Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Hyperbolicity of Small-World and Tree-Like Random Graphs   Hyperbolicity is a property of a graph that may be viewed as being a "soft"
version of a tree, and recent empirical and theoretical work has suggested that
many graphs arising in Internet and related data applications have hyperbolic
properties. We consider Gromov's notion of \delta-hyperbolicity, and establish
several results for small-world and tree-like random graph models. First, we
study the hyperbolicity of Kleinberg small-world random graphs and show that
the hyperbolicity of these random graphs is not significantly improved
comparing to graph diameter even when it greatly improves decentralized
navigation. Next we study a class of tree-like graphs called ringed trees that
have constant hyperbolicity. We show that adding random links among the leaves
similar to the small-world graph constructions may easily destroy the
hyperbolicity of the graphs, except for a class of random edges added using an
exponentially decaying probability function based on the ring distance among
the leaves.
  Our study provides one of the first significant analytical results on the
hyperbolicity of a rich class of random graphs, which shed light on the
relationship between hyperbolicity and navigability of random graphs, as well
as on the sensitivity of hyperbolic {\delta} to noises in random graphs.

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The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys