Abstract not found

Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the non-negative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for large-scale real-world networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially bounded kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasi-random graphs are in a

The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph Gnm are shown to be asymptotically normal if m is neither too large nor too small. At the lower limit m n 3=2 , these numbers are asymptotically log-normal. For Gnp , the numbers are asymptotically log-normal for a wide range of p, including p constant. The same results are obtained for random directed graphs and bipartite graphs. The results are proved using decomposition and projection methods.

Let G be a fixed graph and let XG be the number of copies of G contained in the random graph G(n, p). We prove exponential bounds on the upper tail of XG which are best possible up to a logarithmic factor in the exponent. Our argument relies on an extension of Alon's result about the maximum number of copies of G in a graph with a given number of edges. Similar bounds are proved for the random graph G(n, M) too.

Abstract. Let Tn denote the set of unrooted labeled trees of size n and let M be a particular (finite, unlabeled) tree. Assuming that every tree of Tn is equally likely, it is shown that the limiting distribution as n goes to infinity of the number of occurrences of M as an induced subtree is asymptotically normal with mean value and variance asymptotically equivalent to µn and σ 2 n, respectively, where the constants µ> 0 and σ ≥ 0 are computable. 1.

Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − 1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs. 1

Let Tn denote the set of unrooted unlabeled trees of size n and let M be a particular (finite) tree. Assuming that every tree of Tn is equally likely, it is shown that the number of occurrences Xn of M as an induced sub-tree satisfies E Xn ∼ µn and Var Xn ∼ σ 2 n for some (computable) constants µ> 0 and σ ≥ 0. Furthermore, if σ> 0 then (Xn − E Xn) / √ Var Xn converges to a limiting distribution with density (A + Bt 2)e −Ct2 for some constants A, B, C. However, in all cases in which we were able to calculate these constants, we obtained B = 0 and thus a normal distribution. Further, if we consider planted or rooted trees instead of Tn then the limiting distribution is always normal. Similar results can be proved for planar, labeled and simply generated trees.

Abstract not found

Abstract: We prove a moderate deviation principle for subgraph count statistics of Erdős-Rényi random graphs. This is equivalent in showing a moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an estimation of the log-Laplace transform and the Gärtner-Ellis theorem. We obtain upper bounds on the upper tail probabilities of the number of occurrences of small subgraphs. The method of proof is used to show supplemental moderate deviation principles for a class of symmetric statistics, including non-degenerate U-statistics with independent or Markovian entries. 1.

Let X be the random variable that counts the number of triangles in the random graph G(n, p). We show that for some absolute constant c, the probability that X deviates from its expectation by at least λVar(X) 1/2 is at most e−cλ2, provided that n−1 (lnn) 10 ≤ p ≤ n−1/2 (ln n) −10, λ = ω(lnn) and λ ≤ min{(np) 1/2, n−3/4p−3/2, n1/6}. 1