Distances in scale free networks: between small and ultrasmall

A lot of the properties of large scale free networks are related to the power law exponent \tau of their degree sequence. For instance, if \tau\in(2,3), then supercritical networks are ultrasmall, i.e. vertex distances are typically of order \log\log N, where N denotes Network size. If \tau\in (3,\infty), however typical distances are of logarithmic order (this is the so-called `small word phenomenon’). This article is concerned with the situation for \tau=3 and the difference behaviour in this case depending on whether there is preferential attachment in the network or not.

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Gaussian correlation inequality and persistence of the Rosenblatt process

Recently, Thomas Royen resolved the long standing Gaussian Correlation Conjecture, stating that P(C_1\cap C_2)\geq P(C_1)P(C_2) for any Gaussian measure P on \mathbb{R}^n and any pair C_1,C_2 of convex, centrally symmetric subsets of \mathbb{R}^n. In combination with general results about persistence of stationary increment processes the inequality enabled us to determine the persistence exponent of certain long range correlated non-Gaussian processes. The detailed argument and some considerations related to the persistence problem for a more general class of processes can be found here.

At the Probl@b in Bath

Here is a corrected version of the slides for my talk at the Probl@b Seminar in Bath on 19 March. The talk also included some material about recent work on scale free preferential attachment networks with “critical” power law exponent \tau=3.

Stochastiktage 2012 in Mainz

I am going to speak about random networks (in particular about doubly logarithmic distances in power law random graphs) at the German Probability and Stochastic Days in Mainz (06/03 – 09/03 2012) in the section about random discrete structures and analysis of algorithms. Here are the presentation slides.

Relations between p-mean and pointwise convergence

Every introductory course on integration theory contains an example of a sequence (f_i)_{i\in\mathbb{N}} of functions which converge in the p-th mean but not pointwise. What happens if instead of a sequence one considers a curve f_t,t\in [0,1] of functions? More precisely, is there a continuous (with respect to \int|\cdot|^p d\lambda, where \lambda is Lebesgue measure) curve of functions which is not “pointwise continuous”, i.e. does not allow  pointwise approximation f_{s}(x)\rightarrow f_t(x) as s\rightarrow t at all? Here is a partial answer, obtained together with Vaios Laschos.

Random Networks – Presentation Slides

Here are some links to (old) presentations I gave about distances in random network. In March 2010 I spoke in the Problab Seminar Series in Bath, the slides provide some introductory material about Random Networks. In October 2011, I contributed a talk to the Summer School of the BMS on Random Graphs and Random Motions, entitled “Typical Distances in Ultrasmall Random Networks“.

Typical Distances in Ultrasmall Random Networks

This is joint work with my PhD supervisor Peter Mörters and Steffen Dereich. We show that in preferential attachment models with power-law exponent \tau\in(2,3) the distance between
randomly chosen vertices in the giant component is asymptotically equal to
(4+o(1))\, \frac{\log\log N}{-\log (\tau-2)}, where N denotes the number of nodes. This is
twice the value obtained for the configuration model with the same
power-law exponent.  The extra factor reveals the different structure of typical
shortest paths in preferential attachment graphs.

UPDATE: The article has appeared in Advances in Applied Probability.