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Clustering in a hyperbolic model of complex networksVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

In this talk we consider the clustering coefficient and clustering function in a random graph model proposed by Krioukov et al.~in 2010. In this random graph model, vertices are chosen randomly inside a disk in the hyperbolic plane and two vertices 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 non-vanishing clustering coefficient. The model is specified using three parameters: the number of vertices $n$, which we think of as going to infinity, and $\alpha, \nu > 0$, which we think of as constant. Roughly speaking, the parameter $\alpha$ controls the power-law exponent of the degree distribution and $\nu$ the average degree.

Our results 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 over 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 $\gamma(k)$. 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, $\gamma(k) = \Theta( k^{2 - 4\alpha})$ if $\frac{1}{2} < \alpha < \frac{3}{4}$, $\gamma(k) = \Theta\left( \frac{\log(k)}{k} \right)$ if $\alpha=\frac{3}{4}$ and $\gamma = \Theta\left(k^{-1}\right)$ if $\alpha > \frac{3}{4}$, as $k \rightarrow \infty$. These results contradict a previous claim by Krioukov et al., which stated that the limiting values $\gamma(k)$ should always scale with $k^{-1}$ as we let $k$ grow.

(joint work with: Nikolaos Fountoulakis, Pim van der Hoorn, Tobias M\" uller)

Analysing neural population activity to study information processing in the brainVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

Information in the brain is processed along a cascade of activity that propagates through a network of neurons. Insights into neural information processing can be gained from studying population activity that occurs spontaneously, and in response to sensory stimulation. In the first part of the talk, I will discuss a new way to analyse spontaneous activity. A prominent feature of this activity is the occurrence of spatiotemporal patterns or "neural assemblies". In order to study the properties of these assemblies it is crucial to first be able to reliably identify them, even in the presence of considerable levels of noise. I will present an algorithm to detect neural assemblies in population activity recorded using fluorescent calcium imaging. In this algorithm, the problem is translated into one of graph clustering, to which statistical inference methods can be applied. I will then compare the performance of this algorithm against other related approaches in the field. In the second part of the talk I will discuss the analysis of evoked activity, for which we can use information theory to characterise sensory processing. I will introduce a simple model of sensory processing and demonstrate through a combinatorial argument that there are fundamental limitations to estimating mutual information in large neural populations.

Extending the Scope of Geometric Singular Perturbation TheoryVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

Multi-scale phenomena in e.g. biology, engineering, and neuroscience are frequently described by singularly perturbed ordinary differential equations with solutions varying over vastly separated timescales, making their analysis a challenging problem. In recent decades, significant progress has been made via the development of Geometric Singular Perturbation Theory (GSPT), which provides a powerful theoretical framework for the analysis of such problems. When combined with a geometric method for the desingularization of singularities known as blow-up, GSPT can provide a remarkably detailed and geometrically informative understanding of the dynamics. However, GSPT in its standard form has a number of limitations which restrict the scope of its applicability. Discontinuity, exponential nonlinearities and 'hidden scales' all present significant challenges to the theory. In this talk, we will explore these limitations in the context of a simple electrical oscillator model - the Le Corbeiller oscillator - and show how they can be overcome using a combination of tools that are adapted from GSPT and the theory of piecewise-smooth (PWS) systems. Our aim will be to understand the onset of multi-scale 'relaxation oscillations' which converge to PWS cycles as a singular perturbation parameter tends to zero. Our main analytical tool is the blow-up method, which must be adapted to resolve degeneracy stemming from (i) the loss of smoothness and (ii) the presence of an essential singularity. The talk will conclude with a brief discussion on the systematic implementation of these ideas, which have so far been developed only in the context of applications.

The role of networks and stochasticity in the long-term behavior of epidemic modelsVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

In the spread of epidemics, the connections between individual units (e.g. people, city, etc.) clearly determine the possible routes of disease transmission. The use of networks to describe these interactions patterns represents a major advance in our ability to model more realistic social behaviors. In theoretical network-based models, individuals are represented by nodes, with edges (or links) depicting the interactions between nodes. These models provide a useful tool for understanding the impact of network metrics on outbreak threshold, final epidemic size, and the efficacy of control measures. Moreover, since the epidemic spreading is governed by probabilistic processes, the analysis of epidemic models should also consider its stochastic nature.

In this talk, we analyze some individual-based epidemic models and show how particular regularities in the network connectivity allows us to provide a global stability analysis by means of a lower-dimensional system than the starting one. We also consider the introduction of stochastic fluctuations by considering some model parameters as stochastic processes. We show sufficient conditions that ensure almost surely extinction, and the stochastic permanent of the system.

Remedying non-smoothness in the bifurcation analysis of a stochastically driven limit cycleVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

We consider stochastic differential equations with Sobolev diffusion coefficients with one-dimensional noise and regular drifts in terms of weak differentiability with respect to starting points, in order to derive the bifurcation analysis of a stochastically driven limit cycle.

Models for coupled active--passive population dynamics: mathematical analysis and simulationVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

We study models for coupled active--passive pedestrian dynamics from mathematical analysis and simulation perspectives. This work comes in three main parts, in which I adopt distinct perspectives and conceptually different tools from lattice gas models, partial differential equations, and stochastic differential equations, respectively. In part one, we introduce two lattice models for active--passive pedestrian dynamics. In a first model, using descriptions based on the simple exclusion process, we study the dynamics of pedestrian escape from an obscure room in a lattice domain with two species of particles (pedestrians). In a second model, we consider again a microscopic approach based on a modification of the simple exclusion process formulated for active--passive populations of interacting pedestrians. The model describes a scenario where pedestrians are walking in a built environment and enter a room from two opposite sides.

In part two, we study a fluid-like driven system modeling active--passive pedestrian dynamics in a heterogenous domain. We prove the well-posedness of a nonlinear coupled parabolic system that models the evolution of the complex pedestrian flow by using special energy estimates, a Schauder's fixed point argument and the properties of the nonlinearity's structure. In the third part, we describe via a coupled nonlinear system of Skorohod-like stochastic differential equations the dynamics of active--passive pedestrians dynamics through a heterogenous domain in the presence of fire and smoke. We prove the existence and uniqueness of strong solutions to our model when reflecting boundary conditions are imposed on the boundaries. Furthermore, we study an homogenization setting for a toy model (a semi-linear elliptic equation) where later on our pedestrian models can be studied.

This is joint work with Emilio Cirillo (Rome, Italy), Matteo Colangeli (L’Aquila, Italy) and Adrian Muntean (Karlstad, Sweden).

From a Kac-like particle system to the homogeneous Boltzmann equation for hard potentialsVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

In this talk, we consider the Kac stochastic particle system associated to the spatially homogeneous Boltzmann equation for true hard potentials. We establish a rate of propagation of chaos of the particle system to the unique solution of the Boltzmann equation. More precisely, we estimate the expectation of the squaredWasserstein distance with quadratic cost between the empirical measure of the particle system and the solution to the Boltzmann equation.

A Stochastic Gronwall Lemma and Well-Posedness of Path-Dependent SDEs Driven by Martingale NoiseVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

We show the existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the supremum norm. In this set-up, the usual proof using the ordinary Gronwall lemma together with the Burkholder-Davis-Gundy inequality seems impossible. In order to solve this problem, we present a new and quite general stochastic Gronwall lemma for cadlag martingales using Lenglart's inequality.

Partial Knowledge Differential Privacy and Its Application to Counting Queries(using zoom) (Theresienstr. 39, 80333 München)

Differential privacy offers formal quantitative guarantees for algorithms over datasets, but it assumes attackers that know and can influence all but one record in the database. This assumption often vastly over-approximates the attackers’ actual strength, resulting in unnecessarily poor utility or unnecessarily poor privacy guarantees. In my talk I derive the recent definitions of Active and Passive Partial Knowledge Differential Privacy to systematically characterize attackers with limited influence or only partial background knowledge over the dataset. Relations between the definitions as well as important properties are presented. To show their relevance, these foundations are then used to analyze the privacy guarantees of counting queries.