## Complexity Squared

I decided to give to this blog post an obscure title because today I want to talk about something that in complex network analysis goes under many names, so I did not want to favor any of them. What I am talking about are networks with multiple types of relations in them, the main subject of my PhD Thesis and of a recent article that I published in the World Wide Web Journal. These structures are putting more complexity on top of complex networks, therefore they are complex network squared: hence the fancy blog title.

These networks are referred to in the literature with the following terms:

- Multidimensional (the term that I use in my thesis);
- Multirelational;
- Layered;
- Interdependent;
- Multisliced;
- Multilevel;

and so on and so forth. All these terms refer to the same theoretical object, that is also implemented in many ways. I’ll mention some of them just to sound like the guardian of an obscure cult: labeled multigraphs, hypergraphs, mesostructures and coupling edges.

Despite the confusion that I tried to create with the first paragraphs, the general idea of this line of research is brutally simple: in our everyday life we are not part of only one network. It may look like we are, but when we start thinking harder about our relationships, we realize that we know the people we know for different reasons. This idea is the one behind the fact that every person can belong to different “communities” at the same time, which I already discussed in these pages. But it is deeper than that. It does not only require the more sophisticated, but still traditional, community discovery algorithm that I described in that blog post. It requires a whole new model and mindset.

Before multidimensional networks (forgive me if for clarity I’ll use my term for these structures) the classical complex network analyst would just assume that a single relation represents a particular phenomenon and nothing else can be said about it. Allow me to recycle this picture about my Facebook friends:

Intuitively this looks nice, as we can find communities and central nodes. But is this picture really telling us everything about my Facebook friends? What about a higher order of aggregation among them? What about not only their friendship links but also their common interests? The multidimensional network analyst throws a bunch of new connections on top of it and she tells you: “There’s something more”. In this case:

A visualization that is not nearly as elegant as the previous one, I give you that, but nevertheless it is useful to understand a higher level aggregation of my Facebook friends. On top of the connections between friends, we added edges connecting people if they are part of the same group or if they like the same stuff on Facebook. The two gigantic hairballs are composed by people who are in the same location: there is the cluster of people living in Italy, the one of people living in the US, and connections between them from people travelling between the two countries. So, we saw that adding different types of relations uncovers structural properties that none of the relations by itself would reveal.

I’ll give you another example of a cool real world effect of multidimensional networks. This is not from a work of mine, but it is from the Nature paper “Catastrophic cascade of failures in interdependent networks” by Sergey V. Buldyrev, Roni Parshani, Gerald Paul, H. Eugene Stanley and Shlomo Havlin. Suppose you have a power grid: what happens if one plant is subject to a failure? The classical complex network analyst tells you that we could not care less: the power grid is a scale free network, in which the majority of plants are only connected to a couple other plants. So, a random failure of one plant does not affect the rest of the network too much, unless we are extremely unlucky and we lose a power hub (but that’s really rare, and the classical network guy is an incurable optimist).

A multidimensional network scientist, instead, is way more careful. Why? Because he knows that the power grid network is not independent from everything else, but it is plugged into another network. For example, in a computer network that regulates its functioning. When a power plant goes down, a set of computers cannot work anymore. And what happen to the plants that are connected to those computers? They fail too, triggering another computer failure and God helps us all. It is theoretically proven that two different scale free relations, dependent on each other, are much much **much** more fragile than a single scale free network. This actually happened in Italy (where else?) and the following is a depiction from Buldyrev et al’s paper:

In the first Italy we see one plant going down (in red on the map) taking with it the computers it supplies with energy (in the flying network). This triggers a couple more failures in the second picture that eventually, in the third picture, completely destroy the power supply chain of southern Italy.

So far I gave you the idea that multidimensional networks are not exactly the same animal as classical complex networks. To give you a taste of how to prove this, I’ll spare you the super complicated equations of interdependent network percolation present in the Nature paper. I’ll instead provide another example from community discovery. As I said in my previous post, community discovery is loosely defined as the problem of grouping nodes in a network that are “densely connected”. Naturally, when we deal with **multidimensional** networks, the “densely connected” has to be changed into “**multidimensionally** densely connected”. Why is this challenging? Here I’ll give you an intuition and I promise that in the future I’ll come back with more details. For now, it is sufficient to use two pictures. Here’s the first:

Here we assume that we have two different dimensions and they are represented with solid or dashed edges. Is this set of nodes multidimensionally dense? Of course: everybody is connected with everybody and all dimensions of the network are equally represented. Now consider another situation:

Is this set of nodes multidimensionally dense? Of course: everybody is connected with everybody and all dimensions of the network are equally represented. But the two examples are very different. That’s funny: we just discovered that, in multidimensional networks, density is an ambiguous concept.

And, as conclusion, I’ll add some multidimensional flavor to another classical network problem: link prediction. Link prediction aims at predicting your next Facebook friend. The above mentioned multidimensional network scientist steps in and says: “But why only your next Facebook friend? Why not your next virtual acquaintance tout-court?”. He means that all your social media connections and their different types play a role in determining when and where you’ll connect with somebody. This is exactly what multidimensional link prediction is, and how to do this is a complex problem that currently remains unsolved. But the multidimensional network guy loves complex problems as much as he loves complex words.