Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Finite Percolation at a Multiple of the Threshold   Bond percolation on infinite heavy-tailed power-law random networks lacks a
proper phase transition; or one may say, there is a phase transition at {\em
zero percolation probability}. Nevertheless, a finite size percolation
threshold $q_c(N)$, where $N$ is the network size, can be defined. For such
heavy-tailed networks, one can choose a percolation probability $q(N)=\rho
q_c(N)$ such that $\displaystyle \lim_{N\to \infty}(q-q_c(N)) =0$, and yet
$\rho$ is arbitrarily large (such a scenario does not exist for networks with
non-zero percolation threshold). We find that the critical behavior of random
power-law networks is best described in terms of $\rho$ as the order parameter,
rather than $q$. This paper makes the notion of the phase transition of the
size of the largest connected component at $\rho=1$ precise. In particular,
using a generating function based approach, we show that for $\rho>1$, and the
power-law exponent, $2\leq \tau<3$, the largest connected component scales as
$\sim N^{1-1/\tau}$, while for $0<\rho<1$ the scaling is at most $\sim
N^{\frac{2-\tau}{\tau}}$; here, the maximum degree of any node, $k_{max}$, has
been assumed to scale as N^{1/\tau}$. In general, our approach yields that for
large $N$, $\rho \gg 1$, $2\leq \tau<3$, and $k_{max} \sim N^{1/\tau}$, the
largest connected component scales as $\sim
\rho^{1/(3-\tau)}N^{1-1/\tau}$.Thus, for any fixed but large N, we recover, and
make it precise, a recent result that computed a scaling behavior of
$q^{1/(3-\tau)}$ for "small $q$". We also provide large-scale simulation
results validating some of these scaling predictions, and discuss applications
of these scaling results to supporting efficient unstructured queries in
peer-to-peer networks.

5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: cs
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The categories of this paper are: stat