Journal on Policy and Complex Systems
e = 0 e = 1 e = 2 e = 3
Figure 3 . Networks with increasing preferential attachment in networks of mean degree 2 .
e = 0 e = 1 e = 2 e = 3
Figure 4 . Networks with increasing preferential attachment in networks of mean degree 3 .
its degree over the total sum of degrees in the existing network : the standard pattern for preferential attachment . As e is increased to higher positive values , however , the bias in favor of nodes with higher degree is exaggerated . We can , therefore , generate the extent of preferential attachment by simple adjustment of the preferential attachment exponent e . 2 Figure 3 shows typical networks generated with an e of 0 , 1 , 2 , and 3 .
Each of the networks shown in Figure 3 is generated with an average degree of only two . For networks with higher degrees , a new node will connect to one of those to which it is not already connected with a probability measured by our preferential exponent . For networks with an average degree of 3 , for example , increased preferential attachment will take the form of those networks shown in Figure 4 . With e = 3 and higher we get not one focus of preferential attachment , for example , but two .
When plotted on two axes , our two measures allow us a landscape of networks more or less democratic in the
2 In Barabási and Albert ( 1999 ), the authors note the possibility of using an exponent in this way , but confine their attention entirely to e = 1 on the grounds that it most clearly models the scale-free networks that are their target . On the use of a variable exponent , see also Krapivsky , Redner , and Leyraz ( 2000 ); Dorogovtsev , Mendes , and Samukhin ( 2000 ); and Noble , Davy , and Franks ( 2004 ).
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