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Friday, September 10, 2010

DeGroot Model of Social Influence and the Microfoundations of Eigenvector Centrality...

Most of what follows is culled from M. Jackson's Social and Economic Networks...

Start with the DeGroot model of imitation and social influence.  In this model individuals start with initial opinions on a subject.  These opinions are represented by an n-dimensional vector of probabilities:

p(0)=(p1(0), ..., pn(0))

Each of the p1(0) lies on [0,1] and might be thought of as the probability that a given statement about the world (such as "the economy is an open thermodynamic system") is true.  Alternatively one could interpret the vector of initial opinions as beliefs concerning the quality of a product, or the likelihood that an individual might engage in a given activity (like lending perhaps), etc...

In this model the interactions are captured through a possibly weighted and directed nxn non-negative matrix T.  The matrix T is also a row stochastic matrix, which means that the entries across each row will always sum to one.  The interpretation of Tij is that it represents the amount of weight or trust that agent i places on the current belief of agent j in forming i's belief next period.  Beliefs are updated over time so that 


Illustrative Example: Suppose that we have three banks.  Bank 1 is run by James, Bank 2 is run by Keshav, and Bank 3 is run by David. These three banks have an updating matrix T and a network digram that that looks as follows:
With this matrix T, Bank 1 (run by James) puts equal weight (1/3 each) on Banks 2 and 3 (run by David and Keshav) when forming his beliefs about the world.  Bank 2 weights its own beliefs slightly more, but completely discounts Bank 3 (which given that the bank is run by Keshav is fairly ridiculous).  Bank 3 meanwhile, puts the most weight/trust in its own beliefs (3/4) and then puts a weight of 1/4 on Bank 2.

Now suppose that the initial vector of beliefs is p(0)=(1,0,0).  So Bank 1, run by James, initially believes that some future event (like a financial crisis) will occur with probability 1.  Lets see how these beliefs evolve overtime given this influence network...

Does beliefs converge in this case? The answer is yes.  Standard mathematical results from Markov chain theory show that as long as T is strongly connected (which means that there is a directed path from any node to any other node) and aperiodic (there are no cycles of beliefs; no feedback loops where beliefs flow from one agent through all other agents and then end up influencing the initial agent again) then societal beliefs must converge.  In this case they converge to...

Thus even though we started with a situation in which only James and his bank believed that a financial crisis was going to happen with probability 1, we end up in a situation where all three banks believe that a financial crisis will occur with positive probability.  I think that this type of idea is highly relevant to my research into the relationship between financial network structure and systemic risk (assuming that social influence has something to do with bank lending practices for example). 

Now suppose we want to keep track of how each agent in the social network influences the limiting beliefs.  Let the n-vector of limiting beliefs be defined as p(*)=(p*,p*,...,p*):

To keep track of the limiting influence that each agent has, we want to find a n-vector s whose entries are all on [0,1] and whose entries sum to on, such that p* equals the inner product of s and the vector of initial beliefs (i.e., p*=s.p(0)). If we can find such a vector s, then our limiting/consensus beliefs would be a weighted average of the initial beliefs where the relative weights would be the influence of the various agents on the consensus beliefs.  Now since starting with p(0) or with p(1)=Tp(0) ends up in the same limit, it must be the case that s.p(1)=s.p(0) and that therefore s.(Tp(0))=s.p(0).

Since this equation is required to hold for any vector of initial beliefs p(0) it follows that:


But his just says that s is the left-hand eigenvector of the matrix T whose eigenvalue is 1!  Appealing to the mathematical gods, we find that as long T is strongly connected, aperiodic, and row stochastic then there exists a unique such unit eigenvector and this eigenvector has all non-negative values.  

Finally we have arrived at our connection between a measures of social influence and eigenvector centrality.  Eigenvector centrality measures essentially rank agents based on the size of the respective component of the eigenvector corresponding to the largest eigenvalue of whatever connectivity matrix you are studying...which in this case is equivalent to finding the vector of social influence s and then ranking agents high-to-low based on their contribution to consensus beliefs.
So IF we were able to map a financial network, say a network of banks, and we were able to obtain a matrix of "social influence," say by getting lots of data on balance sheet linkages between the banks, then have a theoretical basis for using eigenvector centrality measures to identify which banks are relatively more important than others.  This type of idea would, I think, be useful for (amongst other things...) identifying banks for idiosyncratic microprudential regulation.

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