## 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.

## 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.