Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Transitions in loopy random graphs with fixed degrees and arbitrary
  degree distributions   We analyze maximum entropy random graph ensembles with constrained degrees,
drawn from arbitrary degree distributions, and a tuneable number of 3-loops
(triangles). We find that such ensembles generally exhibit two transitions, a
clustering and a shattering transition, separating three distinct regimes. At
the clustering transition, the graphs change from typically having only
isolated loops to forming loop clusters. At the shattering transition the
graphs break up into extensively many small cliques to achieve the desired loop
density. The locations of both transitions depend nontrivially on the system
size. We derive a general formula for the loop density in the regime of
isolated loops, for graphs with degree distributions that have finite and
second moments. For bounded degree distributions we present further analytical
results on loop densities and phase transition locations, which, while
non-rigorous, are all validated via MCMC sampling simulations. We show that the
shattering transition is of an entropic nature, occurring for all loop density
values, provided the system is large enough.

19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
2
The categories of this paper are: math
13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th