Two-dimensional small-world networks: navigation with local information

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.Comment: Accepted for publication in Phys. Rev.

Quantum information processing architecture with endohedral fullerenes in a carbon nanotube

A potential quantum information processor is proposed using a fullerene peapod, i.e., an array of the endohedral fullerenes 15N@C60 or 31P@C60 contained in a single walled carbon nanotube (SWCNT). The qubits are encoded in the nuclear spins of the doped atoms, while the electronic spins are used for initialization and readout, as well as for two-qubit operations.Comment: 8 pages, 8 figure