We consider a geometric preferential attachment model of random graphs, and its links to other forms of preferential attachment models, especially preferential attachment with multiplicative fitness. We will discuss both known results and open questions about our main model.

Random mapping models have been studied by various authors since the 1950's and have applications in modelling epidemic processes, the analysis of cryptographic systems (e.g. DES) and of Pollard's algorithm, and random number generation. In this talk we consider random mappings from a new perspective which is inspired, in part, by results for preferential and anti-preferential attachment in other random graph models. Our work shows that both the usual uniform random mapping model and other models (e.g. random mappings with preferential and anti-preferential attachment) are special cases of random mappings with exchangeable in-degrees. It turns out that by viewing random mappings from this perspective, questions related to their asymptotic structure can be tackled by using a new calculus that is based on the moments of the joint distribution of the exchangeable in-degree sequence of the vertices in the (directed) graphical representation of the random mapping. This calculus gives us tools to tackle questions about the component structure of a random mapping which would be more difficult to attack using classical combinatorial approaches such as generating function arguments. In this talk we give an overview of the development of this calculus and of the results which can be obtained using it. In addition, we will explore some natural and attractive connections between random mappings with exchangeable in-degrees and various urn schemes.

This talk is based on joint work with Jerzy Jaworski (Adam Mickiewicz University), who was supported by the Marie Curie Intra-European Fellowship No. 236845 (RANDOMAPP) within the 7th European Community Framework Programme.

We study a class of interacting particle systems on the two-dimensional lattice (with unbounded number of individuals per site) whose dynamics imitates behaviour of ecological populations, as described within the Bolker-Dieckmann-Law-Pacala approach. By stochastically comparing trajectories of the resulting process to a suitably defined subcritical site percolation model in $\mathbb{Z}_+\times\mathbb{Z}^2$, we show that, given regular initial conditions, the behaviour of such BDLP processes is stable in the whole region of biologically relevant parameters.

Based on a joint work with Yu.Kondratiev (Bielefeld), O.Kutovyi (MIT / Bielefeld) and S.Molchanov (UNCC).

The biochemical and molecular mechanisms underlying epistatic phenomena observed in various living organisms are poorly understood. Epistasis, or genetic interactions, refers to functional relationships between genes. It describes the phenotypic effect of perturbing (e.g., knocking down or knocking out) two genes separately versus jointly relative to the unperturbed system. Thus, epistasis is a property of the underlying network of biochemical interactions in the cell.

Interacting biological or biochemical entities are often represented as networks (or graphs), where vertices correspond to components (e.g., genes, proteins, or metabolites) and edges correspond to pairwise interactions (e.g., activation, molecular binding, or chemical reaction). This abstract representation provides the conceptual basis for network biology, which aims at understanding the cell's functional organization and the complex behavior of living systems through biological network analysis. In this work, we introduce a mathematical framework linking epistatic gene interactions to the redundancy of biological networks. Our approach is based on network reliability, an engineering concept that allows for computing the probability of functional network operation under di¤erent network perturbations, such as the failure of specific components, which, in a genetic system, correspond to the knock-out or knock-down of specific genes. Using this framework, we provide a formal definition of epistasis in terms of network reliability and we show how this concept can be used to infer functional constraints in biological networks from observed genetic interactions.

In this talk, we will introduce the concept of epistasis on networks within the framework of proba- bilistic graphs. Furthermore, we will present some basic mathematical properties relating redundancy of the network under consideration and epistasis. Moreover, we will demonstrate, using a concrete experimental data set, how our methodology can be used to infer functional and topological constraints in biological networks from observed genetic interactions. Our formalism might help increase our understanding of the systemic properties of the cell that give rise to observed epistatic patterns.