Synchronizability and Navigability of Small-World Networks Generated by One Dimensional Kleinberg Model

In this paper, the impact of the clustering exponent (α) on synchronizability, average shortest path length and navigability of small-world networks generated by one dimensional Kleinberg model is investigated. It could be seen from the analysis that the synchronizability becomes stronger as the clustering exponent decreases. And the navigability achieves peak at the neighborhood of α = 1, as well as the navigability becomes smaller as the clustering exponent increases. Moreover, the average path length of one dimensional Kleinberg small-world network decreases with respect to increasing clustering exponent. And this phenomenon is verified by numerical simulations on a network of Rossler oscillators. Then, it could be deduced from the phenomenon observed that compared with the low probabilities of longer distance of the edge-adding, the high probabilities of shorter distance of the edge-adding could achieve better synchronizability.