arXiv subjects

Markus Schepers

Publications and source records attributed to Markus Schepers.

4 recordsLinked to original sources

arXiv2026

Blinded sample size review for McNemar's test based on primary and surrogate endpoints

We develop blinded sample size re-estimation strategies for McNemar's test based on paired binary primary and secondary short-term surrogate endpoints. The development is motivated by a prospective randomized clinical trial on childhood glaucoma. A conditional power expression for McNemar's test given the primary endpoint at an interim analysis is derived and complemented by a sample size re-estimation rule. We show that this procedure preserves the type I error rate while allowing the second-stage sample size to be chosen to attain a prespecified target power. In the case where for some patients only a short-term surrogate endpoint is available at interim, we introduce a surrogate-based re-estimation approach that conditions on all possible numbers of primary-endpoint discordant pairs using transition rates from the surrogate to the primary outcome. We show how these transition rates can be estimated from data on a subsample for which both surrogate and primary endpoint are available. We derive the resulting surrogate endpoint-based conditional power and sample size rule and illustrate their use with the example of the motivating trial.

stat.ME↗

arXiv2022

Cover and Hitting Times of Hyperbolic Random Graphs

We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range $(2,3)$. In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is $n(\log n)^2$, the maximum hitting time is $n\log n$, and the average hitting time is $n$. We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in $n$, and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest-like structure.

math.PR↗

arXiv2020

Clustering in a hyperbolic model of complex networks

In this paper we consider the clustering coefficient and clustering function in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been shown that this model has various properties associated with complex networks, e.g. power-law degree distribution, short distances and non-vanishing clustering coefficient. Here we show that the clustering coefficient tends in probability to a constant $\gamma$ that we give explicitly as a closed form expression in terms of $\alpha, \nu$ and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that $c(k)$, the average clustering coefficient over all vertices of degree exactly $k$, tends in probability to a limit $\gamma(k)$ which we give explicitly as a closed form expression in terms of $\alpha, \nu$ and certain special functions. We are able to extend this last result also to sequences $(k_n)_n$ where $k_n$ grows as a function of $n$. Our results show that $\gamma(k)$ scales differently, as $k$ grows, for different ranges of $\alpha$. More precisely, there exists constants $c_{\alpha,\nu}$ depending on $\alpha$ and $\nu$, such that as $k \to \infty$, $\gamma(k) \sim c_{\alpha,\nu} \cdot k^{2 - 4\alpha}$ if $\frac{1}{2} < \alpha < \frac{3}{4}$, $\gamma(k) \sim c_{\alpha,\nu} \cdot \log(k) \cdot k^{-1} $ if $\alpha=\frac{3}{4}$ and $\gamma(k) \sim c_{\alpha,\nu} \cdot k^{-1}$ when $\alpha > \frac{3}{4}$. These results contradict a claim of Krioukov et al., which stated that the limiting values $\gamma(k)$ should always scale with $k^{-1}$ as we let $k$ grow.

math.PR↗

arXiv2019

Hamilton cycles and perfect matchings in the KPKVB model

In this paper we consider the existence of Hamilton cycles and perfect matchings in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, "short distances" and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes $n$, which we think of as going to infinity, and $\alpha, \nu > 0$, which we think of as constant. Roughly speaking $\alpha$ controls the power law exponent of the degree sequence and $\nu$ the average degree. Here we show that for every $\alpha < 1/2$ and $\nu=\nu(\alpha)$ sufficiently small, the model does not contain a perfect matching with high probability, whereas for every $\alpha < 1/2$ and $\nu=\nu(\alpha)$ sufficiently large, the model contains a Hamilton cycle with high probability.

math.PR↗