miércoles, 9 de octubre de 2013

Centralidad en redes sociales: Depende de la métrica

Who is central to a social network? It depends on your centrality measure.

One important feature of networks is the relative centrality of individuals in them.  Centrality is a structural characteristic of individuals in the network, meaning a centrality score tells you something about how that individual fits within the network overall.  Individuals with high centrality scores are often more likely to be leaders, key conduits of information, and be more likely to be early adopters of anything that spreads in a network. 
Low centrality individuals can be termed peripheral.  Being peripheral can have advantages as well by protecting individuals from negative contagion and influence (think of the flu spreading or the ricocheting effects of that really deflating coworker).  Sometimes lower centrality is associated with less work overload in an organization.
The confusing thing about centrality is there is not just one kind.  Social network scientists have invented literally dozens of centrality measures to characterize slightly different aspects of structural positions in networks.  Here we will explain three of the most commonly used centrality measures.

Closeness Centrality

This measure expresses the average social distance from each individual to every other individual in the network.  The concept of social distance is easily understood by considering the common parlor and roadtrip game, six degrees of Kevin Bacon, in which players try to find the shortest set of connections from any actor to Kevin Bacon based on movies that actors have starred in together.  An actor who has costarred with Kevin Bacon is considered to be 1 degree away, while an actor who costarred with a Kevin Bacon costar is 2 degrees away, and so forth.  The same concept can be applied to any social network.
 If we divide 1 by the average shortest path from an individual to all other individuals in the network, then we have calculated their closeness centrality.  In this way an individual with a direct tie to everyone else ends up with a closeness score of 1.  Individuals who connect to most others through many intermediaries get closeness scores that are increasingly nearer to zero.  One property of closeness centrality is that it tends to give high scores to individuals who are near the center of local clusters (aka network communities) in an overall larger network (Figure 1).

Figure 1: Individuals who are highly connected to others within their own cluster will have a high closeness centrality.
Applications: High closeness centrality individuals tend to be important influencers within their local network community.  They may often not be public figures to the entire network of a corporation or profession, but they are often respected locally and they occupy short paths for information spread within their network community.

Betweenness Centrality

Betweenness is another measure that is derived from the concept of counting the shortest paths between individuals in a network.  It has very different properties, however, from closeness centrality.  To calculate betweenness centrality, one starts by finding all the shortest paths between any two individuals in the network.  You then count the number of these shortest paths that go through each individual.  This number is betweenness centrality.
The result of this calculation is finding the individuals who are necessary conduits for information that must traverse disparate parts of the network.  These are usually very different individuals from those with high closeness.  High betweenness individuals often do not have the shortest average path to everyone else, but they have the greatest number of shortest paths that necessarily have to go through them. 
One example of this would be the highway map of the United States.  Cities in the Midwest like St. Louis have higher betweenness centrality than do New York or LA because so many shortest paths that combine any cities on the east and west coast have to pass through them. 
In a social network, high betweenness individuals are often found at the intersections of more densely connected network communities (Figure 2).  They are well positioned to perform brokering roles across these clusters in the sense that brokers connect otherwise disconnected people who yet may benefit from an exchange of information.  The term broker applies quite naturally in this case, as actual brokers, whether real estate, mortgage, or pawn brokers, make a living by connecting otherwise disconnected sets of individuals.  That’s what high betweenness people do within a social network.

Figure 2: Those who act as bridges between clusters in the network have high betweenness centrality.
Applications:  High betweenness individuals are often critical to collaboration across departments and to maintaining the spread of a new product through an entire network.  Because of their locations between network communities, they are natural brokers of information and collaboration.  One difference between high betweenness individuals in a network and actual brokers is the latter usually have a public profile as part of their business, whereas high betweenness individuals often are overlooked.  This occurs because they are not central to any single social clique, and instead reside on the periphery of several such cliques each of which all engender more trust and admiration within rather than outside of the clique.   

Eigenvector Centrality
This measure has a complicated name, but it basically denotes the extent to which an individual is a big fish connected with other big fish in a big pond.  Eigenvector centrality is calculated by assessing how well connected an individual is to the parts of the network with the greatest connectivity (Figure 3).  Individuals with high eigenvector scores have many connections, and their connections have many connections, and their connections have many connections … out to the end of the network. 

Figure 3: Highly connected individuals within highly interconnected clusters, or ‘big fish in big ponds’, have high eigenvector centrality.
There is another way to think about eigenvector centrality that relates more to its name.  The name ‘eigen’ come from German, and can be translated to mean ‘characteristic’.  In a social network, we can observe directly whether individuals are connected or not.  The mathematical notion behind eigenvectors is that we can re-express the matrix of person-to-person connections with a set of ‘eigen (characteristic)’ values that are assigned to each individual.  So, eigenvector centrality scores correspond to the score you get for individuals if you start by constructing the pairwise connections between all individuals in a network (1 for connected, 0 for not), and then assign a single number to each individual while attempting to keep the distances between these new values equal to the distances observed in the social network matrix of connections.  Of course this can’t be done with a single numeric value per individual, but in fact you can always represent a set of social connections or distances by assigning as many vectors (strings of numbers for each individual) as there are individuals in the network.
Applications:  High eigenvector centrality individuals are leaders of the network.  They are often public figures with many connections to other high-profile individuals.  Thus, they often play roles of key opinion leaders and shape public perception.  A related example of this is Google’s page rank algorithm, which is closely related to eigenvector centrality calculated on websites based on links to them. 
High eigenvector centrality individuals, however, cannot necessarily perform the roles of high closeness and betweenness.  They do not always have the greatest local influence and may have limited brokering potential.  Like an aloof king in his court or CEO in her boardroom, they may at times be isolated from peripheral individuals and smaller network communities that have limited connectivity with the most densely connected parts of the network. 

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