The Michelangelo Principle — almost certainly apocryphal — states that:

If you want to transform a marble block into David, just chip away the stone that doesn’t look like David.

This seems to be a great idea to solve literally every problem in the world: if you want to fix evil, just remove from the world everything that is evil. But I don’t want to solve every problem in the world. I want to publish papers on network analysis. And so I apply it to network analysis. In this case, I use it to answer the question whether a directed network has a hierarchical structure or not. The result has been published in the paper “Using arborescences to estimate hierarchicalness in directed complex networks“, which appeared last month in PLoS One.

To determine whether a network is hierarchical is useful in a number of applications. Maybe you want to know how resilient your organization is to the removal of key elements in the team. Or you want to prove that a system you mapped does indeed have a head, instead of being a messy blob. Or you want to know whether it is wise to attempt a takedown on the structure. In all these scenarios, you really desire a number between 0 and 1 that tells you how hierarchical the network is. This is the aim of the paper.

Following the Michelangelo Principle, I decide to take the directed network and chip away from it everything that does not look like a perfect hierarchy. Whatever I’m left with is, by definition, a perfect hierarchy. If I’m left with a significant portion of the original network, then it was a pretty darn good hierarchy to begin with. If I’m left with nothing, well, then it wasn’t a hierarchy at all. Easy. Let’s translate this into an algorithm. To do so, we need to answer a couple of questions:

1. What’s a “perfect hierarchy”?
2. How do we do the chipping?
3. How do we quantify the amount we chipped?

The first question is the one where we have most wiggle room. People might agree or disagree with the definition of perfect hierarchy that I came up with. Which is: a perfect hierarchy is a system where everybody answers to a single boss, except the boss of them all, who answers to no one. I like this definition because it solves a few problems.

Consider the picture above. In the leftmost example, if we assume nodes 1 and 2 give contradictory orders, node 5 doesn’t really know what to do, and the idea of a hierarchy breaks down. In the example in the middle, we don’t even know who’s the boss of them all: is it node 0 or node 3? The rightmost example leaves us no doubt about who’s boss, and there’s no tension. For those curious, network scientists call that particular topology an “arborescence“, and that’s the reason this exotic word is in the paper title. Since this is a well defined concept, we know exactly what to remove from an arbitrary directed network to make it into an arborescence.

Time to chip! Arbitrary directed networks contain strongly connected components: they have paths that can lead you back to your origin if you follow the edge directions. An arborescence is a directed acyclic graph, meaning that it cannot have such components. So our first step is to collapse them (highlighted in yellow above) into a single node. Think of strongly connected components as headless teams where all the collaborators are at the same level. They are a node in the hierarchy. We don’t care how a node organizes itself internally. As long as it answers to a boss and gives direction to its underlings, it can do it in whichever way it wants.

Second chipping step: in an arborescence, all nodes have at most one incoming connection, and only one node has zero. So we need to remove all offending remaining edges (highlighted in orange above). Once we complete both steps, we end up with an arborescence, and we’re done. (There are edge cases in which you’ll end up with multiple weakly connected components. That’s fine. If you follow the procedure, each of these components is an arborescence. Technically speaking, this is an “arborescence forest”, and it is an acceptable output)

We can now answer the final question: quantifying how much we chipped. I decide to focus on the number of edges removed. Above, you see that the original graph (left) had twenty edges, and that (right) nine edges survived. So the “hierarchicalness” of the original graph is 9 / 20 = .45.

Now the question is: why would someone use this method to estimate a network’s degree of hierarchicalness and not one of the many already proposed in the literature? The other methods all have small downsides. I build some toy examples where I can show that arborescence is the way to go. For instance, you can’t find a more perfect hierarchy than a balanced tree (leftmost example above). However, Global Reach Centrality would fail to give it a perfect score — since it thinks only a star graph is a perfect hierarchy. Agony and Flow Hierarchy aren’t fooled in this case, but give perfect scores in many other scenarios: a wheel graph with a single flipped edge (example in the middle), or a case where there are more bosses than underlings (rightmost example). Those who have been in a team with more bosses than workers know that the arrangement could be described in many ways, but “perfect” ain’t one of them.

Arborescence is also able to better distinguish between purely random graphs and graphs with a hierarchy — such as a preferential attachment with edges going from the older to the newer nodes (details in the paper). Before you start despairing, it’s not that I’m saying we’ve been doing hierarchy detection wrong for all these years. In most real world scenarios, these measures agree. But, when they disagree, arborescence is the one that most often sides with the domain experts, who have well informed opinions whether the system represented by the network should be a hierarchy or not.

To conclude, this method has several advantages over the competitors. It’s intuitive: it doesn’t give perfect ratings to imperfect hierarchies and vice versa. It’s graphic: you can easily picture what’s going on in it, as I did in this blog post. It’s conservative: it doesn’t make the outlandish claim that “everyone before me was a fool”. It’s rich: it gives you not only a score, but also a picture of the hierarchy itself. So… Give it a try! The code is freely available, and it plays nicely with networkx.

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Networks are a fantastic tool for understanding an interconnected world. But, to paraphrase Spider Man, with networks’ expressive power come great headaches. Networks lure us in with their promise of clearly representing complex phenomena. However, once you start working with them, all you get is a tangled mess. This is because, most of the time, there’s noise in the data and/or there are too many connections: you need to weed out the spurious ones. The process of shaving the hairball by keeping only the significant connections — the red ones in the picture below —  is called “network backboning”. The network backbone represents the true relationships better and will play much nicer with other network algorithms. In this post, I describe a backboning method I developed with Frank Neffke, from the paper “Network Backboning with Noisy Data” accepted for publication in the International Conference on Data Engineering (the code implementing the most important backbone algorithms is available here).

Network backboning is as old as network analysis. The first solution to the problem was to keep edges according to their weight. If you want to connect people who read the same books, pairs who have few books in common are out. Serrano et al. pointed out that edge weight distributions can span many orders of magnitude — as shown in the figure below (left). Even with a small threshold, we are throwing away a lot of edges. This might not seem like a big deal — after all we’re in the business of making the network sparser — except that the weights are not distributed randomly. The weight of an edge is correlated with the weights of the edges sharing a node with it — as shown by the figure below (right). It is easy to see why: if you have a person who read only one book, all its edges can have at most weight one.

Their weights might be low in comparison with the rest of the network, but they are high for their nodes, given their propensity to connect weakly. Isolating too many nodes because we accidentally removed all their edges is a no-no, so Serrano and coauthors developed the Disparity Filter (DF): a technique to estimate the significance of one node’s connections given its typical edge weight, regardless of what the rest of the network says.

This sounds great, but DF and other network backboning approaches make imprecise assumptions about the possibility of noise in our estimate of edge weights. In our book example, noise means that a user might have accidentally said that she read a book she didn’t, maybe because the titles were very similar. One thing DF gets wrong is that, when two nodes are not connected in the raw network data, it would say that measurement error is absent. This is likely incorrect, and it screams for a more accurate estimate of noise. I’m going to leave the gory math details in the paper, but the bottom line is that we used Bayes’ rule. The law allows us to answer the question: how surprising is the weight of this edge, given the weights of the two connected nodes? How much does it defy my expectation?

The expectation here can be thought of as an extraction without replacement, much like Bingo (which statisticians — notorious for being terrible at naming things — would call a “hypergeometric” one). Each reader gets to extract a given number of balls (n, the total number of books she read), drawing from a bin in which all balls are the other users. If a user read ten books, then there are ten balls representing her in the bin. This is a good way to have an expectation for zero edge weights (nodes that are not connected), because we can estimate the probability of never extracting a ball with a particular label.

I highlighted the words one and two, because they’re a helpful key to understand the practical difference between the approaches. Consider the toy example below. In it, each edge’s thickness is proportional to its weight. Both DF and our Noise Corrected backbone (NC) select the black edges: they’re thick and important. But they have different opinions about the blue and red edges. DF sees that nodes 2 and 3 have mostly weak connections, meaning their thick connection to node 1 stands out. So, DF keeps the blue edges and it drops the red edge. It only ever looks at one node at a time.

NC takes a different stance. It selects the red edge and drops the blue ones. Why? Because for NC what matters more is the collaboration between the two nodes. Sure, the blue connection is thicker than the red one. But node 1 always has strong connections, and its blue edges are actually particularly weak. On the other hand, node 3 usually has weak connections. Proportionally speaking, the red edge is more important for it, and so it gets saved.

To sum up, NC:

1. Refines our estimate of noise in the edge weights;
2. Sees an edge as the collaboration between two nodes rather that an event happening to one of them;
3. Uses a different model exploiting Bayes’ law to bake these aspects together.

How does that work for us in practice? Above you see some simulations made with artificial networks, of which we know the actual underlying structure, plus some random noise — edges thrown in that shouldn’t exist. The more noise we add the more difficult it is to recover the original structure. When there is little noise, DF (in blue) is better. NC (in red) starts to shine as we increase the amount of noise, because that’s the scenario we were targeting.

In the paper we also show that NC backbones have a comparable stability with DF, meaning that extracting the backbone from different time snapshots of the same phenomenon usually does not yield wildly different results. Coverage — the number of nodes that still have at least one edge in the network — is also comparable. Then we look at quality. When we want to predict some future relationship among the nodes, we expect noisy edges to introduce errors in the estimates. Since a backbone aims at throwing them away, it should increase our predictive power. The table below (click it to enlarge) shows that, in different country-country networks, the predictive quality (R²) using an NC backbone is higher than the one we get using the full noisy network. The quality of prediction can get as high as twice the baseline (the table reports the quality ratio: R² of the backbone over R² of the full network, for different methods).

The conclusion is that, when you are confident about the measurement of your network, you should probably extract its backbone using DF. However, in cases of uncertainty, NC is the better option. You can test it yourself!

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A few weeks ago I had to honor to speak at my group’s  “Global Empowerment Meeting” about my research on data science and economic development. I’m linking here the Youtube video of my talk and my transcript for those who want to follow it. The transcript is not 100% accurate given some last minute edits — and the fact that I’m a horrible presenter 🙂 — but it should be good enough. Enjoy!

We think that the big question of this decade is on data. Data is the building blocks of our modern society. We think in development we are not currently using enough of these blocks, we are not exploiting data nearly as much as we should. And we want to fix that.

Many of the fastest growing companies in the world, and definitely the ones that are shaping the progress of humanity, are data-intensive companies. Here at CID we just want to add the entire world to the party.

So how do we do it? To fix the data problem development literature has, we focus on knowing how the global knowledge building looks like. And we inspect three floors: how does knowledge flow between countries? What lessons can we learn inside these countries? What are the policy implications?

To answer these questions, we were helped by two big data players. The quantity and quality of the data they collect represent a revolution in the economic development literature. You heard them speaking at the event: they are MasterCard – through their Center for Inclusive Growth – and Telefonica.

Let’s start with MasterCard, they help us with the first question: how does knowledge flow between countries? Credit card data answer to that. Some of you might have a corporate issued credit card in your wallet right now. And you are here, offering your knowledge and assimilating the knowledge offered by the people sitting at the table with you. The movements of these cards are movements of brains, ideas and knowledge.

When you aggregate this at the global level you can draw the map of international knowledge exchange. When you have a map, you have a way to know where you are and where you want to be. The map doesn’t tell you why you are where you are. That’s why CID builds something better than a map.

We are developing a method to tell why people are traveling. And reasons are different for different countries: equity in foreign establishments like the UK, trade partnerships like Saudi Arabia, foreign greenfield investments like Taiwan.

Using this map, it is easy to envision where you want to go. You can find countries who have a profile similar to yours and copy their best practices. For Kenya, Taiwan seems to be the best choice. You can see that, if investments drive more knowledge into a country, then you should attract investments. And we have preliminary results to suggest whom to attract: the people carrying the knowledge you can use.

The Product Space helps here. If you want to attract knowledge, you want to attract the one you can more easily use. The one connected to what you already know. Nobody likes to build cathedrals in a desert. More than having a cool knowledge building, you want your knowledge to be useful. And used.

There are other things you can do with international travelers flows. Like tourism. Tourism is a great export: for many countries it is the first export. See these big portion of the exports of Zimbabwe or Spain? For them tourism would look like this.

Tourism is hard to pin down. But it is easier with our data partners. We can know when, where and which foreigners spend their money in a country. You cannot paint pictures as accurate as these without the unique dataset MasterCard has.

Let’s go to our second question: what lessons can we learn from knowledge flows inside a country? Telefonica data is helping answering this question for us. Here we focus on a test country: Colombia. We use anonymized call metadata to paint the knowledge map of Colombia, and we discover that the country has its own knowledge departments. You can see them here, where each square is a municipality, connecting to the ones it talks to. These departments correlate only so slightly with the actual political boundaries. But they matter so much more.

In fact, we asked if these boundaries could explain the growth in wages inside the country. And they seem to be able to do it, in surprisingly different ways. If you are a poor municipality in a rich state in Colombia, we see your wage growth penalized. You are on a path of divergence.

However, if you are a poor municipality and you talk to rich ones, we have evidence to show that you are on a path of convergence: you grow faster than you expect to. Our preliminary results seem to suggest that being in a rich knowledge state matters.

So, how do you use this data and knowledge? To do so you have to drill down at the city level. We look not only at communication links, but also at mobility ones. We ask if a city like Bogota is really a city, or different cities in the same metropolitan area. With the data you can draw four different “mobility districts”, with a lot of movements inside them, and not so many across them.

The mobility districts matter, because combining mobility and economic activities we can map the potential of a neighborhood, answering the question: if I live here, how productive can I be? A lot in the green areas, not so much in the red ones.

With this data you can reshape urban mobility. You know where the entrance barriers to productivity are, and you can destroy them. You remodel your city to include in its productive structure people that are currently isolated by commuting time and cost. These people have valuable skills and knowhow, but they are relegated in the informal sector.

So, MasterCard data told us how knowledge flows between countries. Telefonica data showed the lessons we can learn inside a country. We are left with the last question: what are the policy implications?

So far we have mapped the landscape of knowledge, at different levels. But to hike through it you need a lot of equipment. And governments provide part of that equipment. Some in better ways than others.

To discover the policy implications, we unleashed a data collector program on the Web. We wanted to know how the structure of the government in the US looks like. Our program returned us a picture of the hierarchical organization of government functions. We know how each state structures its own version of this hierarchy. And we know how all those connections fit together in the union, state by state. We are discovering that the way a state government is shaped seems to be the result of two main ingredients: where a state is and how its productive structure looks like.

We want to establish that the way a state expresses its government on the Web reflects the way it actually performs its functions. We seem to find a positive answer: for instance having your environmental agencies to talk with each other seems to work well to improve your environmental indicators, as recorded by the EPA. Wiring organization when we see positive feedback and rethinking them when we see a negative one is a direct consequence of this Web investigation.

I hope I was able to communicate to you the enthusiasm CID discovered in the usage of big data. Zooming out to gaze at the big picture, we start to realize how the knowledge building looks like. As the building grows, so does our understanding of the world, development and growth. And here’s the punchline of CID: the building of knowledge grows with data, but the shape it takes is up to what we make of this data. We chose to shape this building with larger doors, so that it can be used to ensure a more inclusive world.

By the way, the other presentations of my session were great, and we had a nice panel after that. You can check out the presentations in the official Center for International Development Youtube channel. I’m embedding the panel’s video below:

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The time has finally come! The NetSci conference—the place to be if you are interested in complex networks—is happening next week, from June 1st to June 5th. The venue is in Zaragoza, Spain. You can get all the information you need about the event from the official website. For the third year, I am co-organizing one of its satellite events: the Multiple Network Modeling, Analysis and Mining symposium, this year held jointly with Networks of Networks. The satellite will take place on June 2nd. As I previously said, unfortunately I am not going to be physically present in Span, and that makes me sad, because we have a phenomenal program this year.

We have four great invited speakers: Giovanni Sansavini, Rui Carvalho, Arunabha Sen and Katharina Zweig. It is a perfect mix between the infrastructure focus of the networks of networks crowd and the multidisciplinary approach of multiple networks. Sansavini works on reliability and risk engineering, while Carvalho focuses on characterizing and modeling networks in energy. Sen and Zweig provide their outstanding experience in the fields of computer networks and graph theory.

Among the contributed talks I am delighted to see that many interesting names from the network analysis crowd decided to send their work to be presented in our event. Among the highlights we have a contribution from the group of Mason Porter, who won last year’s Erdos Prize as one of the most outstanding young network scientists. I am also happy to see contributions from the group of Cellai and Gleeson, with whom I share not only an interest on multiplex networks, but also on internet memes. Contributions from groups lead by heavyweights like Schweitzer and Havlin are another sign of the attention that this event has captured.

I hope many of you will attend this seminar. You’ll be in good hands: Gregorio D’Agostino, Przemyslaw Kazienko and Antonio Scala will be much better hosts than I can ever be. I am copying here the full program of the event. Enjoy Spain!

NoN’15 Program

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Maybe it’s the new year, maybe it’s the fact that I haven’t published anything new recently, but today I wanted to take a look at my publication history. This, for a scientist, is something not unlike a time machine, bringing her back to an earlier age. What was I thinking in 2009? What sparked my interest and what were the tools further refined to get to the point where I am now? It’s usually a humbling (not to say embarrassing) operation, as past papers always look so awful – mine, at least. But small interesting bits can be found, like the one I retrieved today, about shortest paths in communication networks.

A shortest path in a network is the most efficient way to go from one node to another. You start from your origin and you choose to follow an edge to another node. Then you choose again an edge and so on until you get to your destination. When your choices are perfect and you used the minimum possible number of edges to follow, that’s a shortest path (it’s A shortest path and not THE shortest path because there might be alternative paths of the same length). Now, in creating this path, you obviously visited some nodes in between, unless your origin and destination are directly connected. Turns out that there are some nodes that are crossed by a lot of shortest paths, it’s a characteristic of real world networks. This is interesting, so scientists decided to create a measure called betweenness centrality. For each node, betweenness centrality is the share of all possible shortest paths in the network that pass through them.

Intuitively, these nodes are important. Think about a rail network, where the nodes are the train stations. High betweenness stations see a lot of trains passing through them. They are big and important to make connections run faster: if they didn’t exist every train would have to make detours and would take longer to bring you home. A good engineer would then craft rail networks in such a way to have these hubs and make her passengers happy. However, it turns out that this intuitive rule is not universally applicable. For example some communication networks aren’t willing to let this happen. Michele Berlingerio, Fosca Giannotti and I stumbled upon this interesting result while working on a paper titled Mining the Temporal Dimension of the Information Propagation.

We built two communication networks. One is corporate-based: it’s the web of emails exchanged across the Enron employee ecosystem. The email record has been publicly released for the investigation about the company’s financial meltdown. An employee is connected to all the employees she emailed. The second is more horizontal in nature, with no work hierarchies. We took users from different email newsgroups and connected them if they sent a message to the same thread. It’s the nerdy version of commenting on the same status update on Facebook. Differently from most communication network papers, we didn’t stop there. Every edge still carries some temporal information, namely the moment in which the email was sent. Above you have an extract of the network for a particular subject, where we have the email timestamp next to each edge.

Here’s where the magic happens. With some data mining wizardry, we are able to tell the characteristic reaction times of different nodes in the network. We can divide these nodes in classes: high degree nodes, nodes inside a smaller community where everybody replies to everybody else and, yes, nodes with high betweenness centrality, our train station hubs. For every measure (characteristic), nodes are divided in five classes. Let’s consider betweenness. Class 1 contains all nodes which have betweenness 0, i.e. those through which no shortest path passes. From class 2 to 5 we have nodes of increasing betweenness. So, nodes in class 3 have a medium-low betweenness centrality and nodes in class 5 are the most central nodes in the network. At this point, we can plot the average reaction times for nodes belonging to different classes in the two networks. (Click on the plots to enlarge them)

The first thing that jumps to the eye is that Enron’s communications (on the left) are much more dependent on the node’s characteristics (whether the characteristic is degree or betweenness it doesn’t seem to matter) than Newsgroup’s ones, given the higher spread. But the interesting bit, at least for me, comes when you only look at betweenness centrality – the dashed line with crosses. Nodes with low (class 2) and medium-low (class 3) betweenness centrality have low reaction times, while more central nodes have significantly higher reaction times. Note that the classes have the same number of nodes in them, so we are not looking at statistical aberrations*. This does not happen in Newsgroups, due to the different nature of the communication in there: corporate in Enron versus topic-driven in Newsgroup.

The result carries some counter intuitive implications. In a corporate communication network the shortest path is not the fastest. In other words, don’t let your train pass through the central hub for a shortcut, ’cause it’s going to stay there for a long long time. It looks like people’s brains are less elastic than our train stations. You can’t add more platforms and personnel to make more things passing through them: if your communication network has large hubs, they are going to work slower. Surprisingly, this does not hold for the degree (solid line): it doesn’t seem to matter with how many people you interact, only that you are the person through which many shortest paths pass.

I can see myself trying to bring this line of research back from the dead. This premature paper needs quite some sanity checks (understatement alert), but it can go a long way. It can be part of the manual on how to build an efficient communication culture in your organization. Don’t overload people. Don’t create over-important nodes in the network, because you can’t allow all your communications to pass through them. Do keep in mind that your team is not a clockwork, it’s a brain-work. And brains don’t work like clocks.

* That’s also the reason to ditch class 1: it contains outliers and it is not comparable in size to the other classes.

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Earlier this month I had the pleasure of being invited to hold a workshop with Isabel Meirelles on complex network visualization and analysis at the Digital Humanities 2014 Forum, held at Kansas University, Lawrence. I figured that this is a good occasion to report on my experience, since it was very interesting and, being quite different from my usual venues, it adds a bit of diversity. The official page of the event is useful to get an overall picture of what was going on. It will be helpful also for everything I will not touch upon in this post.

I think that one of the main highlights of the event was the half of our workshop curated by Isabel, with the additional keynote that she gave. Isabel is extremely skilled both in the know-how and in the know-what about information visualization: she is not only able to create wonderful visualizations, but she also has a powerful critical sense of what works and what doesn’t. I think that the best piece of supporting evidence for this statement is her latest book, which you can find here. As for my part of the workshop, it was focused on a very basic introduction to the simplest metrics of network analysis. You can take a glance at the slides here, but if you are already somewhat proficient in network terminology do not expect your world to be shattered by it.

The other two keynotes were equally fascinating. The first one was from Steven Jones. His talk gravitated around the concept of the eversion of the virtual into reality. Many works of science fiction imagined human beings ending up in some more or less well defined “virtual reality”, where everything is possible as long as you can program it. See for example Gibson’s “Neuromancer”, or “The Matrix”, just to give a popular example that most people would know. But what is happening right now, observes Jones, is exactly the opposite. We see more and more examples of virtual reality elements being introduced, mostly playfully, into reality. Think about qonqr, where team of people have to physically “fight” to keep virtual control of an actual neighborhood. A clever artistic way to depict eversion is also:

The last keynote was from Scott Weingart. Scott is a smart guy and he is particularly interested in studying the history of science. In the (too few!) interactions we had during the forum we touched upon many topics also included in his talk: ethic responsibility of usage of data about people, the influence of the perspective you use to analyze human activities and, a must exchange between a historian of science and yours truly formed as scientist in Pisa, Galileo Galilei. I feel I cannot do justice to his very eloquent and thought-provoking keynote in this narrow space. So I redirect you to its transcript, hosted on Scott’s blog. It’s a good read.

Then, the contributed talks. Among all the papers you can explore from the official forum page, I’d like to focus particularly on two. The first is the Salons project, presented by Melanie Conroy. The idea is to map the cultural exchange happening in Europe during the Enlightenment years. A great role for this exchange was played by Salons, where wealthy people were happy to give intellectuals a place for gathering and discussing. You can find more information on the Salons project page. I liked it because it fits with the idea of knowledge creation and human advancement as a collective process, where an equal contribution is given by both intellect and communication. By basing itself on richly annotated data, projects like these can help us understanding where breakthroughs come from, or to understand that there is no such a thing as a breakthrough, only a progressive interconnection of ideas. Usually, we realize it only after the fact, and that’s why we think it happened all of a sudden.

Another talk I really enjoyed was from Hannah Jacobs. Her talk described a visualization tool to explore the evolution of the concept of “New Woman”, one of the first examples of feminism. I am currently unable to find an online link to the tool. What I liked about it was the seamless way in which different visualizations are used to tell the various points of view on the story. The whole point of information visualization is that when there is too much data to show at the same time, one has to select what to highlight and what to discard. But in this framework, with a wise choice of techniques, one can jump into different magnifying glasses and understand one part of the story of the term “New Woman” at a time.

Many other things were cool, from the usage of the Unity 3D engine to recreate historic view, to the charming visualizations of “Enchanters of Men”. But my time here is up, and I’m left with the hope of being invited also to the 2015 edition of the forum.

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Complex networks can come in different flavors. As you know if you follow this blog, my signature dish is multilayer/multidimensional networks: networks with multiple edge types. One of the most popular types is bipartite networks. In bipartite networks, you have two types of nodes. For example, you can connect users of Netflix to the movies they like. As you can see from this example, in bipartite networks we allow only edges going from one type of nodes to the other. Users connect to movies, but not to other users, and movies can’t like other movies (movies are notoriously mean to each other).

Many things (arguably almost everything) can be represented as a bipartite network. An occupation can be connected to the skills and/or tasks it requires, an aid organization can be connected to the countries and/or the topics into which it is interested, a politician is connected to the bills she sponsored. Any object has attributes. And so it can be represented as an object-attribute bipartite network. However, most of the times you just want to know how similar two nodes of the same type are. For example, given a movie you like, you want to know a similar movie you might like too. This is called link prediction and there are two ways to do this. You could focus on predicting a new user-movie connection, or focus instead on projecting the bipartite network to discover the previously unknown movie-movie connections. The latter is the path I chose, and the result is called “Network Map”.

It is clearly the wrong choice, as the real money lies in tackling the former challenge. But if I wanted to get rich I wouldn’t have chosen a life in academia anyway. The network map, in fact, has several advantages over just predicting the bipartite connections. By creating a network map you can have a systemic view of the similarities between entities. The Product Space, the Diseasome, my work on international aid. These are all examples of network maps, where we go from a bipartite network to a unipartite network that is much easier to understand for humans and to analyze for computers.

Creating a network map, meaning going from a user-movie bipartite network to a movie-movie unipartite network, is conceptually easy. After all, we are basically dealing with objects with attributes. You just calculate a similarity between these attributes and you are done. There are many similarities you can use: Jaccard, Pearson, Cosine, Euclidean distances… the possibilities are endless. So, are we good? Not quite. In a paper that was recently accepted in PLoS One, Muhammed Yildirim and I showed that real world networks have properties that make the general application of any of these measures quite troublesome.

For example, bipartite networks have power-law degree distributions. That means that a handful of attributes are very popular. It also means that most objects have very few attributes. You put the two together and, with almost 100% probability, the many objects with few attributes will have the most popular attributes. This causes a great deal of problems. Most statistical techniques aren’t ready for this scenario. Thus they tend to clutter the network map, because they think that everything is similar to everything else. The resulting network maps are quite useless, made of poorly connected dense areas and lacking properties of real world networks, such as power-law degree distributions and short average path length, as shown in these plots:

Of course sometimes some measure gets it right. But if you look closely at the pictures above, the only method that consistently give the shortest paths (above, when the peak is on the left we are good) and the broadest degree distributions (below, the rightmost line at the end in the lower-right part of the plot is the best one) is the red line of “BPR”. BPR stands for “Bipartite Projection via Random-walks” and it happens to be the methodology that Muhammed and I invented. BPR is cool not only because its network maps are pretty. It is also achieving higher scores when using the network maps to predict the similarity between objects using ground truth, meaning that it gives the results we expect when we actually already know the answers, that are made artificially invisible to test the methodology. Here we have the ROC plots, where the highest line is the winner:

So what makes BPR so special? It all comes down to the way you discount the popular attributes. BPR does it in a “network intelligent” way. We unleash countless random walkers on the bipartite network. A random walker is just a process that starts from a random object of the network and then it jumps from it to one of its attributes. The target attribute is chosen at random. And then the walker jumps back to an object possessing that attribute, again choosing it at random. And then we go on. At some point, we start from scratch with a new random walk. We note down how many times two objects end up in the same random walk and that’s our similarity measure. Why does it work? Because when the walker jumps back from a very popular attribute, it could essentially go to any object of the network. This simple fact makes the contribution of the very popular attributes quite low.

BPR is just the latest proof that random walks are one of the most powerful tools in network analysis. They solve node ranking, community discovery, link prediction and now also network mapping. Sometimes I think that all of network science is founded on just one algorithm, and that’s random walks. As a final note, I point out that you can create your own network maps using BPR. I put the code online (the page still bears the old algorithm’s name, YCN). That’s because I am a generous coder.

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New year, old topic. I could make a lot of resolutions for this new year, but for sure to stop talking about community discovery is not among them. At least this time I tried to turn it up a notch in the epicness of the title. My aim is to give some substance to one of the many typical filler phrases in science writing. The culprit sentence in this case is “different application scenarios demand different approaches”. Bear with me for a metaphoric example.

When presenting a new toaster, it is difficult to prove that it toasts everything better under any point of view, under any circumstances. It usually does most toasts okay, and for one kind of toasts it really shines. Or its toasts really suck, but it can toast underwater. That’s fine. We are all grown up here, we don’t believe in the fairy tales of the silver bullets any more. At this point, our toaster salesman is forced to say it. “Different application scenarios demand different approaches”. In some cases this is a shameful fig leaf, but in many others it is simply true. Problem is: nobody really checks.

I decided to check. At least one of them. Teaming up with Diego Pennacchioli and Dino Pedreschi, I put the spotlight on one of the strongest dichotomies in community discovery.  As you may remember, community discovery algorithms can force every node to belong to just one community, or allow them to be in many of them. The former approach is called “graph partitioning”, whilst the latter aims to find an “overlapping coverage”. Are these two strategies yielding interesting, yet completely different, results? This question has been dissected in the paper: “Overlap Versus Partition: Marketing Classification and Customer Profiling in Complex Networks of Products“, that will be presented in one workshop of the 2014 edition of the International Conference of Data Engineering.  Let me refresh your mind about overlaps and partitions.

Above you have the nec plus ultra scenario for a partitioning algorithm. If a partitioning algorithm sees the graph on the left, it would just die of happiness. In the graph, in fact, it appears very clearly that each node belongs to a very specific community. And it can’t belong to any other. If we assume that our algorithm works on edge strength (e.g. the inverse of the edge betweenness), then what the algorithm really sees is the graph on the right. It then proceeds to group together the nodes for which the edge strength is maximal, et voilà.

Here we have an example that’s a bit more complex. The picture has too many overlapping parts, so let me describe the connection pattern. In the graph on the left there are several groups of 6 nodes, each node connected to all other members of the group. In practice, each diagonal is completely connected to the two neighbouring diagonals. Clearly, here there is no way we can put each node in a disjoint group. Why put together nodes 0,1,2 with 3,4,5 and not with 9,10,11? But at that point, why 9,10,11 should be in a community with them and not with 6,7,8? The correct approach is just to allow every completely connected group to be a community, thus letting nodes to be part of more than a community. Some overlapping algorithms see the graph as it has been depicted on the right, with an edge colour per densely connected group.

Time to test which one of these approaches is The Right One! For our data quest we focused on supermarket transactions. We created a network of products that you can buy in supermarkets. To be connected, two products have to be bought together by the same customers in a significant number of times. What does that mean? By pure intuition, bread and water aren’t going to be connected: both of them are bought very frequently, but they have little to do with each other, thus they are expected to be in the same shopping cart by chance. Eggs and flour are too very popular, but probably more than chance, since there are a lot of things you can do with them together. Therefore they are connected. Other specific pairs of products, say bacon flavoured lipstick and liquorice shoelaces, may ended up in the same, quite weird, shopping cart. But we don’t connect them, as their volume of sales is too low (or at least I hope so).

Here are some of the facts we found. First. The overlapping approach* tends to return relatively more communities with a larger amount of nodes than the partition approach**. In absolute terms that’s obvious, since the same node is counted more than once, but here the key term is “relatively”. See the plot above on the right, where we graph the probability (y axis) of finding a community with a given number of nodes (x axis). Second. The overlapping approach returns more “messy” communities. Our messiness measure checks how many different product categories are grouped together on average in the same community. Again, larger communities are expected to be messier, but the messiness measure that we used controls for community size. See the plot on the right, again the probability (y axis) of finding a community with a given entropy (x axis, “entropy” is the fancy scientific term for “messiness”). Third. The partition approach returned denser communities, whose link strength (the number of people buying the products together) is higher.

What is the meaning of all this? In our opinion, the two algorithms are aiming to do something completely different. The partition approach is aiming to create a new marketing classification. It more or less coincides with the established one (thus lower messiness), most customers buy those products together (high link strength) and there are very few giant categories (most communities are small). The overlapping approach, instead, wants to do customer profiling. A customer rarely buys all products of a marketing category (thus increasing its messiness), it has specific needs (that not many people have, thus lowering edge weight) and she usually needs a bunch of stuff (thus larger communities, on average).

Who’s right? That’s the catch: both. The fact that two results are incompatible, in this case, does not mean that one is right and one is wrong. They are just different applications. Which was exactly what I wanted to prove, in this narrow and very specific, probably unsurprising, scenario. Now you should feel better: I gave you a small proof that the hours you spend to choose the perfect toaster for you are really worth your time!

* As overlapping approach, we used the Hierarchical Link Clustering.

** As partitioning approach, we used Infomap.

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The four of you who follow this blog regularly will know that I have a thing for something called “community discovery“. That’s because no matter how you call it, it always sounds damn cool. “Discovering Communities” or “Detecting the functional modules” or “Uncovering node clusters”. These are all names given to the task of finding groups of nodes in a network that are very similar to each other. And they make you feel like some kind of wizard. Adding to that, there are countless applications in epidemiology, sociology, immunology, marketing.

Far from being original, I share this passion with at least a thousand researchers. Being as smart as they are, they quickly realized that there are many ways in which you can group nodes based on their similarity. On the one hand, this is good news, as we basically have an algorithm for any possible community you want to find in your network. On the other hand, this made a lot of people freak out, as too many algorithms and too different solutions are usually a big red flag in computer science. A flag that says: “You have no idea what you are doing!” (although a computer scientist would put it in the cold and rational “Your problem is not formally defined”: it means the same).

Yes, my signature “Community Discovery Picture” strikes again!

I personally think that the plus side is more predominant than the minus side, and you can get rid of the latter with a bit of work. Work that I have done with Dino Pedreschi and Fosca Giannotti in our paper “A classification for Community Discovery Methods in Complex Networks“. The trick is very simple. It just consists in noticing what’s wrong with the starting point. “Finding groups of nodes in a network that are very similar to each other”. Exactly what is “similar“? It is an umbrella term that can be interpreted in many different ways. After all, we already do this outside of network science. People can be very similar because they look alike. Or because they like the same things. So why can’t we just have different definitions of communities, based on how we intend similarity?

Well, because at the beginning of community discovery we thought that the problem was well defined. The first definition of community was something like: “A community is a group of nodes that are densely connected, and they have few edges connecting them to nodes outside the community”. Which is fine. In some cases. In others, we discovered that it doesn’t really make sense. For example, we discovered that many social networks have a pervasive overlap. It means that nodes are densely connected with many different groups, disproving the definition: now, the area outside the community could be just as dense as the community itself! And this is just one example: you take a hundred community discovery algorithms in literature and you’ll get a hundred different community results on the same network.

Overlap in the infamous Zachary Karate Club network, you can even win a prize if you mention it!

So now researchers in the community discovery… well… community were divided in three factions. We had those who thought that the problem was ill defined, thus everything done so far was just a royal mess. Then there were those who still thought that the problem was well defined, because their definition of community was the only one standing on solid ground and everybody else was just running around like a headless chicken. And then there were people like me and Sune Lehmann (whom I thank for the useful discussions). Our point was that there were many different definitions of communities, and the incompatible results are just the output of incompatible definitions of community.

This is the main take-away message of the paper. We then moved on and tried to actually spot and categorize all different community definitions (for 90s kids: think of a Pokédex for algorithms). Some choices were easy, some others weren’t. I personally think that more than an established classification, this is just a conversation starter. Also because the boundaries between community definitions are at least as fuzzy as the boundaries between the communities themselves. Algorithms in one category may also satisfy conditions imposed by another category. And to me that’s fine: I don’t really like to put things in separate boxes, I just want to have an insight about them.

I put tags, not classes.

So here you go, the classification we made includes the following “community types” (names are slightly changed from the paper, but it should be obvious which is which):

• Common Features: in this definition, each node has a number of attributes. If we are in a social network and the nodes are people, these attributes may well be the social connections, the movies you like, the songs you listen to. Communities are groups of nodes with similar attributes.
• Internal Density: the classical starting point of community discovery. Here we are interested in just maximizing the number of edges inside the communities.
• External Sparsity: a subtle variant of the Internal Density class. The focus of this definition is on considering communities as islands of nodes, not necessarily densely connected.
• Action Communities: this is a very dynamic definition of communities. Nodes are not just static entities, but they perform actions. Again, in a social network you not only like a particular artist: you listen to her songs. If your listening happens with the same, or similar, dynamics of other people, then you might as well form a community with them.
• Proximal Nodes: here we want the edges inside the communities to make it easy for a node to be connected to all other nodes in the community. Or: to get to any other node in the community I have to follow just a few edges.
• Fixed Structure: this is a very demanding community definition. It says that the algorithm knows what a community looks like and it just has to find that structure in the network.
• Link Communities: one of my favorites, because it revolutionizes the idea of community. Here we think that we need to group the edges, not the nodes. In a social network, we know different people for different reasons: family, work, free time, … The reason why you know somebody is the community. And you belong to many of them: to all the communities your edges belong to.
• Others: in any decent classification there must be a miscellaneous category! Some algorithms do not really follow a particular definition, whether because they just add features to other community discovery algorithms or because they let the user define their communities and then try to find them.

And now just a shortlist of readily available community discovery algorithms you can find on the Web:

• DEMON (by yours truly)
• HLC
• Infomap
• Label Propagation, Walktrap and Modularity Maximization (in the igraph library, either source code in C or for R or Python)
• BigClam (in the SNAP network library)
• k-Clique (in the Networkx Python package)
• CONGA/Peacock
• Edge Clustering
• PMM
• Fast Unfolding Modularity
• Conclude

That’s it! I hope I created a couple of new community discovery aficionados!

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“It’s a bit sad that some among the most brilliant minds of our generation are working tirelessly on strategies to increase clicks on online ads” popped up on my Facebook stream some days ago (I don’t remember who wrote it, so you are welcome to contact me to restore credit where credit is due 🙂 ). But the point still remains. I actually don’t find it that bad. Yes, it’s bad, but it could be worse. It reminds me of other “wrong” reasons to do incredible improvements in science and stuff. For example, war is responsible for many technology advancements. Even if the aim of online marketing is just to increase revenues, what it actually requires is to understand human psychology, behavior and social interactions. In practice, that’s philosophy of the human mind at its best: how does the brain work? How does a collection of brains work? What drives our behavior and needs?

When you put together many minds in the real world, you have to deal with complex networks. We are not connected with one another at random, and the eyes of our friends are the channel through which we observe the world. This fact is studied in complex network analysis, in the sub-branch of cascade behaviors. Cascade behaviors happen when a person in a social network decides to modify her behavior according to the behavior of the people she is connected to. As a consequence, there are some people in the network who are in a very particular position: given the people they know and their prominence among them, they can modify their behavior and they will modify their friends’ behavior and so on an so forth, changing forever how every node in the network behaves. And that’s the cascade. If you find a way to identify these prominent actors in the network, you can control the behavior of the entire system. Now you can see why there is a mountain of work about it. In the computer science approach, we have threshold models simulating the cascade for many starting nodes and thus identify the practical leaders (for example Jon Kleinberg’s work); in physics we have models, aiming at understanding the degree of controllability of complex systems (I’ll go with Laszlo Barabasi in this).

Visualization of network cascade, from my good friend Mauro Martino. The red dots at the bottom are the “drivers”, who influence the collection of green nodes they are attached to.

Genuinely curious about the topic, I started my own track of research on it. One thing that Diego Pennacchioli, Giulio Rossetti, Luca Pappalardo, Dino Pedreschi, Fosca Giannotti and me found curious is that everybody working on social prominence was looking at it from a monodimensional perspective. That means: the only thing they are interested in is how to maximize the number of nodes influenced by the leaders. The bigger this number, the better. All fun and games, but why? I can think about several scenarios where the final total number is not the most important thing. For example:

• What if I want people to buy a product? The total number of people knowing about the product is nice, but I want them to be strongly committed, strongly enough to buy it.
• What if I am actually looking to reach a particular person? Then I care how deeply my message can go through the network.
• What if I just care about my friends? Then screw their friends (and everybody else), as long as I can influence a wide range of my direct connections!

To calculate our measure we need to infer the diffusion trees. So from the left, where the number on each arrow gives you the action time of the node at the base of the arrow, we go to the right by selecting the lowest possible combination of arrows.

Strength, depth and width of social prominence. That’s why our paper is called “The Three Dimensions of Social Prominence” (check it out). Strength is how committed the people you influenced are to keep doing what you influenced them to do. Depth is how many degrees of separation (or, how far) the cascade of influence that you triggered can go. Width is simply the ratio of your friends that you are able to influence. By analyzing how much a user in Last.fm (a social website based on music) is able to influence her friends in listening to new artists, we found a collection of very interesting facts.

For example, it is well known that in social networks there are some nodes that are structurally very important. They are the central users, the ones that keep the network connected. Intuitively, they are the only way, or the easiest way, through which a signal (in our case social influence) can go from one part of the network to the other. Guess what: they can’t do it. We found a significant anti-correlation between centrality and width and depth. That is bad news, because those nodes are the ones in the only position with a theoretical ability of controlling the network and a practical inability in doing so. I like to call it “The Paradox of Social Controllability” (hence, the post title).

The anti-correlation between depth and strength.

Another piece of food for thought is the trade off between strength and depth. While width is unrelated to both, we found that if you want to go deeply into the network, then you can’t expect that the people you touch will be extremely committed to your message.

The third big thing is the distribution of connections per leader. We found that the leaders showing highest values of strength, depth and width were those who used Last.fm with average frequency. The highly connected and very active users (hubs, in network lingo) scored poorly, as we saw. So did the occasional users, the ones with just two or three connections (that is the majority of the system). The people who have control over the network are the mildly engaged. They are you, in practice: chances are that you are not a record label, nor a music fanatic, but just a person with her tastes and preferences. So you have control. Problem is, the control is scattered equally on the vast set of people like you.

To conclude, we saw what wonderful things network cascades are: they could empower us to do a lot of good. We also saw how there are theoretical results about the possibility of identifying people who can trigger them. But my unfortunate conclusion is about the paradox between theory and practice. Those who theoretically should, apparently can’t.

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