95 research outputs found
Contractors for flows
We answer a question raised by Lov\'asz and B. Szegedy [Contractors and
connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a
contractor for the graph parameter counting the number of B-flows of a graph,
where B is a subset of a finite Abelian group closed under inverses. We prove
our main result using the duality between flows and tensions and finite Fourier
analysis. We exhibit several examples of contractors for B-flows, which are of
interest in relation to the family of B-flow conjectures formulated by Tutte,
Fulkerson, Jaeger, and others.Comment: 22 pages, 1 figur
Generalized quasirandom graphs
AbstractWe prove that if a sequence of graphs has (asymptotically) the same distribution of small subgraphs as a generalized random graph modeled on a fixed weighted graph H, then these graphs have a structure that is asymptotically the same as the structure of H. Furthermore, it suffices to require this for a finite number of subgraphs, whose number and size is bounded by a function of |V(H)|
On the number of B-flows of a graph
We exhibit explicit constructions of contractors for the graph parameter counting the number of B-flows
of a graph, where B is a subset of a finite Abelian group closed under inverses. These constructions are of
great interest because of their relevance to the family of B-flow conjectures formulated by Tutte, Fulkerson,
Jaeger, and others.Junta de Andalucía FQM-016
The large deviation principle for the Erd\H{o}s-R\'enyi random graph
What does an Erdos-Renyi graph look like when a rare event happens? This
paper answers this question when p is fixed and n tends to infinity by
establishing a large deviation principle under an appropriate topology. The
formulation and proof of the main result uses the recent development of the
theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma
from graph theory. As a basic application of the general principle, we work out
large deviations for the number of triangles in G(n,p). Surprisingly, even this
simple example yields an interesting double phase transition.Comment: 24 pages. To appear in European J. Comb. (special issue on graph
limits
Estimating and understanding exponential random graph models
We introduce a method for the theoretical analysis of exponential random
graph models. The method is based on a large-deviations approximation to the
normalizing constant shown to be consistent using theory developed by
Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000-1017]. The theory
explains a host of difficulties encountered by applied workers: many distinct
models have essentially the same MLE, rendering the problems ``practically''
ill-posed. We give the first rigorous proofs of ``degeneracy'' observed in
these models. Here, almost all graphs have essentially no edges or are
essentially complete. We supplement recent work of Bhamidi, Bresler and Sly
[2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS)
(2008) 803-812 IEEE] showing that for many models, the extra sufficient
statistics are useless: most realizations look like the results of a simple
Erd\H{o}s-R\'{e}nyi model. We also find classes of models where the limiting
graphs differ from Erd\H{o}s-R\'{e}nyi graphs. A limitation of our approach,
inherited from the limitation of graph limit theory, is that it works only for
dense graphs.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1155 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Graph limits and exchangeable random graphs
We develop a clear connection between deFinetti's theorem for exchangeable
arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph
limits (work of Lovasz and many coauthors). Along the way, we translate the
graph theory into more classical probability.Comment: 26 page
Bounds on the mod 2 homology of random 2-dimensional simplicial complexes
As a first step towards a conjecture of Kahle and Newman, we prove that if
is a random -dimensional determinantal hypertree on vertices, then
converges to zero in probability.
Confirming a conjecture of Linial and Peled, we also prove the analogous
statement for the -out -complex.
Our proof relies on the large deviation principle for the Erd\H{o}s-R\'enyi
random graph by Chatterjee and Varadhan
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