Scale-free property of local-world networks and their community structures

The scale-free property and community structures of complex networks formed by local events have been studied theoretically, numerically, and empirically. We showed analytically and numerically that the degree distribution function P(k) of the local-world evolving network exhibits a crossover from an exponential to power-law form by increasing the local-world size M. For M much larger than the crossover local-world size Mco, the distribution function P(k) has a power-law form for any degree k(≫1). Below Mco, however, P(k) obeys a power law for 1≪k≪kco and decays exponentially for k≫kco. The crossover size Mco and the crossover degree kco have been also elucidated. In addition, we constructed the drug prescription network (DPN) as a real local-world network, in which the local-world subsets are definitely specified, to reveal how the local-world nature affects properties of real-world networks. We found that the community structure of the DPN strongly correlates with the local worlds.