendship, in the Boolean sense of Facebook, but we might expect that some noticeable proportion have this property. Naturally, there will be cliques to which both belong. One or more of these might be the reason they met in the first place, and others (eg parents at children’s schools) might have developed over the course of their relationship. However, as we’ve explained, this doesn’t narrow things down much.

(We need not be constrained by this heteronormative scenario. However, as the authors point out in a footnote, there are challenges in collecting data because of the large number of ironic relationship listings on Facebook, mainly among the undergraduate and younger community. This problem is particularly obstructive in the case of same-sex marriage, owing to the smaller numbers of genuine pairings, and larger numbers of false listings for this setting.)

The crucial observation is that if we look at the couple’s mutual friends, we expect to see large parts of the most important cliques from both husband and wife’s lives. Among these mutual friends, there will be some overlap, that is cliques of which both are an integral member. But among the rest, there will be a natural partition into friends who really originate from the husband, and friends who were introduced via the wife. So the induced graph on these mutual friends is likely to split into three classes of vertices, with very poor connectivity between two of them.

This is, up to sorting out scaling and so on, precisely the definition of *dispersion*, introduced by the authors. The dispersion between two vertices is high if the induced graph on their mutual neighbourhood has poor connectivity. Modulo exact choice of definition, they then exhibit data showing that this is indeed a good metric for determining marriages from the network topology, with success rate of around 50% over a wide range of users.

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