How Stable is Democracy ? Suggestions from Artificial Social Networks
2012 ) and broad , equal , protected , and mutually binding consultation ( Parris , 2008 ). In the network on the right , more people are in communication with more people , approaching the ideal of a New England town meeting . Higher mean degree , we propose , offers a second dimension along which we can measure increasingly “ democratic ” networks .
Figure 2 . A less democratic network ( left ) and more democratic ( right ) in terms of higher node degree and wider contact .
We consider the idea of more and less democratic communicative networks a natural one , with a characteristic degree and preferential attachment as natural measures . Our formal treatment of the first will be in terms of mean node degree : the higher the mean node degree of a network , the more democratic we take it to be in the second sense . Our formal treatment of how preferential the attachment structure of a network is will be in terms of the preferential exponent .
Consider the prospect of adding a new node to an existing network of nodes x i
… x n
( Barabási & Albert , 1999 ; Newman , 2005 ). The probability that the new node will be connected to a specific node x j can be given as :
Here d j represents the degree of node j and represents the sum of degrees of all nodes , but in each case those degrees are raised to our preferential attachment exponent e . Where e = 0 , ( d j
) e for any node = 1 , and thus a new node attaches to existing nodes with no preference between them in terms of relative degrees . The result approaches a random network . 1 Where e = 1 , ( d j
) e is simply ( d j
), and a new node attaches to an existing node simply as the ratio of
1 “ Approaches ” because nodes formed early in the process do have increased chances of being connected to by newly added nodes . We can create a continuum from more truly random networks to those of higher preferential attachment by eliminating the assumption that our network is formed node by node . In that case we regard all nodes in the network as formed ab initio and proceed either node by node or by random choice of node , applying the exponential function above to the totality of n nodes .
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