Natural selection begets adaptation through the exponential amplification of initially rare, highly fit lineages. As such, it bears strong ressemblance with topics such as extreme value theory (dominance of extreme events) and coarsening dynamics (dominance of large structures); like them, natural selection mathematical theory is structured by a basic limit theorem. I will present that theorem and discuss its potential application to detect statistical signatures of selection in empirical data.

With access to more than 40 million chemical reactions from 1779 up to the present, which account for more than 20 million substances; historical and fundamental questions of chemistry can be addressed in a mathematical framework. The first question is on the nature of the network $N$ resulting from linking chemical reactions and on its structure and characterisation. Here some results on early explorations of parts of $N$ will be discussed, which address the issue of representing reactions in terms of graphs by highlighting the importance of hypergraphs and of directed bipartite graphs. A novel network curvature approach and its possible application and adaptation to analyse $N$ will be discussed. Results on how to mathematically approach the reduction of $N$ and its associated graph $G$ to a network with graph $G'$ relating classes of similar substances will be shown, where elements of category theory and partially ordered sets in the form of contexts of Formal Concept Analysis are highlighted.Delving into the molecular description level for substances, a reaction is the transformation of educts into products, which is understood as the graph rewrite rule (graph grammar) of the graph of educts to the corresponding one of products. It will be shown how graph grammars work and it will be discussed an approach to look for new rules in $N$. The partially ordering of rules will be discussed as an approach to find central grammars for chemistry. Finally, the historical aspect of $N$ and its temporal change will be stressed as a mathematical and computational approach to the history of chemical reactivity.

The notion of ‘genetic distance’ between individuals and between populations plays a central role in population genetic analysis of diversity and differentiation. It has also been a focal point of some recent interest as empirical and theoretical studies have often made somewhat contradictory claims. I will trace the source of some of the confusion and suggest new perspectives for capturing the relevance of genetic distances to work on ‘ancestry inference’ and to models of phenotypic distance, and present a model that allows assessing the veracity of conclusions stemming from empirical analysis. In particular, I will show how the inferential leaps from genetic to phenotypic distance in the study of ‘population structure’ is hardly straightforward, and is highly contingent on simplifying assumptions with regard to the quantitative trait model and the putative selection pressure on the target trait.

Inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs making the joint distribution Markovian. Here we argue why causal inference is also possible when only single observations are present.
We state that similarities between two objects x and y indicate a causal link whenever their algorithmic mutual information is sufficiently high. It is defined as the number of bits that can be saved when optimally compressing the pair (x,y) jointly compared to compressing them independently. To infer causal graphs among n objects, we replace the notion of conditional *stochastic* independence in the causal Markov condition with the one of conditional *algorithmic* mutual information and describe the corresponding causal inference rules.
In contrast to causal inference methods that rely on statistical dependences, our theory implies rules for distinguishing between the causal hypotheses X --> Y and Y --> X for two random variables X,Y. This is because the causal graphs relating the individual observations of the statistical ensemble induce different sets of algorithmic independences for the two cases. Therefore, our theory provides a foundation for a new type of methods for inferring causal structure from statistical data.
For details see our recent preprint http://aps.arxiv.org/abs/0804.3678

Adaptive Networks combine dynamics ON a network with topological evolution OF the network. Recently this class of systems has received rapidly increasing attenion in different disciplines. Adaptive networks are therefore currently attacked from many different directions with the tools different disciplines have established. Despite the different angles, repeated themes frequently appear in these investigations. In particular it is found that adaptive networks can self-organize toward complex topologies and dynamics based on simple local rules. In this talk I will review some recent breakthrouhs and present three very different approaches to the analysis of adaptive networks. These will be illustrated by application to an neural model and cooperative games on adaptive networks.

Alterations in transcriptional regulation are considered major driving forces in divergent evolution. This is reflected in different species by the variable architecture of regulatory networks controlling highly conserved metabolic switches. The switch from glycolysis to gluconeogenesis and back that serves as an adaptive mechanism in response to changes in nutrient or oxygen supply in yeast is controlled by a conserved set of protein kinases and their downstream effectors. Apparentl, the wiring of these regulators has changed gradually during evolution. The goal of this project recently submitted to DFG is to uncover sequential steps in regulatory modules and, in the long run, the evolution of the transcriptional regulatory network controlling gluconeogenesis.

Due to the limited size of available datasets, performing proper inference of interactions between sites in related biological sequences (protein families, regulatory sites, etc...) is a key aspect in various recurrent problems in bioinformatics, such as gene prediction, modelling of transcription factor binding sites, protein classification, etc...
In 2001, S.I. Amari introduced a very general framework for decomposing stochastic interactions between sites in a hierarchical manner when the joint distribution is given; he published later a statistical extension of this framework, allowing to assess the presence of significant interactions as reflected by some data, using a frequentist approach: interactions of increasing order are iteratively tested.
On the other hand, bayesian statistics have proven their efficiency at performing such model selection tasks both theoretically and practically, for instance in the framework of text compression where the structure of a context tree has to be inferred. My project is to combine the merits of Amari's approach with bayesian model selection procedures.
After an introduction to the biological motivation of this work, I will present bayesian model selection procedures and comment on their rational; future lines of work aimed at applying this theory to Amari's model will conclude the lecture.

Although there are a number of quite different types of stem cells (such as embryonic, tissue, or cancer stem cells) all of them are characterised by the ability to ”decide” about the realisation of two generally different developmental pathways: self-renewing their current state or differentiating into (multiple) distinct mature cell types. Even though it is widely accepted that transcription factor regulation plays an important role in this decision process, neither the topology nor the dynamics of the regulatory networks are known at the moment.
Beside a general introduction into the biology of stem cells, my talk will focus on the discussion of an approach to model the dynamics of transcription factor interactions in stem cells on the basis of biologically motivated, small-scale networks, which are represented by systems of ordinary differential equations. I will present results on the application of such an approach to two stem cell systems, elaborating on the process of lineage specification of haematopoietic stem cells (HSC) and on the process of maintaining the undifferentiated, pluripotent ”ground state” of embryonic stem cells (ESC).

We report work in progress on a new class of vertex routing models and on recursive clique graphs, which are both motivated by our long-term research on cognitive system theory.
Information arriving to a vertex B arriving from a vertex A is routed to vertex C in a vertex routing model. We define a class of random vertex routing models with and without a memory trace and the notion of information centrality based on the long-term information flow in terms of cyclic attractors. We find that the information centrality is extensive/sub-extensive for models with/without a memory trace.
A clique is a maximal fully connected subgraph and a clique graph is the meta-network of cliques. We study numerically the possible occurrence of a phase-transition in recursive clique graphs with respect to their growth properties, starting from Erdös-Renyi graphs, as a function of the initial connectivity.

The main advantage of peer-to-peer (P2P) systems versus traditional client/server systems lies in achieving scalability and dependability by decentralized organization. Wireless mesh networks as well as ad hoc extensions of the Internet will be rapidly emerging in the commercial market as access networks, since these wireless multihop networks can perform data transfers both several orders of magnitude faster and cheaper than 3G cellular networks. Due to mobility and limited resources, P2P systems for multihop wireless networks require fundamentally new concepts than the ones known for the wired Internet.
This talk will present presents a novel congestion control algorithm for TCP over multihop IEEE 802.11 wireless networks implementing rate-based scheduling of transmissions within the TCP congestion window. The novel TCP variant is denoted as TCP with Adaptive Pacing (TCP-AP). A comprehensive mesurement study in a testbed shows that TCP-AP achieves up to 83% more goodput than the widely deployed TCP NewReno, provides excellent fairness in almost all scenarios, and is highly responsive to changing traffic conditions. Finally, an outline will be presented on how mobile P2P data sharing, e.g. photos and videos in a social network, can be effectively implemented in wireless multi-hop networks via epidemic data dissemination and TCP-AP. The talk concludes with sketching current research projects funded by DFG and BMBF.

How is living matter regulated so reliably, despite the molecular and fluctuating nature of their central information processing circuits? I will discuss this problem from two perspectives, in a toy model of stochastic dynamical networks, and subsequently in the context of biological examples. Mathematical toy models can teach us about the conditions of when and how stochastic networks can self-synchronize into a stable and reproducable dynamical pattern. The central question is how a network of unreliable elements can function reliably, for example to perform computations. This points us to possible construction principles of real biochemical regulatory networks which we will discuss for some known biological molecular networks.

While for text strings many complexity measures exist, for a graph (binary, unweighted) still it is unclear how to define complexity: especially if the complexity measure shall discard both regularity and randomness, and if it itself shall be computeable in low order polynomial time. This seems to be an open problem of applied graph theory for which solutions are more prohibitive than intuitively expected. Here I discuss an entropy-type property for the distribution of degree-differences between the nodes [Physica A 375, 365]. While being nonadditive, it goes beyond the network assortativity and shows some of the desired properties: vanishing for regular lattices, taking low values for random graphs, and significantly higher values for scalefree graphs and biological networks. Finally I will survey the strengths and limitations of the existing approaches.