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Not Everything is Triadic Closure

The simplicity and power of the notion of triadic closure makes it a candidate for analyzing almost anything that can be represented by nodes and edges. Pick three things, beg, borrow and steal until you get relationships between any two of them, and now you have a case for the third relationship that closes the triangle. Wonderful.

Now, for about fifteen seconds, switch your browser tab from this one to the one right next to it. The one that says Facebook. Take a glance at your news feed, and switch right back. If you’ve got Twitter, or really, any other social network that aggregates information across users, take a raincheck and go peek there too.

It used to be that all the information we received on our activity feeds was a direct derivative of the people with whom we shared a formal relationship. This still is true if you’re a new entrant to the social network (but I’m told even babies have Facebook accounts these days, so there), because your footprint on the network is still being constructed. The consequence of companies making a business out of social interaction is that we are now exposed to information from a wide variety of sources—sources that we do not necessarily know personally. Some of this information is excellent, some of it passes right under our nose because we have become operantly conditioned to ignore the “noise” on our news feeds, and some are outright nebulous.

Let’s put this in context. We’ll start off with a more generic example, then move on to bring out the big guns.

I happen to have a good friend called John. His last name’s Doe, though I don’t think that’s really important at this time. John’s a great guy. His entire presence on Facebook is built around “sharing” information which comes from friends of his, friends of friends (at any depth down this chain) of his, and even from news aggregation sites around the Internet.

One day John shares something from Jane. Jane’s last name is also Doe, but it turns out that they’re not really related—this is important. Reading what Jane has written, I decide to follow her, because well, if someone writes nice, intellectual content and puts it in the public domain, that makes for good after-dinner reading.

I’ve just formed a relationship that wasn’t induced by triadic closure.

In the past, this wouldn’t have been very common. The amount, and more importantly, the flow of information was not as liberal as it is today. However, we’re now doing this all. the. time. Social networks even assist this process by broadcasting to all my friends that “Kenneth has added Jane Doe as a friend”, and thence, triadic closure takes over.

Not convinced? Here’s a better example. In Summer of 2014, The ALS Association launched a campaign on social media, now known (and immortalized) as “The ALS Ice Bucket Challenge”. The premise was simple—dunk a bucket of cold water over your head and donate ten dollars to raise awareness for the cause, then task three other people with the same challenge. Fail to do it, and donate a hundred dollars for the cause. But we’re not interested in the hundred dollars. We’re interested in the propagation effect of this campaign over a network.

Let’s set things up a little more rigorously this time. Instead of setting up edges in the network based on relationships, we’ll set up edges in the network based on information flow. Our invariant for each node, assuming the node represents a participating person, is that it’s going to have an out-degree of three. Where could these edges lead? Anywhere. Any node is fair game, but it’s also reasonable to expect that edges will normally be formed within a relatively small radius, say a path distance of two or three, because the challenge is only fun when you can see people you know get drenched.

Wait. Who doesn’t want to see Justin Bieber get drenched?

As it turns out, I could very well challenge Bill Gates, Lady Gaga and Mark Zuckerberg to do the challenge. Alone, I wouldn’t be very successful, but if everyone thinks the same way I do (which I’m counting on), the aforementioned three people are suddenly going to have a very large in-degree of edges. Why will these edges reach them? The social network spreads it for me. People like John Doe spread it for me. And because these people have so many followers to begin with, when I cross over to the original social network (connected by relationships rather than information flow), my reach has increased very significantly.

This brings us to the crux of the matter. Triadic closure works as an excellent link predictor when the network (or subset of the network) is in the early stages. Once past a certain point, information transmission takes over, creating “shortcuts” that bypass normal link formation behavior. People who broker information on social networks number far fewer than people who form links purely based on relationships, but the former group are far more influential simply because they are in a better position to facilitate the formation of shortcuts. These shortcuts are a good way to spread information, but the shortcuts alone cannot create awareness through the network (more commonly described by the concept of being “viral”). Instead, these shortcuts serve the purpose of spreading information from relatively distant, badly connected nodes, to nodes with a large number of connections (or friendships). From there, it’s a home run, because normal user behavior takes over, and I have a tendency to “trust” what I see from a friend.

What does this mean? This means that if one were choosing to model information spread throughout a network, one would ideally adopt a hybridized model that “swaps over” from triadic closure to shortcut formation once the network evolves past a certain point. Beyond just treating every node as equal, and every edge as “strong” or “weak”, we would also have to consider which of the nodes are information epicenters, and which of the nodes are friendship epicenters. If one wished to maximize awareness, one would come up with a scheme that used information epicenters to “bounce” to friendship epicenters.

As a closing note, it would be interesting to see, in the near future, literature that attempts to model the spread of the ALS Ice Bucket Challenge through the social network. How would we predict the people who are challenged? How would we estimate the number of people who are willing and able to respond to a challenge? How many people, really, did the ALS Ice Bucket Challenge reach?

L. Weng, J. Ratkiewicz, N. Perra, B. Goncalves, C. Castillo, F. Bonchi, R. Schifanella, F. Menczer, and A. Flammini. “The Role of Information Diffusion in the Evolution of Social Networks”. Proc. 19th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. CoRR, abs/1302.6276, 2013.

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