22 research outputs found
On Approximating the Number of -cliques in Sublinear Time
We study the problem of approximating the number of -cliques in a graph
when given query access to the graph.
We consider the standard query model for general graphs via (1) degree
queries, (2) neighbor queries and (3) pair queries. Let denote the number
of vertices in the graph, the number of edges, and the number of
-cliques. We design an algorithm that outputs a
-approximation (with high probability) for , whose
expected query complexity and running time are
O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log
n,1/\varepsilon,k).
Hence, the complexity of the algorithm is sublinear in the size of the graph
for . Furthermore, we prove a lower bound showing that
the query complexity of our algorithm is essentially optimal (up to the
dependence on , and ).
The previous results in this vein are by Feige (SICOMP 06) and by Goldreich
and Ron (RSA 08) for edge counting () and by Eden et al. (FOCS 2015) for
triangle counting (). Our result matches the complexities of these
results.
The previous result by Eden et al. hinges on a certain amortization technique
that works only for triangle counting, and does not generalize for larger
cliques. We obtain a general algorithm that works for any by
designing a procedure that samples each -clique incident to a given set
of vertices with approximately equal probability. The primary difficulty is in
finding cliques incident to purely high-degree vertices, since random sampling
within neighbors has a low success probability. This is achieved by an
algorithm that samples uniform random high degree vertices and a careful
tradeoff between estimating cliques incident purely to high-degree vertices and
those that include a low-degree vertex
Conjectures on Convergence and Scalar Curvature
Here we survey the compactness and geometric stability conjectures formulated
by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar
Curvature and Convergence}. We have tried to survey all the progress towards
these conjectures as well as related examples, although it is impossible to
cover everything. We focus primarily on sequences of compact Riemannian
manifolds with nonnegative scalar curvature and their limit spaces. Christina
Sormani is grateful to have had the opportunity to write up our ideas and has
done her best to credit everyone involved within the paper even though she is
the only author listed above. In truth we are a team of over thirty people
working together and apart on these deep questions and we welcome everyone who
is interested in these conjectures to join us.Comment: Please email us any comments or corrections. 57 pages, 20 figures,
IAS Emerging Topics on Scalar Curvature and Convergenc
New Loop Representations for 2+1 Gravity
Since the gauge group underlying 2+1-dimensional general relativity is
non-compact, certain difficulties arise in the passage from the connection to
the loop representations. It is shown that these problems can be handled by
appropriately choosing the measure that features in the definition of the loop
transform. Thus, ``old-fashioned'' loop representations - based on ordinary
loops - do exist. In the case when the spatial topology is that of a two-torus,
these can be constructed explicitly; {\it all} quantum states can be
represented as functions of (homotopy classes of) loops and the scalar product
and the action of the basic observables can be given directly in terms of
loops.Comment: 28pp, 1 figure (postscript, compressed and uuencoded), TeX,
Pennsylvania State University, CGPG-94/5-