Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex