Synchronizability of Small-World Networks Generated from a Two-Dimensional Kleinberg Model

Yi Zhao,Jingyi Wang,Jianwen Feng

doi:10.1155/2013/451960

Abstract

This paper investigates the synchronizability of small-world networks generated from a two-dimensional Kleinberg model, which is more general than NW small-world network. The three parameters of the Kleinberg model, namely, the distance of neighbors, the number of edge-adding, and the edge-adding probability, are analyzed for their impacts on its synchronizability and average path length. It can be deduced that the synchronizability becomes stronger as the edge-adding probability increases, and the increasing edge-adding probability could make the average path length of the Kleinberg small-world network go smaller. Moreover, larger distance among neighbors and more edges to be added could play positive roles in enhancing the synchronizability of the Kleinberg model. The lorentz oscillators are employed to verify the conclusions numerically.

Highlights

A complex network is a large set of nodes connected by a set of links such as coupled biological and chemical system, neural networks, social interacting species, the Internet, and the World Wide Web
This paper investigates the synchronizability of small-world networks generated from a two-dimensional Kleinberg model, which is more general than NW small-world network
In the research articles [10,11,12,13], the synchronizability of a smallworld network generated by randomly adding a fraction of long-range shortcuts to a ring network is investigated. It can be deduced from the theoretical analysis and numerical simulation that the synchronizability of the small-world network becomes stronger as the edge-adding probability p grows larger

Summary

Introduction

A complex network is a large set of nodes (or vertices) connected by a set of links (or edges) such as coupled biological and chemical system, neural networks, social interacting species, the Internet, and the World Wide Web. In the research articles [10,11,12,13], the synchronizability of a smallworld network generated by randomly adding a fraction of long-range shortcuts to a ring network is investigated It can be deduced from the theoretical analysis and numerical simulation that the synchronizability of the small-world network becomes stronger as the edge-adding probability p grows larger. It can be deduced that the decreasing in the average path length may result in the increasing synchronizability These phenomena are interesting, and a natural question is that whether other small-world networks have similar properties, which motivates us to take a two-dimensional Kleinberg small-world network [19] as an example and investigate the impact factors of such network. We could get some conclusions about impact factor on the synchronizability and the average path length of the Kleinberg small-world network, which are complementary to the studies on the synchronizability of the small-world networks

Preliminaries

Influencing Factors the Synchronizability of a Network

Numerical Simulation for Lorentz Oscillators

Conclusion