Journal on Policy and Complex Systems
The measure we use is the democratic mean , more sensitive for our purposes than Gini coefficient and easily applicable to networks before and after rewiring . Ours are connected networks , without isolated nodes . A network with a high preferential attachment will have a wide range in degree — many nodes will have few connections ; a very few will have many . The ratio of the mean degree over the span between least and highest degree will , therefore , be a small number . A network with a low preferential attachment will have a much smaller range in degree , with the result that the mean degree over the difference between highest and lowest will be relatively large . The democratic mean is the ratio of a mean degree over the highest degree : the higher the democratic mean , the less the preferential attachment of a communication network and the more democratic the network is in that sense .
More formally , where dmax is the degree of the most connected node and dmin the degree of the least connected , we take the degree spread D as dmax − dmin . With dm as our mean degree , the democratic mean of a network is ( dm − dmin )/ D . For a 50-node network with extremely high preferential attachment — 49 nodes with 1 connection and 1 node with 49 , for example — the democratic mean will approach . 96 / 48 or . 02 . In a random network with a normal distribution between 1 and 3 connections , on the other hand , the democratic mean will approach . 5 . Sample network distributions typical of those considered here , with corresponding democratic means , are shown in Figure 11 .
How will a network rewire in response to volatility ? We start with a network generated with a particular preferential exponent , then rewire in response to volatility . Will democratic communication increase , as measured in terms of a democratic mean , or not ?
Volatility , as above , is stimulated by the direct change in a random 10 % of our nodes at regular intervals over the course of a run . Here we count as “ volatile ” those nodes that change more often than they are directly changed : nodes that change opinion at least 1.5 times as often as they are directly changed by the program . These volatile nodes are those that are vulnerable to opinion change from changes elsewhere in the network as well . In all cases , we assume that it will be the volatile nodes that break links , replacing them with links to nodes . We consider each of the following as possible patterns for rewiring in response to
Random Reaction : Rewiring at Random within the Network
There is something about the reinforcement pattern of volatile nodes that makes them unstable . They , therefore , break a link at random and establish a replacement link to another node in the network , chosen at random . In the end , we can expect volatility to die down , but with a newly structured network in its wake .
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