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:

Inhomogenous long-range percolation beyond the weak decay regime. This is a project within the DFG-Priority Programme SPP 22065 Random Geometric Systems. The aim is to investigate the properties of infinite clusters in long-range percolation models with dependencies. This includes their topological properties, network metrics such as clustering coefficients etc. and the behaviour of stochastic process on them, for instance random walks or the contact process.

Structure and Dynamics of directed scale-free networks.  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.

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.

Current and former collaborators

Frank Aurzada (Darmstadt), Stein Andreas Bethuelsen (Bergen), Steffen Dereich (Münster), Peter Gracar (Leeds), Lisa Hartung (Mainz), Markus Heydenreich (Augsburg), Christian Hirsch (Aarhus), Benedikt Jahnel (Braunschweig/WIAS Berlin), Antal A. Járai (Bath), Andreas Klippel (Darmstadt), Vaios Laschos (WIAS Berlin), Ben Lees (Leeds), Lukas Lüchtrath (WIAS Berlin), Peter Mörters (Cologne), Amr Rizk (Hannover), Lorenzo Taggi (Rome), and Florian Völlering (Leipzig)

Publications

Preprints

  1. Loop vs. Bernoulli percolation on trees: strict inequality of critical values, with Andreas Klippel and Ben Lees. arXiv:2503.03319
  2. Phase transitions for contact processes on one-dimensional networks, with Benedikt Jahnel and Lukas Lüchtrath. arXiv:2501.16858
  3. Inhomogeneous long-range percolation in the strong decay regime: recurrence in one dimension. arXiv2408.06918
  4. The mean field stubborn voter model, with Lisa Hartung and Florian Völlering. arXiv:2405.08202
  5. The critical window in activated random walk on the complete graph, with Antal A. Járai and Lorenzo Taggi. arXiv2304.10169
  6. Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension, with Peter Gracar and Lukas Lüchtrath.  arXiv:2203.11966

Peer reviewed publications in journals

  1. A very short proof of Sidorenko’s inequality for counts of homomorphism between graphs, with Lukas Lüchtrath, to appear in Bulletin of the Australian Mathematical Society. arXiv:2408.01478
  2. Inhomogeneous long-range percolation in the weak decay regime. Probability Theory and Related Fields 189, 3-4 (2024), pp. 1129–1160. MR4771112
  3. Self-similar co-ascent processes and Palm calculus. Stochastic Processes and their Applications 174, (2024) paper no. 104378, 10 pp. MR4746578
  4. DAG-type Distributed Ledgers via Young-age Preferential Attachment, with Amr Rizk. Stochastic Systems 13.3 (2023), pp. 377-397. MR4650338
  5. 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
  6. Universality for persistence exponents of local times of self-similar processes with stationary increments. Journal of Theoretical Probability, 35 (2022), pp. 1842–1862. MR4488560
  7. 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
  8. Distances and large deviations in the spatial preferential attachment model, with Christian Hirsch. Bernoulli 26.2 (2020), pp. 927--947. MR4058356
  9. 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
  10. 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
  11. 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
  12. 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 book chapters

  1. The directed Age-dependent Random Connection Model with arc reciprocity, with Lukas Lüchtrath. Modelling and Mining Networks. 19th International Workshop, WAW 2024, Warsaw, Poland, June 3–6, 2024, Proceedings. Lecture Notes in Computer Science, vol 14671, 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. 18th International Workshop, WAW 2023, Toronto, ON, Canada, May 23–26, 2023, Proceedings. 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. 17th International Workshop, WAW 2020, Warsaw, Poland, September 21–22, 2020, Proceedings. Lecture Notes in Computer Science, vol 12091, 2020.

Working papers, extended abstracts and theses

  1. Law of Large Numbers for an elementary model of Self-organised Criticality, with Antal A. Járai and Lorenzo Taggi. Working paper, 2023.
  2. Conditionally Poissonian random digraphs. Working paper, 2017.
  3. Persistence of activity in critical scale free Boolean networks. Tagungsbericht/ extended abstract, Oberwolfach Report 12 (2015), pp. 2020–2023.
  4. Distances in preferential attachment networks. PhD thesis, December 2013, supervised by Prof. Peter Mörters, University of Bath.
  5. 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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