129,445 research outputs found
Synchronization in random networks with given expected degree sequences
Synchronization in random networks with given expected degree sequences is studied. We also investigate in details the synchronization in networks whose topology is described by classical random graphs, power-law random graphs and hybrid graphs when N goes to infinity. In particular, we show that random graphs almost surely synchronize. We also show that adding small number of global edges to a local graph makes the corresponding hybrid graph to synchroniz
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
Limits of Random Trees. II
Local convergence of bounded degree graphs was introduced by Benjamini and Schramm. This result was extended further by Lyons to bounded average degree graphs. In this paper we study the convergence of random tree sequences with given degree distributions. Denote by (Formula presented.) the set of possible degree sequences of a labeled tree on n nodes. Let Dn be a random variable on (Formula presented.) and T(Dn) be a uniform random labeled tree with degree sequence Dn. We show that the sequence T(Dn) converges in probability if and only if (Formula presented.), where (Formula presented.) and D(1) is a random variable on (Formula presented.)
Push is Fast on Sparse Random Graphs
We consider the classical push broadcast process on a large class of sparse
random multigraphs that includes random power law graphs and multigraphs. Our
analysis shows that for every , whp rounds are
sufficient to inform all but an -fraction of the vertices.
It is not hard to see that, e.g. for random power law graphs, the push
process needs whp rounds to inform all vertices. Fountoulakis,
Panagiotou and Sauerwald proved that for random graphs that have power law
degree sequences with , the push-pull protocol needs
to inform all but vertices whp. Our result demonstrates that,
for such random graphs, the pull mechanism does not (asymptotically) improve
the running time. This is surprising as it is known that, on random power law
graphs with , push-pull is exponentially faster than pull
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