ARS 101: Centralidad en una estructura de red

Introducción a la centralidad en una estructura de red

ARS 101: Modularidad

Modularidad en redes

La modularidad es una medida de la estructura de las redes o grafos. Fue diseñado para medir la fuerza de la división de una red en módulos (también llamados grupos, agrupamientos o comunidades). Las redes con alta modularidad tienen conexiones sólidas entre los nodos dentro de los módulos, pero escasas conexiones entre nodos en diferentes módulos. La modularidad se utiliza a menudo en los métodos de optimización para la detección de la estructura comunitaria de las redes. Sin embargo, se ha demostrado que la modularidad sufre un límite de resolución y, por lo tanto, no es capaz de detectar pequeñas comunidades. Las redes biológicas, incluidos los cerebros animales, presentan un alto grado de modularidad.

Motivación

Muchos problemas científicamente importantes se pueden representar y ser empíricamente estudiados a través de redes. Por ejemplo, los patrones biológicos y sociales, la World Wide Web, las redes metabólicas, redes tróficas, redes neuronales y redes patológicos son algunos ejemplos de los problemas del mundo real que pueden ser representados matemáticamente y topológicamente estudiaron para revelar algunas características estructurales inesperadas. [1] La mayoría de estas redes poseen una cierta estructura de comunidad que tiene considerable importancia en la construcción de un entendimiento respecto a la dinámica de la red. Por ejemplo, una comunidad social estrechamente relacionado implicará una mayor velocidad de transmisión de información o rumor entre ellos de una comunidad mal conectados. Por lo tanto, si una red está representado por un número de nodos individuales conectadas por enlaces que significan un cierto grado de interacción entre los nodos, las comunidades se definen como grupos de nodos densamente interconectados que son sólo escasamente conectados con el resto de la red. Por lo tanto, puede ser imprescindible identificar comunidades en las redes ya que las comunidades pueden tener propiedades muy diferentes, tales como grado, coeficiente de agrupamiento, intermediación, centralidad nodal [2], etc, respecto a la media de la red. La modularidad es una de tales medidas, que, cuando se maximiza, sino que conduce a la aparición de comunidades en una red dada.

Definición

La modularidad es la fracción de los enlaces que caen dentro de grupos dados menos el valor esperado que dicha fracción hubiese recibido si los enlaces se hubiesen distribuido al azar. El valor de la modularidad se encuentra en el intervalo [-.5,1). Es positivo si el número de enlaces dentro de los grupos supera el número esperado sobre la base de pura casualidad. Para una determinada división de vértices de la red en algunos módulos, la modularidad refleja la concentración de los nodos dentro de los módulos en comparación con distribución al azar de los enlaces entre todos los nodos, independientemente de los módulos.
Existen diferentes métodos para el cálculo de la modularidad. [1] En la versión más común del concepto, la aleatorización de los enlaces se realiza con el fin de preservar el grado de cada vértice. Veamos un grafo con  nodos y  enlaces (enlaces) de tal manera que el grafo se puede dividir en 2 comunidades usando una variable de membrecía   Si un nodo   pertenece a la comunidad 1,   o si  pertenece a la comunidad 2,   Sea que la matriz de adyacencia de la red estará representado por A, donde  significa que no hay ninguna ventaja (sin interacción) entre los nodos i y j y  significa que hay un enlace entre los dos. También por simplicidad consideramos una red no dirigida. Así   (Es importante tener en cuenta que pueden existir múltiples enlaces entre los nodos 2 y aquí evaluamos el caso más simple).
La modularidad Q se define entonces como la fracción de enlaces que caen dentro del grupo 1 o 2, menos el número esperado de enlaces dentro del grupo 1 y 2 para un grafo aleatorio con el mismo grado de distribución como el nodo de red determinado.
El número esperado de enlaces se calcula utilizando el concepto de modelos de configuración. [3] El modelo de configuración es una realización aleatoria de una red en particular. Dada una red con n nodos, donde cada nodo i tiene un grado nodal ${\displaystyle k_{v}}$, el modelo de configuración corta cada enlace en 2 mitades, y luego cada media punta, llamados a trozo, es reconectado al azar con cualquier otro ramal en la red, incluso permitiendo auto bucles. Por lo tanto, a pesar de que la distribución de grado de nodo del grafo permanece intacta, los resultados del modelo de configuración en una red completamente al azar.

Número esperado de enlaces entre nodos

Si el número total de ramales es ln


Ahora, si se selecciona al azar dos nodos i y j con grados nodo ki y kj respectivamente, y recablear los ramales a estos 2 nodos y, a continuación,

Número esperado de enlaces [enlaces completos entre i y j] = (enlaces completos entre i y j) / (número total de posibles reconexiones) (2)

Número total de reconexiones posibles = número de ramales que quedan después de la elección de un determinado ramal = ln- 1 = ln para n grande

Por lo tanto, se esperaba [Número de enlaces completos entre i y j] = (ki * kJ) / ln = (ki kJ) / 2m.
Por lo tanto, el número real de enlaces entre i y j menos el número esperado de enlaces entre ellos es Aij - (ki kj) / 2 m. Por lo tanto el uso de [1]



Es importante tener en cuenta que (3) es válido para la partición en sólo 2 comunidades. La partición jerárquica (es decir, partición en 2 comunidades, a continuación, las 2 sub- comunidades con particiones más en 2 comunidades más pequeñas sub sólo para maximizar Q) es un posible enfoque para identificar múltiples comunidades en una red. Además, (3) puede ser generalizada para la partición de una red en comunidades c. [4]


eii donde es la fracción de los enlaces con ambas vértices finales de la misma comunidad i:


y ai es la fracción de enlaces con al menos un extremo de vértice en la comunidad i :

Ejemplo de detección múltiple comunidad

Consideramos una red no dirigida con 10 nodos y 12 enlaces y la siguiente matriz de adyacencia.

Fig. 1. Red de muestra correspondiente a la matriz de adyacencia con 10 nodos, 12 enlaces.

Fig. 2. Particiones de red que maximizan Q. máxima Q = 0.4895

Nodo ID	1	2	3	4	5	6	7	8	9	10
1	0	1	1	0	0	0	0	0	0	1
2	1	0	1	0	0	0	0	0	0	0
3	1	1	0	0	0	0	0	0	0	0
4	0	0	0	0	1	1	0	0	0	1
5	0	0	0	1	0	1	0	0	0	0
6	0	0	0	1	1	0	0	0	0	0
7	0	0	0	0	0	0	0	1	1	1
8	0	0	0	0	0	0	1	0	1	0
9	0	0	0	0	0	0	1	1	0	0
10	1	0	0	1	0	0	1	0	0	0

Las comunidades en la gráfica están representados por los grupos de nodos de color rojo, verde y azul en la figura 1. Las particiones de la comunidad óptimos se representan en la figura 2.

Formulación matricial

Una formulación alternativa de la modularidad, útil en particular en algoritmos de optimización espectrales, es como sigue. [1] Definir Sir sea 1 si vértice i pertenece al grupo de r y cero en caso contrario. entonces


y por lo tanto


donde S es teniendo la matriz (no cuadrados) elementos Sir y B es la llamada matriz de modularidad, que tiene elementos



Todas las filas y columnas de la matriz de modularidad suma a cero, lo que significa que la modularidad de una red indivisa también es siempre cero.
Para redes divididas en sólo dos comunidades, se puede definir como alternativa si = ± 1 para indicar que la comunidad a la que pertenece el nodo i, que a su vez conduce a

,
donde s es el vector columna con elementos de SI. [1]
Esta función tiene la misma forma que el hamiltoniano de un vidrio de espín Ising, una conexión que ha sido explotada para crear algoritmos computacionales simples, por ejemplo mediante el recocido simulado, para maximizar la modularidad. La forma general de la modularidad para un número arbitrario de las comunidades es equivalente a un vidrio de espín Potts y algoritmos similares puede ser desarrollado para este caso también. [5]

Límite de resolución

La modularidad compara el número de enlaces dentro de un grupo con el número esperado de enlaces que uno encontraría en el grupo si la red fuera una red aleatoria con el mismo número de nodos y donde cada nodo mantiene su grado, pero los enlaces son lo contrario adjunta al azar. Este modelo nulo al azar asume implícitamente que cada nodo puede quedar unido a cualquier otro nodo de la red. Tal suposición es razonable sin embargo si la red es muy grande, como el horizonte de un nodo incluye una pequeña parte de la red, haciendo caso omiso de la mayor parte de ella. Por otra parte, esto implica que el número esperado de enlaces entre dos grupos de nodos disminuye si el tamaño de la red aumenta. Así, si una red es lo suficientemente grande, el número esperado de enlaces entre dos grupos de nodos en modelo nulo de modularidad puede ser menor que uno. Si esto sucede, un solo borde entre los dos grupos sería interpretado por la modularidad como un signo de una fuerte correlación entre los dos grupos, y la optimización de la modularidad conduciría a la fusión de los dos grupos, independientemente de las características de la agrupación. Por lo tanto, incluso débilmente interconectado gráficos completos, que tienen la densidad más alta posible de los enlaces internos, y representan las mejores comunidades identificables, se fusionara por optimización modularidad si la red es suficientemente grande. [6] Por este motivo, la optimización de la modularidad en redes grandes sería un fracaso para resolver las pequeñas comunidades, incluso cuando están bien definidos. Este sesgo es inevitable para los métodos como la optimización de la modularidad, que se basan en un modelo nulo global. [7]

Métodos multirresolución

Hay dos enfoques principales que tratan de resolver el límite de resolución en el contexto modularidad : la adición de un r resistencia a todos los nodos, en la forma de un auto -loop, lo que aumenta (r> 0) o disminuye (r < 0) la aversión de los nodos a las comunidades de formulario ; [8] o la adición de un parámetro γ > 0 en frente del término nulo caso en la definición de la modularidad, que controla la importancia relativa entre los enlaces internos de las comunidades y el modelo nulo. [5] modularidad Optimización para los valores de estos parámetros en sus respectivos rangos adecuados, es posible recuperar toda la mesoescala de la red, desde la macroescala en la que todos los nodos pertenecen a la misma comunidad, a la microescala en el que cada nodo forma su propia comunidad, por lo tanto, los métodos de varias resoluciones de nombres. Sin embargo, se ha demostrado recientemente que estos métodos son intrínsecamente deficiente y su uso no producirán soluciones fiables. [9]

Referencias

1.  Jump up to:Newman, M. E. J. (2006). "Modularity and community structure in networks". Proceedings of the National Academy of Sciences of the United States of America 103 (23): 8577–8696. arXiv:physics/0602124. Bibcode:2006PNAS..103.8577N.doi:10.1073/pnas.0601602103. PMC 1482622. PMID 16723398.
2. Jump up Newman, M. E. J. (2007). "Mathematics of networks". In Palgrave Macmillan, Basingstoke. The New Palgrave Encyclopedia of Economics (2 ed.).
3. Jump up Remco van der Hofstad (2009). "7". Random Graphs and Complex Networks. 
4. Jump upClauset, Aaron and Newman, M. E. J. and Moore, Cristopher (2004). "Finding community structure in very large networks". Phys. Rev. E 70 (6): 066111. arXiv:cond-mat/0408187. Bibcode:2004PhRvE..70f6111C. doi:10.1103/PhysRevE.70.066111.
5. Joerg Reichardt and Stefan Bornholdt (2006). "Statistical mechanics of community detection". Physical Review E 74 (1): 016110. arXiv:cond-mat/0603718. Bibcode:2006PhRvE..74a6110R. doi:10.1103/PhysRevE.74.016110.
6. Jump up Santo Fortunato and Marc Barthelemy (2007). "Resolution limit in community detection". Proceedings of the National Academy of Sciences of the United States of America 104 (1): 36–41. arXiv:physics/0607100. Bibcode:2007PNAS..104...36F.doi:10.1073/pnas.0605965104. PMC 1765466. PMID 17190818.
7. Jump upJ.M. Kumpula, J. Saramäki, K. Kaski , and J. Kertész (2007). "Limited resolution in complex network community detection with Potts model approach". European Physical Journal B 56 (1): 41–45. arXiv:cond-mat/0610370. Bibcode:2007EPJB...56...41K.doi:10.1140/epjb/e2007-00088-4.
8. Jump up Alex Arenas, Alberto Fernández and Sergio Gómez (2008). "Analysis of the structure of complex networks at different resolution levels". New Journal of Physics 10 (5): 053039. arXiv:physics/0703218. Bibcode:2008NJPh...10e3039A. doi:10.1088/1367-2630/10/5/053039.
9. Jump up Andrea Lancichinetti and Santo Fortunato (2011). "Limits of modularity maximization in community detection". Physical Review E 84: 066122. arXiv:1107.1155. Bibcode:2011PhRvE..84f6122L. doi:10.1103/PhysRevE.84.066122.

Wikipedia

Minando Twitter con R para construir datos para análisis de sentimiento

Mining Twitter with R - Tutorial 1: Building a corpus from Twitter data

This video shows you how to access Twitter API using R Studio and how to create 3 different data sets for analysis. Data sets include tweets with #Rangers, #Athletics (an official twitter for OaklandA's), and #MLB (Major League Baseball).

Tutorial 1 - Building a corpus from Twitter data:http://youtu.be/mJVcANlkxU8
Tutorial 2 - Scoring tweets: http://youtu.be/S1y3PxULNaQ
Tutorial 3 - Histogram (): http://youtu.be/KDYZTFLBrZk 

References
Breen, J. O. (2012). Mining twitter for airline consumer sentiment. Practical Text Mining and Statistical Analysis for Non-structured Text Data Applications, 133.
Jeffrey Breen's Blog: http://jeffreybreen.wordpress.com/201...

La viralización de textos en los 1800s

Here’s How Memes Went Viral — In the 1800s

• BY GREG MILLER - WIRED.COM

Thicker lines indicate more content-sharing between 19th century newspapers. Image: Ryan Cordell / Infectious texts project

The story had everything — exotic locale, breathtaking engineering, Napoleon Bonaparte. No wonder the account of a lamplit flat-bottom boat journey through the Paris sewer went viral after it was published — on May 23, 1860.

At least 15 American newspapers reprinted it, exposing tens of thousands of readers to the dank wonders of the French city’s “splendid system of sewerage.”

Twitter is faster and HuffPo more sophisticated, but the parasitic dynamics of networked media were fully functional in the 19th century. For proof, look no further than the Infectious Texts project, a collaboration of humanities scholars and computer scientists.

The project expects to launch by the end of the month. When it does, researchers and the public will be able to comb through widely reprinted texts identified by mining 41,829 issues of 132 newspapers from the Library of Congress. While this first stage focuses on texts from before the Civil War, the project eventually will include the later 19th century and expand to include magazines and other publications, says Ryan Cordell, an assistant professor of English at Northeastern University and a leader of the project.

Some of the stories were printed in 50 or more newspapers, each with thousands to tens of thousands of subscribers. The most popular of them most likely were read by hundreds of thousands of people, Cordell says. Most have been completely forgotten. “Almost none of those are texts that scholars have studied, or even knew existed,” he said.

Yellow dots represent hotspots of recycled content. The shading indicates the number of publications in a region recorded by the 1840 census. Image: Ryan Cordell / Infectious texts project

The tech may have been less sophisticated, but some barriers to virality were low in the 1800s. Before modern copyright laws there were no legal or even cultural barriers to borrowing content, Cordell says. Newspapers borrowed freely. Large papers often had an “exchange editor” whose job it was to read through other papers and clip out interesting pieces. “They were sort of like BuzzFeed employees,” Cordell said.

Clips got sorted into drawers according to length; when the paper needed, say, a 3-inch piece to fill a gap, they’d pluck out a story of the appropriate length and publish it, often verbatim.

Fast forward a century and a half and many of these newspapers have been scanned and digitized. Northeastern computer scientist David Smith developed an algorithm that mines this vast trove of text for reprinted items by hunting for clusters of five words that appear in the same sequence in multiple publications (Google uses a similar concept for its Ngram viewer).

The project is sponsored by the NULab for Texts, Maps, and Networks at Northeastern and the Office of Digital Humanities at the National Endowment for the Humanities. Cordell says the main goal is to build a resource for other scholars, but he’s already capitalizing on it for his own research, using modern mapping and network analysis tools to explore how things went viral back then.

Counting page views from two centuries ago is anything but an exact science, but Cordell has used Census records to estimate how many people were living within a certain distance of where a particular piece was published and combined that with newspaper circulation data to estimate what fraction of the population would have seen it (a quarter to a third, for the most infectious texts, he says).

He’s also interested in mapping how the growth of the transcontinental railroad — and later the telegraph and wire services — changed the way information moved across the country. The animation below shows the spread of a single viral text, a poem by the Scottish poet Charles MacKay, overlaid on the developing railroad system. The one at the very bottom depicts how newspapers grew with the country from the colonial era to modern times, often expanding into a territory before the political boundaries had been drawn.

Another approach takes advantage of the same type of network analysis tools social scientists use to map the flow of information in Twitter and other social media. Cordell has found that certain cities that don’t get much attention from scholars were actually important hubs in the 19th century information economy. Nashville is one of them. “It makes sense if you think about it,” Cordell said. “It was the center of the country at the time.”

Some of the texts that went viral in the 1800s aren’t all that different from the things people post on Facebook today, Cordell says. Political rants were popular, for example, as were recipes and travel stories.

Poems also turn up frequently, as well as another type of writing Cordell calls vignettes. These are sentimental stories that are presented as if they’re real, but aren’t attributable to an author and lack details that would make it possible to verify them. One example is a letter, supposedly tucked into a book by a dying woman and found by her husband after her death. She urges him to remember her fondly and live a good life after she’s gone. “These are fascinating to me because they blur the line between fact and fiction, which sort of exemplifies the 19th century newspaper,” Cordell said.

The vignettes often had a moral to them. On popular variety, temperance stories, were aimed at getting drunks to sober up. Cordell likens these cautionary tales to the email you’ve probably gotten from a concerned aunt or uncle that turns out to be based on a bogus urban legend when you look it up on Snopes.

The media may have changed, but a century and a half later many of the messages are surprisingly familiar.

ARS ayuda al marketing digital

How network analysis can make us better digital marketers
by Andrew Lamb

Social ‘networks’, computer ‘networks’, ‘networking’ events, agency ‘networks’ – even the ‘Internet’. ‘Networks’ of one form or another are everywhere, but what do all these things have in common? And how can an understanding of networks help us become better digital marketers?

I want to go beyond our often unthinking use of the word network, and explore how a deeper understanding of networkscan help us identify influencers and communities online, creating content that really resonates and is more likely to be shared.

I’ll use an example of a network generated from Twitter, showing the follow relationships between people talking about Econsultancy, and sharing links from the site, over an eight day period in October 2013.

A network map of 3,930 Twitter users talking about and sharing Econsultancy links over an eight day period in October 2013

What is a 'network'?

At its most basic, a network can be defined as a collection of ‘nodes’ (things like people, products, or web pages) connected by relationships or associations of some kind (known as ‘edges’).

Mathematicians have been wrestling with networks for hundreds of years (through 'graph theory'), and more recently economists, sociologists and computer scientists have applied the mathematicians' insights to solve real-world problems.

There are many ways that understanding networks can help us become better digital marketers. A network algorithm, PageRank, underlies Google’s ordering of search results and allowed the company to power ahead of less network-savvy search engines such as Lycos, AltaVista and Yahoo! in the late 90s. 

Networks are also central to the way that Facebook allows users to see and now find content, for example through Graph Search.

Network analysis also helps us understand patterns of social interaction between users on blogs, forums, and social media sites.

The Econsultancy network

In our example Econsultancy Twitter network we queried the Twitter API to find the users who mentioned, or shared a link from, Econsultancy over an 8-day period, and then worked out the relationships between them. 

A detail of the network map of 3,930 users sharing Econsultancy.com content on Twitter

An increasing range of tools – including BlueNod, BottleNose, and various social media listening tools – allow us to gather Twitter network data. There’s also a range of free and paid software (NodeXL, yEd, Gephi, Mathematica) that allows us to analyse and visualise this data once we have it.

The ‘edges’ in our Econsultancy network are follower relationships between the users. Twitter networks made of other kinds of edges are possible. For example, we could decide to create an edge when one user mentions another, or retweets something they have tweeted. 

But a network of follow relationships contains the richest and most complete data – because users follow so many more users than they interact directly with, and a ‘follow’ is a vote of confidence – or at least an expression of interest – in the user who is followed.

It shows that someone cares about their opinion and wants to hear it. It also gives us a clue about the direction in which information flows through this group of people, from the followed to the follower.

The Econsultancy 'community of interest'

Just by looking at the Econsultancy network we can see something fairly remarkable.

The vast majority of users (99.26% to be exact, or 3,901 of 3,930) are bound together in a single group – or network ‘component’, to use the technical term.

Every user in this group is connected, either directly or indirectly through others, to every other user. Why should this large group of people who seemingly share nothing apart from having mentioned Econsultancy on Twitter be bound together like this? 

There are a few arcane mathematical reasons that make it more likely, but the fundamental reason is that the people in the network are a ‘community of interest’, defined by a shared enthusiasm for digital marketing in general, and Econsultancy.com in particular.

Real-life communities tend to be bound together by geography (a school, a workplace, a city), but online communities of interest are much more frictionless.

The connectedness of all these people talking about Econsultancy is pretty remarkable, when you think about it, and should give us a clue that something important is going on just below the surface.

In future posts I’ll be showing how we can look at the Econsultancy network to work out who is most influential, and to identify communities and cliques. I’ll also show how a Twitter network model can help us tailor content so that it really resonates, and is disseminated as deep into the network as possible.

Econsultancy

Detección de comunidades en Twitter italiano

Estados Generales del Centro-Norte (italiano): Detección de la comunidad Twitter
Publicado el 11 de diciembre 2012 por Alessandro Zonin


En referencia al reciente mensaje a los Estados Generales del Centro-Norte: Análisis de la comunicación en Twitter es una información útil sobre las comunidades regionales que se han reunido en el Barcamp y han interactuado entre sí y con otros usuarios de Twitter.

8 regiones cuentan con un usuario de Twitter específico asociados con el evento: @ iclazio, iclombardia @, @ icmarche, ICToscana @, @ ICTrentinoAA, @ ItaliaCampLigur, itcampumbria @, @ PiemonteCamp. A partir de la gráfica completa de la red relacional informaron caso es posible, usando algoritmos de detección de la comunidad, para identificar a los usuarios que entran en contacto uno con el otro y pasar a formar la sub-comunidad real, entre-veces.

Las fotos adjuntas son miniaturas y se pueden ampliar con un clic.


GraphInsight @ PiemonteCamp zoom de red relacional
@ PiemonteCamp

GraphInsight @ itcampumbria red relacional zoom
@ Itcampumbria

GraphInsight @ ItaliaCampLigur red relacional zoom
@ ItaliaCampLigur

GraphInsight @ ICTrentinoAA red relacional zoom
@ ICTrentinoAA

GraphInsight @ ICToscana red relacional zoom
@ ICToscana

GraphInsight @ ICMarche red relacional zoom
@ ICMarche

GraphInsight @ ICLombardia red relacional zoom
@ ICLombardia

GraphInsight @ ICLazio red relacional zoom
@ ICLazio


Social Network Analysis

Glattfelder: "¿Quien controla el Mundo?"

"When the crisis came, the serious limitations of existing economic and financial models immediately became apparent." "There is also a strong belief, which I share, that bad or oversimplistic and overconfident economics helped create the crisis."

Now, you've probably all heard of similar criticism coming from people who are skeptical of capitalism. But this is different. This is coming from the heart of finance. The first quote is from Jean-Claude Trichet when he was governor of the European Central Bank. The second quote is from the head of the U.K. Financial Services Authority. Are these people implyingthat we don't understand the economic systems that drive our modern societies? It gets worse. "We spend billions of dollars trying to understand the origins of the universe while we still don't understand the conditions for a stable society, a functioning economy, or peace."

What's happening here? How can this be possible? Do we really understand more about the fabric of reality than we do about the fabric which emerges from our human interactions?Unfortunately, the answer is yes. But there's an intriguing solution which is coming from what is known as the science of complexity.

To explain what this means and what this thing is, please let me quickly take a couple of steps back. I ended up in physics by accident. It was a random encounter when I was young, and since then, I've often wondered about the amazing success of physics in describing the reality we wake up in every day. In a nutshell, you can think of physics as follows. So you take a chunk of reality you want to understand and you translate it into mathematics. You encode it into equations. Then predictions can be made and tested.We're actually really lucky that this works, because no one really knows why the thoughts in our heads should actually relate to the fundamental workings of the universe. Despite the success, physics has its limits. As Dirk Helbing pointed out in the last quote, we don't really understand the complexity that relates to us, that surrounds us. This paradox is what got me interested in complex systems. So these are systems which are made up of many interconnected or interacting parts: swarms of birds or fish, ant colonies, ecosystems, brains, financial markets. These are just a few examples. Interestingly, complex systems are very hard to map into mathematical equations, so the usual physics approach doesn't really work here.

So what do we know about complex systems? Well, it turns out that what looks like complex behavior from the outside is actually the result of a few simple rules of interaction.This means you can forget about the equations and just start to understand the system by looking at the interactions, so you can actually forget about the equations and you just start to look at the interactions. And it gets even better, because most complex systems have this amazing property called emergence. So this means that the system as a wholesuddenly starts to show a behavior which cannot be understood or predicted by looking at the components of the system. So the whole is literally more than the sum of its parts. And all of this also means that you can forget about the individual parts of the system, how complex they are. So if it's a cell or a termite or a bird, you just focus on the rules of interaction.

As a result, networks are ideal representations of complex systems. The nodes in the network are the system's components and the links are given by the interactions. So what equations are for physics, complex networks are for the study of complex systems.

This approach has been very successfully applied to many complex systems in physics, biology, computer science, the social sciences, but what about economics? Where are economic networks? This is a surprising and prominent gap in the literature. The study we published last year called "The Network of Global Corporate Control" was the first extensive analysis of economic networks. The study went viral on the Internet and it attracted a lot of attention from the international media. This is quite remarkable, because, again, why did no one look at this before? Similar data has been around for quite some time.

What we looked at in detail was ownership networks. So here the nodes are companies, people, governments, foundations, etc. And the links represent the shareholding relations,so Shareholder A has x percent of the shares in Company B. And we also assign a value to the company given by the operating revenue. So ownership networks reveal the patterns of shareholding relations. In this little example, you can see a few financial institutions with some of the many links highlighted.

Now you may think that no one's looked at this before because ownership networks arereally, really boring to study. Well, as ownership is related to control, as I shall explain later,looking at ownership networks actually can give you answers to questions like, who are the key players? How are they organized? Are they isolated? Are they interconnected? And what is the overall distribution of control? In other words, who controls the world? I think this is an interesting question.

And it has implications for systemic risk. This is a measure of how vulnerable a system is overall. A high degree of interconnectivity can be bad for stability, because then the stress can spread through the system like an epidemic.

Scientists have sometimes criticized economists who believe ideas and concepts are more important than empirical data, because a foundational guideline in science is: Let the data speak. Okay. Let's do that.

So we started with a database containing 13 million ownership relations from 2007. This is a lot of data, and because we wanted to find out who rules the world, we decided to focus on transnational corporations, or TNCs for short. These are companies that operate in more than one country, and we found 43,000. In the next step, we built the network around these companies, so we took all the TNCs' shareholders, and the shareholders' shareholders, etc., all the way upstream, and we did the same downstream, and ended up with a network containing 600,000 nodes and one million links. This is the TNC network which we analyzed.

And it turns out to be structured as follows. So you have a periphery and a center which contains about 75 percent of all the players, and in the center there's this tiny but dominant core which is made up of highly interconnected companies. To give you a better picture,think about a metropolitan area. So you have the suburbs and the periphery, you have a center like a financial district, then the core will be something like the tallest high rise building in the center. And we already see signs of organization going on here. Thirty-six percent of the TNCs are in the core only, but they make up 95 percent of the total operating revenue of all TNCs.

Okay, so now we analyzed the structure, so how does this relate to the control? Well, ownership gives voting rights to shareholders. This is the normal notion of control. And there are different models which allow you to compute the control you get from ownership. If you have more than 50 percent of the shares in a company, you get control, but usually it depends on the relative distribution of shares. And the network really matters. About 10 years ago, Mr. Tronchetti Provera had ownership and control in a small company, which had ownership and control in a bigger company. You get the idea. This ended up giving him control in Telecom Italia with a leverage of 26. So this means that, with each euro he invested, he was able to move 26 euros of market value through the chain of ownership relations.

Now what we actually computed in our study was the control over the TNCs' value. This allowed us to assign a degree of influence to each shareholder. This is very much in the sense of Max Weber's idea of potential power, which is the probability of imposing one's own will despite the opposition of others.

If you want to compute the flow in an ownership network, this is what you have to do. It's actually not that hard to understand. Let me explain by giving you this analogy. So think about water flowing in pipes where the pipes have different thickness. So similarly, the control is flowing in the ownership networks and is accumulating at the nodes. So what did we find after computing all this network control? Well, it turns out that the 737 top shareholders have the potential to collectively control 80 percent of the TNCs' value. Now remember, we started out with 600,000 nodes, so these 737 top players make up a bit more than 0.1 percent. They're mostly financial institutions in the U.S. and the U.K. And it gets even more extreme. There are 146 top players in the core, and they together have the potential to collectively control 40 percent of the TNCs' value.

What should you take home from all of this? Well, the high degree of control you saw is very extreme by any standard. The high degree of interconnectivity of the top players in the core could pose a significant systemic risk to the global economy and we could easily reproduce the TNC network with a few simple rules. This means that its structure is probably the result of self-organization. It's an emergent property which depends on the rules of interaction in the system, so it's probably not the result of a top-down approach like a global conspiracy.

Our study "is an impression of the moon's surface. It's not a street map." So you should take the exact numbers in our study with a grain of salt, yet it "gave us a tantalizing glimpseof a brave new world of finance." We hope to have opened the door for more such research in this direction, so the remaining unknown terrain will be charted in the future. And this is slowly starting. We're seeing the emergence of long-term and highly-funded programs which aim at understanding our networked world from a complexity point of view. But this journey has only just begun, so we will have to wait before we see the first results.

Now there is still a big problem, in my opinion. Ideas relating to finance, economics, politics, society, are very often tainted by people's personal ideologies. I really hope that this complexity perspective allows for some common ground to be found. It would be really great if it has the power to help end the gridlock created by conflicting ideas, which appears to be paralyzing our globalized world. Reality is so complex, we need to move away from dogma. But this is just my own personal ideology.


Thank you.


(Applause)