There are many reasons why knowing whether a network follows this type of scale or is scale free is important. One of the first and easiest reasons is to imagine a virus. If there was a virus spreading through a network that had a scale effect, the most sensible thing would be to quarantine those few hubs that have the most ties. If those few get infected by the virus then an exponentially higher amount of other ties would be effected, than if an average actor in the network was exposed.
Another reason is for matters of prediction. By knowing how a network grows and operates, one could predict the behavior of a network. For an example the flow of information, or of money would be vastly different depending on whether the network was scale free or not. The robustness of a network is also affected by this. If a network is scale-free it so much more interconnected with a normal distribution. It would be much harder to pick apart, or take out key actors. One could removed many actors from the network and see hardly any difference. However if the network was on a scale and the main actors were removed the network could quite easily collapse.
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