The human brain has a huge number of neurons connected to each other, forming a multitude of networks; notably, such connectivity typically exhibits a small-world structure. However, the brains of persons with autism spectrum disorder (ASD) reportedly have what has been termed as 'local over-connectivity.' The neural activity of the ASD brain is also atypical; resting-state EEG signals in the ASD brain have lower complexity and enhanced power at low and high frequency oscillations. In this study, we used a small-world network model based on the model proposed by Watts and Strogatz to investigate the relationship between the degree of local over-connectivity and neural activity. We controlled the degree of local over-connectivity in the model according to the parameters laid out by Watts and Strogatz. We assessed connectivity using graph-theoretical approaches, and analyzed the complexity and frequency spectrum of the activity. We found that an ASD-like network with local over-connectivity (i.e., a high clustering coefficient and a high degree of centrality) would have excessively high power in the high frequency band, and less complexity than that of a network without local over-connectivity. This result supports the idea that local over-connectivity could underlie the characteristic brain electrical activity in persons with ASD.