26,178 research outputs found
Quasi-randomness and algorithmic regularity for graphs with general degree distributions
We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to express how much a given graph āresemblesā a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we introduce a concept of regularity that takes into account vertex weights, and show that if satisfies a certain boundedness condition, then admits a regular partition. In addition, building on the work of Alon and Naor [Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, ACM, New York, 2004, pp. 72ā80], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph in polynomial time. As an application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs without ādense spots.
Asymptotics for Sparse Exponential Random Graph Models
We study the asymptotics for sparse exponential random graph models where the
parameters may depend on the number of vertices of the graph. We obtain exact
estimates for the mean and variance of the limiting probability distribution
and the limiting log partition function of the edge-(single)-star model. They
are in sharp contrast to the corresponding asymptotics in dense exponential
random graph models. Similar analysis is done for directed sparse exponential
random graph models parametrized by edges and multiple outward stars.Comment: 20 page
Right-convergence of sparse random graphs
The paper is devoted to the problem of establishing right-convergence of
sparse random graphs. This concerns the convergence of the logarithm of number
of homomorphisms from graphs or hyper-graphs \G_N, N\ge 1 to some target
graph . The theory of dense graph convergence, including random dense
graphs, is now well understood, but its counterpart for sparse random graphs
presents some fundamental difficulties. Phrased in the statistical physics
terminology, the issue is the existence of the log-partition function limits,
also known as free energy limits, appropriately normalized for the Gibbs
distribution associated with . In this paper we prove that the sequence of
sparse \ER graphs is right-converging when the tensor product associated with
the target graph satisfies certain convexity property. We treat the case of
discrete and continuous target graphs . The latter case allows us to prove a
special case of Talagrand's recent conjecture (more accurately stated as level
III Research Problem 6.7.2 in his recent book), concerning the existence of the
limit of the measure of a set obtained from by intersecting it with
linearly in many subsets, generated according to some common probability
law.
Our proof is based on the interpolation technique, introduced first by Guerra
and Toninelli and developed further in a series of papers. Specifically, Bayati
et al establish the right-convergence property for Erdos-Renyi graphs for some
special cases of . In this paper most of the results in this paper follow as
a special case of our main theorem.Comment: 22 page
On the strong chromatic number of random graphs
Let G be a graph with n vertices, and let k be an integer dividing n. G is
said to be strongly k-colorable if for every partition of V(G) into disjoint
sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex
k-coloring of G with each color appearing exactly once in each V_i. In the case
when k does not divide n, G is defined to be strongly k-colorable if the graph
obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly
k-colorable. The strong chromatic number of G is the minimum k for which G is
strongly k-colorable. In this paper, we study the behavior of this parameter
for the random graph G(n, p). In the dense case when p >> n^{-1/3}, we prove
that the strong chromatic number is a.s. concentrated on one value \Delta+1,
where \Delta is the maximum degree of the graph. We also obtain several weaker
results for sparse random graphs.Comment: 16 page
An theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
We introduce and develop a theory of limits for sequences of sparse graphs
based on graphons, which generalizes both the existing theory
of dense graph limits and its extension by Bollob\'as and Riordan to sparse
graphs without dense spots. In doing so, we replace the no dense spots
hypothesis with weaker assumptions, which allow us to analyze graphs with power
law degree distributions. This gives the first broadly applicable limit theory
for sparse graphs with unbounded average degrees. In this paper, we lay the
foundations of the theory of graphons, characterize convergence, and
develop corresponding random graph models, while we prove the equivalence of
several alternative metrics in a companion paper.Comment: 44 page
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