Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Majority Dynamics: The Power of One   Consider $n=\ell+m$ individuals, where $\ell\le m$, with $\ell$ individuals
holding an opinion $A$ and $m$ holding an opinion $B$. Suppose that the
individuals communicate via an undirected network $G$, and in each time step,
each individual updates her opinion according to a majority rule (that is,
according to the opinion of the majority of the individuals she can communicate
with in the network). This simple and well studied process is known as
"majority dynamics in social networks". Here we consider the case where $G$ is
a random network, sampled from the binomial model $\mathbb{G}(n,p)$, where
$(\log n)^{-1/16}\le p\le 1-(\log n)^{-1/16}$. We show that for $n=\ell+m$ with
$\Delta=m-\ell\le(\log n)^{1/4}$, the above process terminates whp after three
steps when a consensus is reached. Furthermore, we calculate the
(asymptotically) correct probability for opinion $B$ to "win" and show it is
\[\Phi\bigg(\frac{p\Delta\sqrt{2}}{\sqrt{\pi p(1-p)}}\bigg) + O(n^{-c}),\]
where $\Phi$ is the Gaussian CDF. This answers two conjectures of Tran and Vu
and also a question raised by Berkowitz and Devlin.
  The proof technique involves iterated degree revelation and analysis of the
resulting degree-constrained random graph models via graph enumeration
techniques of McKay and Wormald as well as Canfield, Greenhill, and McKay.

36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs