Institute of Stochastics

Stochastic networks

Lecturer

Prof. Dr. Volker Schmidt

Exercises

Dipl-Math. Christian Hirsch

Time and place

Lecture
TBA

Exercise session
TBA

Type

2 hours lecture + 2 hours exercises

Credit points: TBA

Prerequisites

Probability Calculus

Intended Audience

Master students in "Mathematik", "Wirtschaftsmathematik" and "Mathematische Biometrie"; Master students in Finance

Contents

Stochastic networks and, in particular, random geometric graphs have been studied since more than 50 years. This part of stochastic geometry continues to produce intriguing questions that prove to be a fruitful topic of ongoing mathematical research. Another reason for the success of these networks is due to their importance in applications - at the macroscopic as well as the microscopic scale. Road systems, telecommunication and electricity networks, pore space and solid phase of advanced energy materials (used e.g. in Li-ion batteries, fuel/solar cells), as well as biological structures (such as cytoskeleton and keratin networks) can be modelled by appropriate types of geometric graphs.

Participants of this lecture will learn the basics of percolation theory both in the case of discrete as well as continuum state space. After being familiar with the important notion of phase transition, classical topics such as relationships between bond and site percolation, uniqueness of the infinite cluster or cluster size behaviour in the subcritical case will be treated and basic results on dependent percolation will be considered. A solid understanding of these topics allows us to proceed to the more complex case of continuum percolation, where many analogues of the previously encountered results can be derived. 

Later on we can proceed to more advanced topics such as Erdős–Rényi graphs, oriented percolation, first passage percolation, minimal spanning forests or (random) Euclidean optimization problems.

Requirements and Exam

In order to become accredited for the written exam, one has to earn 50% of all homework credits.

Problem sets

 

Lecture Notes

 

  • Non-percolation of an asymmetric lilypond model <link fileadmin website_uni_ulm mawi.inst.110 mitarbeiter hirsch lilypond.pdf download>pdf, arXiv

Papers in preparation

The following projects are close to completion; preprints available soon.

  • Prediction of regionalized insurance risks based on control variates, joint with M.C. Christiansen and V. Schmidt. We show how regional prediction of car insurance risks can be improved by combining explanatory modeling with phenomenological models from practice.  Motivated by the control variates technique,  we define a specific combined predictor. We  provide explicit conditions which imply that the mean squared error is smaller than the mean squared error of the standard predictor currently used in industry and smaller than predictors from explanatory modeling. We also discuss how a non-parametric random forest approach may be used to construct such predictors in practice and consider an application to German car insurance data.
  • Stationary Apollonian packings, joint with G.W. Delaney and V. Schmidt. We introduce stationary Apollonian packings as mathematical formalizations of so-called random Apollonian packings and rotational random Apollonian packings which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire euclidean space, we provide asymptotics of the grain diameter. Finally, we observe that grains arrange in clusters and deduce percolation-type properties.
  • Asymptotic properties of a dense packing algorithm, joint with G. Gaiselmann and V. Schmidt.We analyze asymptotic properties of collective-rearrangement algorithms being a class of dense packing algorithms. They transform finite systems of (potentially overlapping) particles into non-overlapping configurations by collective rearrangement of particles in finitely many steps.  We consider the effects of applying these algorithms to not-necessarily finite input data. More precisely, we derive sufficient conditions guaranteeing convergence of this algorithm when a stationary process of particles is used as input. We also provide numerical results and present an application of the algorithm in materials science.
  • First-passage percolation on random geometric graphs and an application to shortest-path trees, joint with D. Neuhäuser, C. Gloaguen and V. Schmidt. We consider Euclidean first-passage percolation on a large family of connected fibre processes in $\R^d$ encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on fibre processes which are generated by point processes. This result comprehends two special cases which are of independent interest. In particular, we consider the event that the growth of shortest-path lengths between two (end-) points of the path does not admit a linear upper bound, where our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Moreover, we get that the shortest-path length between two points at fixed distance admits a sub-exponential tail.
    For a wide class of stationary and isotropic fibre processes in $\R^d$, we show that our linear growth property implies a shape theorem for the Euclidean first-passage model defined by such fibre processes.
    Finally we show how our shape theorem can be
    used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in a typical shortest-path tree if the intensity of network stations converges to zero.

 

Literature

 

Kontakt

Lecturer

  • Office hours on appointment
  • Phone: +49 (0)731/50-23527
  • Homepage

Teaching assistant

  • Office hours on appointment
  • Phone: +49 (0)731/50-31083
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