Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Degree centrality and root finding in growing random networks   We consider growing random networks $\{\mathcal{G}_n\}_{n \ge 1}$ where, at
each time, a new vertex attaches itself to a collection of existing vertices
via a fixed number $m \ge 1$ of edges, with probability proportional to a
function $f$ (called attachment function) of their degree. It was shown in
\cite{BBpersistence} that such network models exhibit two distinct regimes: (i)
the persistent regime, corresponding to $\sum_{i=1}^{\infty}f(i)^{-2} <
\infty$, where the top $K$ maximal degree vertices fixate over time for any
given $K$, and (ii) the non-persistent regime, with
$\sum_{i=1}^{\infty}f(i)^{-2} = \infty$, where the identities of these vertices
keep changing infinitely often over time. In this article, we develop root
finding algorithms using the empirical degree structure and local network
information based on a snapshot of such a network at some large time. In the
persistent regime, the algorithm is purely based on degree centrality. For a
given error tolerance $\varepsilon \in (0,1)$, there exists $K_{\varepsilon}
\in \mathbb{N}$ such that for any $n \ge 1$, the confidence set for the root in
$\mathcal{G}_n$, which contains the root with probability at least $1 -
\varepsilon$, consists of the top $K_{\varepsilon}$ maximal degree vertices. In
particular, the size of the confidence set is stable in the network size. Upper
and lower bounds on $K_{\varepsilon}$ are explicitly characterized in terms of
the error tolerance $\varepsilon$ and the attachment function $f$. In the
non-persistent regime, for an appropriate choice of $r_n \rightarrow \infty$ at
a rate much smaller than the diameter of the network, the neighborhood of
radius $r_n$ around the maximal degree vertex is shown to contain the root with
high probability, and a size estimate for this set is obtained.

34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph