20 September 2019 ~ 0 Comments

## Who will Cluster the Cluster Makers?

If you follow this blog, you know that I periodically talk about community discovery. The problem seems so deceptively simple: finding groups of nodes densely connected in a network. If it is so simple, why have I been talking about it since 2012? The reason is that it isn’t so simple, and people have tried to organize the literature explaining how the thousands of different algorithms work, how they perform, what definition of “community” they use. After all that work, I’m still left with one question. Which two algorithms return the same gosh darned communities in a network? That’s what we’re going to discover today.

What I want to do is to take as many community discovery algorithms as I can, test them on a set of networks, and compare their results to estimate how similar they are. This gives me a similarity matrix of algorithms, which I can transform into a network by keeping only similarities that are statistically significant. Once I have the network, I can discovery groups of algorithms returning a coherent set of results, because they’re all significantly related to each other. How do I find these groups? Well, by ehm…. er… performing… community… discovery? on the… network of… community discovery… algorithms… This is exactly the outline of my paper Discovering Communities of Community Discovery, which I presented at ASONAM last month.

“As many community discovery algorithms as I can” turned out to be 73, all implemented in different languages, taking input in different ways, and providing different output formats. I ran them on more than 1,500 networks (real world ones from ICON and synthetic benchmarks). It was a… difficult month for me. I compared their partitions by estimating their mutual information: given that I know the results of algorithm A, how much can I infer about the results of algorithm B? For each network where two algorithms result in a lot of mutual information, I increase their similarity count by one. Once I have all similarity counts, I can extract the backbone of this matrix, controlling for the fact that some algorithms tend to be more peculiar than others, while others tend to be more mainstream.

And this is the result (click to enlarge):

I love this network, because it has well defined groups and they all make sense. There’s the group of modularity maximization algorithms (in green), there’s the ones based on percolation / random walks (in blue), and the ones using neighbor similarity (in purple) as the guiding principle.

Then there’s a lump of algorithms that allow communities to share nodes (in red). The only thing these algorithms have in common is that they allow communities to share nodes, which is not a good enough common characteristics. The ways they find communities are as diverse as the ones you find in the rest of the network. But that’s the beauty of my approach: I can select a subset of nodes — say all overlapping community discovery algorithms — and re-apply the test of statistical significance with a more stringent threshold. This allows me to zoom in and see if there are meaningful structures inside the community. Lo and behold:

Here you can see meaningful groups of overlapping algorithms. There’s the ones achieving overlap by clustering edges instead of nodes (in blue), and the ones applying the percolation / random walk strategy (in green), but allowing for node sharing.

Why is this work significant? First, because it proves that there really are different — and valid — definitions of what communities are in complex networks. If there weren’t, this network would be more homogeneous, without distinct groups.

Second, as I mentioned, some of the networks I tested the algorithms on are standard benchmarks: LFR networks. These benchmarks grow a network with a planted community structure: the real latent structure the algorithm is supposed to find. Yet, this “ground truth” is well embedded in one of the clusters: the percolation/random walk community (in blue). LFR benchmarks follow that specific definition and not others. If you are developing a new community discovery algorithm which has a different community definition, you should not use the LFR benchmark to test it. Moreover, if you are developing a percolation/random walk algorithm and you’re correctly testing it on an LFR benchmark, you cannot test it against algorithms that are not part of the blue community. Otherwise the test would be unfair, because those algorithms are looking for something else: of course they’ll perform poorly on LFR benchmarks!

You can get the full list of algorithms that I tested, with proper references, from the official page of the project. From there, you can also download the network and use it for your purposes. This is necessarily an eternal work in progress: there are more than 73 community discovery algorithms out there. But I am but a man[citation needed] and I cannot spend my entire time scouting for implementations on the web. I got to put bread (or, preferably, pasta) on the table as well. Thus, if you think I really should have included your algorithm in this structure, you can mail me and send me a working implementation of it, and I’ll gladly run it on my benchmarks.

What’s next? I’d be delighted to inaugurate the field of meta-research. So join me, as I develop new projects such as:

• Predicting links of link predictors: which link prediction algorithms will become more similar in the future?
• Spreading epidemics of epidemic spreading: which researchers will cave in to peer pressure and publish a paper studying the diffusion of some phenomenon?
• Modeling the growth of growth models: how has the Barabasi and Albert model evolved over time? Which features were added? What about Watts & Strogatz’s model?

(In case you were wondering, I’m joking. In network science, sometimes that might be hard to tell.)

(Or am I?)

(It’s settled: the best among these papers will receive the hereby instituted Escher prize, awarded by Douglas Hofstadter himself)

20 March 2014 ~ 0 Comments

## When Dimensions Collide

First of all: communities do overlap. When trying to detect them, many researchers realized that hard partitions, where each node can belong to one and only one community, are not always a good idea. Most of them found this a problem. Others were actually very happy: the problem gets harder! Nice! (Researchers are weird). Blinded by their enthusiasm, they started developing algorithms to deal with this overlap. Not many asked the question I am trying to answer here: why do communities overlap? As a result, some of these algorithms detect this overlap, but using approaches that do not really mean anything in real life, it’s just a mathematical trick. Others, instead, build the algorithm around a core hypothesis.

This hypothesis is nothing unheard of. Communities overlap because people have complex lives. Some of your college mates also attend your yoga class. And you know your significant other’s colleagues, which puts you in their community. All these communities have you as common member, and probably some more people too. The beauty of this is that it is not only intuitive: it works well in finding communities in real world social networks. So well that it is the assumption of my approach and of many others outstanding algorithms (this and this are the first two that pop into my mind, but there are probably many more). Another beautiful thing about it is that it is almost obvious, and so it is probably true. But here we hit a wall.

The fact that it is simple, reasonable and it works well in practice proves nothing about its property of being true. There are things that are not simple nor reasonable, but nevertheless true (hello quantum physics!). And there is practical knowledge that does not quite correspond to how things work (in my opinion, most computer science is a patch and nobody really knows why it works). Unless we test it, we cannot say that this nice practical principle actually corresponds to something happening in reality. So how do we go on and prove it? In the paper I proposed a first step.

This brings me back to another old love of mine. Multidimensional networks. They are networks in which we put multiple relations in a cage together in mating season and see what happens (research is fun). The idea behind the paper is that multidimensional networks give us the perfect tool to test the hypothesis. In monodimensional networks you have no clue why two people are connecting besides the obvious “they know each other”. In a multidimensional network, you know why they know each other, it’s information embedded in the type of the relation. So, the hypothesis is that different types of relations are the cause of the community overlap, and with multidimensional networks we can look at how communities distribute over relations. First, let us take a look at what two overlapping communities look like in a multidimensional network.

We collected a multidimensional social network putting together relationships between users in Facebook, Twitter and Foursquare. We then used DEMON to extract overlapping communities from each dimension. We then took two communities with extensive overlap in the Facebook dimension (picture below).

We then looked at the very same set of nodes, but now in the Foursquare network. In the picture below, we kept the edges, and the node positioning, of the Facebook network to make the comparison easier, but keep in mind that the edges in the Foursquare dimension are different, and they are the ones that decide to what community the nodes belong.

Very interesting. The communities look a lot alike, although the shared (and non shared) nodes are slightly different. Now node 7369 is shared (it wasn’t in Facebook) while node 8062 isn’t (whereas it was before). Let’s put another nail on the coffin and see the communities these nodes belong in Twitter (same disclaimer applies):

Surprise surprise, in Twitter there is actually only one community, which brings together the majority of the nodes of the two communities. So here’s where our overlap comes from: common affiliations in different dimensions! Now, I’m going to deal with that voice in your head that is screaming “Anecdotal! Anecdotal!”. (You don’t hear it? Did I already mention that researchers are weird? In any case “Anecdotal” refers to a type of evidence that bears no value in scientifically proving a point if not backed by more solid proofs). Put in a more general way: the more two communities overlap in some dimension, the more likely it is we can find a dimension in which these communities are actually a single community. This involves boring details you can find in the paper which ultimately generate this plot:

Does this plot prove our theory without leaving out any reasonable doubt? Maybe, or not really. There are still things to check. But science is made by tiny steps forward. And this is certainly one.