Power Law Degree Distributions Can Fit Averages of Non-Power Law Distributions

Many complex systems have been modeled as networks. Examples of such systems that are of interest to cognitive psychologists include the mental lexicon for spoken word recognition and semantic memory. A frequent finding in such studies is that the frequency distribution of the number of connections for each node in the network follows a power law. This finding has been interpreted to mean that the network grows through a process similar to preferential attachment: when a node is added to the network, it attaches to other nodes with a probability proportional to the number of connections those other nodes already have. Power-law degree distributions, however, may also well describe degree distributions that result when averaging across multiple individual degree distributions, none of which follows a power-law. Further, each of these individual distributions may reflect a random growth process rather than the more systematic process suggested by preferential attachment.