Dr. Christian Mönch — Research

Research areas

Complex networks

  • scale-free networks
  • spatially extendend networks
  • directed random graphs
  • stochastic processes on (random) graphs

For an introduction to/ overview of this area, see this course taught in winter 2019/20 and the associated webpage.

Some current projects:

Structure and Dynamics of directed scale-free networks. This is a project within the DFG-Priority Programme SPP 22065 Random Geometric Systems. The goal is to investigate directed scale-free network models and stochastic processes on such networks. Considering directed networks, instead of undirected ones, adds additional layers of complexity to the models. The description of networks becomes considerably more involved, even locally, due to the appearance of arbitrary indegree-outdegree correlations. More importantly, the dynamics on directed networks are inherently irreversible, which renders many tools commonly used for the analysis of processes on networks ineffective. Therefore, mathematical results for
directed networks are scarce and the effects emerging from introducing directed edges are, in
general, poorly understood.

The aim of this project is to contribute to the mathematical theory of directed
networks, both from a structural perspective, i.e. in terms of network topology and percolation,
and from the complementary process perspective, i.e. regarding both dynamics on networks
and dynamical network formation.

Soft constraint models for DAG-Type distributed ledgers. This project is joint work with Amr Rizk (Universität Duisburg-Essen). We formulate a simple Directed Acyclic Graph model and study its asymptotic behaviour (connectedness, degree distribution, etc.) as the graph size diverges. Our aim is to illuminate the structural properties of large scale distributed ledger databases outside the classical Blockchain-setting.

Recurrence and transience in spatial scale-free networks (with Peter Gracar, Peter Mörters (both Universität zu Köln) and Markus Heydenreich (LMU München). Spatial scale-free networks combine the inhomogeneity present in many real world networks with classical percolation theory. We focus in this project on infinite limits of networks. Many instances of these limiting models display a connectivity phase transition just as in classical percolation: In the subcritical regime, all connected components of the network are finite, whereas in the supercritical regime there is a unique infinite component. Our goal is to obtain a definitive answer to the following question: For which instances of the underlying network does a random walk started in a given point of the infinite component never return to its starting point?

Resolute voter model (with Lisa Hartung and Florian Völlering (Universität Leipzig)). We study a variation of the classical voter model. The voters sit in the sites of a (large) graph and copy the opinion of a random neighbouring voter whenever their clock rings. However, unlike in the classical voter model, the distribution of the clock process depends on the voter: The rate at which a clock rings is itself the inverse of a heavy tailed random variable, i.e. there is an inhomogeneous population of irresolute voters changing their opinions all the time, and resolute voters who change their opinions very rarely. We investigate what effect this inhomogneity has on the fixation of the system.

Persistence probabilities

This area of probability theory is concerned with problems of the following type: Given a real valued stochastic process whose range almost surely contains (a dense subset of) the whole real line, how rare is the event that it stays, say, below a given threshold for a certain period of time? For Markov processes the answer is given by classical fluctuation theory, but for non-Markovian processes the picture is very far from completion.

For an introduction / survey of this field see this article by F. Aurzada & T. Simon.

Current Projects:

Strong order of the persistence probability of Fractional Brownian Motion. A well known result of G. Molchan states that the probability P(T) that standard fractional Brownian motion with Hurst index H does not hit a fixed level befor time T satisfies log(P(T))/log(T)~-(H-1)  as T gets large. Our aim is to prove that there is also a power function f such that P(T)/f(T) converges.

 

Preprints

  1. Law of Large Numbers for an elementary model of Self-organised Criticality, with Antal A. Járai and Lorenzo Taggi. Preprint, 2023.
  2. Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension, with Peter Gracar and Lukas Lüchtrath. Preprint, 2022.
  3. Self-similar co-ascent processes and Palm calculus. Preprint, 2019.
  4. Conditionally Poissonian random digraphs. Preprint, 2017.

Peer reviewed publications in journals

  1. Inhomogeneous long-range percolation in the weak decay regime. To appear in Probability Theory and Related Fields (2024+).
  2. DAG-type Distributed Ledgers via Young-age Preferential Attachment, with Amr Rizk. Stochastic Systems 13.3 (2023), pp. 377-397
  3. Recurrence versus transience for Weight-dependent Random Connection Models, with Peter Gracar, Markus Heydenreich, and Peter Mörters. Electronic Journal of Probability 27 (2022), paper no. 60, 31 pp. MR4417198
  4. Universality for persistence exponents of local times of self-similar processes with stationary increments. Journal of Theoretical Probability, 35 (2022), pp. 1842–1862. MR4488560
  5. Quenched invariance principle for random walks on dynamically averaging random conductances, with Stein Andreas Bethuelsen and Christian Hirsch. Electronic Communications in Probability 26 (2021), paper no. 69, 13 pp. MR4346873
  6. Distances and large deviations in the spatial preferential attachment model, with Christian Hirsch. Bernoulli 26.2 (2020), pp. 927--947. MR4058356
  7. Persistence probabilities and a decorrelation inequality for the Rosenblatt process and Hermite processes, with Frank Aurzada. Теория вероятностей и ее применения 63 (2018), pp. 817-826 and Theory of Probability and its Applications 63.4 (2019), pp. 664-670. MR3869634
  8. Distances in scale-free networks at criticality, with Steffen Dereich and Peter Mörters. Electronic Journal of Probability 22 (2017), paper no. 77, 38 pp. MR3710797
  9. Relations between L^p- and pointwise convergence of families of functions indexed by the unit interval, with Vaios Laschos. Real Analysis Exchange, 38.1 (2012/13) pp. 177–192. MR3083205
  10. Typical distances in ultrasmall random networks, with Steffen Dereich and Peter Mörters. Advances in Applied Probability, 44.2 (2012), pp. 583–601. MR2977409

Peer reviewed contributions to conferences and books

  1. The directed Age-dependent Random Connection Model with arc reciprocity, with Lukas Lüchtrath. To appear in Algorithms and Models for the Web Graph. WAW 2024. Lecture Notes in Computer Science, 2024.
  2. The emergence of a giant component in one-dimensional inhomogeneous networks with long-range effects, with Peter Gracar and Lukas Lüchtrath. Algorithms and Models for the Web Graph. WAW 2023. Lecture Notes in Computer Science, vol 13894, 2023.
  3. Transience Versus Recurrence for Scale-Free Spatial Networks, with Peter Gracar, Markus Heydenreich and Peter Mörters. Algorithms and Models for the Web Graph. WAW 2020. Lecture Notes in Computer Science, vol 12091, 2020.

Further publications

  1. Persistence of activity in critical scale free Boolean networks. Tagungsbericht/ extended abstract, Oberwolfach Report 12 (2015), pp. 2020–2023.
  2. Distances in preferential attachment networks. PhD thesis, December 2013, supervised by Prof. Peter Mörters, University of Bath.
  3. Large deviations for the empirical pair measure of tree indexed Markov chains. Diploma thesis, April 2009, supervised by Prof. Heinrich von Weizsäcker, Technische Universität Kaiserslautern.

 

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