Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2006-04-11
PHYSICAL REVIEW E 73, 056111 (2006)
Physics
Condensed Matter
Disordered Systems and Neural Networks
Accepted for publication in Phys. Rev. E
Scientific paper
10.1103/PhysRevE.73.056111
Navigation process is studied on a variant of the Watts-Strogatz small world network model embedded on a square lattice. With probability $p$, each vertex sends out a long range link, and the probability of the other end of this link falling on a vertex at lattice distance $r$ away decays as $ r^{-\alpha}$. Vertices on the network have knowledge of only their nearest neighbors. In a navigation process, messages are forwarded to a designated target. For $\alpha <3$ and $\alpha \neq 2$, a scaling relation is found between the average actual path length and $pL$, where $L$ is the average length of the additional long range links. Given $pL>1$, dynamic small world effect is observed, and the behavior of the scaling function at large enough $pL$ is obtained. At $\alpha =2$ and 3, this kind of scaling breaks down, and different functions of the average actual path length are obtained. For $\alpha >3$, the average actual path length is nearly linear with network size.
Chen Jian-Zhen
Liu Wei
Zhu Jian-Yang
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