Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A probabilistic view of latent space graphs and phase transitions   We study random graphs with latent geometric structure, where the probability
of each edge depends on the underlying random positions corresponding to the
two endpoints. We focus on the setting where this conditional probability is a
general monotone increasing function of the inner product of two vectors; such
a function can naturally be viewed as the cumulative distribution function of
some independent random variable. We consider a one-parameter family of random
graphs, characterized by the variance of this random variable, that smoothly
interpolates between a random dot product graph and an Erd\H{o}s--R\'enyi
random graph. We prove phase transitions of detecting geometry in these graphs,
in terms of the dimension of the underlying geometric space and the variance
parameter of the conditional probability. When the dimension is high or the
variance is large, the graph is similar to an Erd\H{o}s--R\'enyi graph with the
same edge density that does not possess geometry; in other parameter regimes,
there is a computationally efficient signed triangle statistic that
distinguishes them. The proofs make use of information-theoretic inequalities
and concentration of measure phenomena.

37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio