4,793 research outputs found
Enumeration of graphs with a heavy-tailed degree sequence
In this paper, we asymptotically enumerate graphs with a given degree
sequence d=(d_1,...,d_n) satisfying restrictions designed to permit
heavy-tailed sequences in the sparse case (i.e. where the average degree is
rather small). Our general result requires upper bounds on functions of M_k=
\sum_{i=1}^n [d_i]_k for a few small integers k\ge 1. Note that M_1 is simply
the total degree of the graphs. As special cases, we asymptotically enumerate
graphs with (i) degree sequences satisfying M_2=o(M_1^{ 9/8}); (ii) degree
sequences following a power law with parameter gamma>5/2; (iii) power-law
degree sequences that mimic independent power-law "degrees" with parameter
gamma>1+\sqrt{3}\approx 2.732; (iv) degree sequences following a certain
"long-tailed" power law; (v) certain bi-valued sequences. A previous result on
sparse graphs by McKay and the second author applies to a wide range of degree
sequences but requires Delta =o(M_1^{1/3}), where Delta is the maximum degree.
Our new result applies in some cases when Delta is only barely o(M_1^ {3/5}).
Case (i) above generalises a result of Janson which requires M_2=O(M_1) (and
hence M_1=O(n) and Delta=O(n^{1/2})). Cases (ii) and (iii) provide the first
asymptotic enumeration results applicable to degree sequences of real-world
networks following a power law, for which it has been empirically observed that
2<gamma<3.Comment: 34 page
Uniform generation of random graphs with power-law degree sequences
We give a linear-time algorithm that approximately uniformly generates a
random simple graph with a power-law degree sequence whose exponent is at least
2.8811. While sampling graphs with power-law degree sequence of exponent at
least 3 is fairly easy, and many samplers work efficiently in this case, the
problem becomes dramatically more difficult when the exponent drops below 3;
ours is the first provably practicable sampler for this case. We also show that
with an appropriate rejection scheme, our algorithm can be tuned into an exact
uniform sampler. The running time of the exact sampler is O(n^{2.107}) with
high probability, and O(n^{4.081}) in expectation.Comment: 50 page
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