Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sublinear Time Estimation of Degree Distribution Moments: The Degeneracy
  Connection   We revisit the classic problem of estimating the degree distribution moments
of an undirected graph. Consider an undirected graph $G=(V,E)$ with $n$
vertices, and define (for $s > 0$) $\mu_s = \frac{1}{n}\cdot\sum_{v \in V}
d^s_v$. Our aim is to estimate $\mu_s$ within a multiplicative error of
$(1+\epsilon)$ (for a given approximation parameter $\epsilon>0$) in sublinear
time. We consider the sparse graph model that allows access to: uniform random
vertices, queries for the degree of any vertex, and queries for a neighbor of
any vertex. For the case of $s=1$ (the average degree),
$\widetilde{O}(\sqrt{n})$ queries suffice for any constant $\epsilon$ (Feige,
SICOMP 06 and Goldreich-Ron, RSA 08). Gonen-Ron-Shavitt (SIDMA 11) extended
this result to all integral $s > 0$, by designing an algorithms that performs
$\widetilde{O}(n^{1-1/(s+1)})$ queries.
  We design a new, significantly simpler algorithm for this problem. In the
worst-case, it exactly matches the bounds of Gonen-Ron-Shavitt, and has a much
simpler proof. More importantly, the running time of this algorithm is
connected to the degeneracy of $G$. This is (essentially) the maximum density
of an induced subgraph. For the family of graphs with degeneracy at most
$\alpha$, it has a query complexity of
$\widetilde{O}\left(\frac{n^{1-1/s}}{\mu^{1/s}_s} \Big(\alpha^{1/s} +
\min\{\alpha,\mu^{1/s}_s\}\Big)\right) =
\widetilde{O}(n^{1-1/s}\alpha/\mu^{1/s}_s)$. Thus, for the class of bounded
degeneracy graphs (which includes all minor closed families and preferential
attachment graphs), we can estimate the average degree in $\widetilde{O}(1)$
queries, and can estimate the variance of the degree distribution in
$\widetilde{O}(\sqrt{n})$ queries. This is a major improvement over the
previous worst-case bounds. Our key insight is in designing an estimator for
$\mu_s$ that has low variance when $G$ does not have large dense subgraphs.

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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph
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The categories of this paper are: math
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg