# Programme NET2009

### Monday 28. September 2009

Theme: **Network Topologies ** Time Table

Network topologies are the most basic graph structure guiding the way through highly complex networks. Nearly any complex system can be as a first modeling step visualised as a graph, either undirected (component A is in relation with component B), or directed (component A is influencing component B). The latter will establish the static basis for the further analysis of dynamical concepts, like feedback. There are fundamentally different network architectures, like exponential or scale-free networks. Small-world networks establish an architecture where information can potentially spread more easily over network paths than in any other type of network.

### Tuesday 29. September 2009

Theme: **Evolving Network Topologies** Time Table

Many (natural) processes, most dominantly evolution, are constantly establishing new components and relations in any (biological) network. Mathematically we need to look at (stochastic) processes that are describing the dynamics of such evolving network topologies accurately. Understanding evolving networks is also essential for understanding the creation process of observed static networks, like road systems, or any other type of infrastructure network. Similar transport networks can be found in biological systems, for example the cell. Morphogenetic processes build up networks of blood vessels or neuronal tissue in the developing embryo.

### Wednesday 30. September 2009

Theme: **Dynamics on Networks, Modularisation and Flows** Time Table

The study of dynamical processes defined on network topologies starts with the analysis of equilibrium conditions, where the flows into and out of the network equilibrate each other, obeying Kirchhoff's law in the network interior at each node. Such a situation can under standard conditions be associated with mathematical tools based on linearity principles. On this level the network dynamics can also be modularised successfully, relying on the seperability of linear processes. Flows defind on graphs have many applications, and can be successfully used to optimise real world processes.

### Thursday 1. October 2009

Theme: **Dynamics on Networks, Attractors, Feedback and Bifurcations ** Time Table

The generic class of processes describing biological (and most other) phenomena are non-linear. In general much less strong mathematical results are availability when compared to theorems based on linearity and conservation principles. Nevertheless mathematics has produced in the last decades a rich literature on attractors, bifurcation theory and feedback. Bifurcation theory is currently combined with the concept of networks and network architecture. More refined methods can be obtained if the given non-linear system defined on the network obeys symmetry principles. For biology non-linear processes defined on networks are the essential modelling technique to understand genetic regulation, metabolic pathways, or neuronal activity.

### Friday 2, October 2009

Theme: **Network Inference ** Time Table

Constructing network models (often unconciously as sketches) is usually the first step in the conceptual part of the scientific process. This is due to the fact that any system can be dissected into components and the relations the components have with each other. But such models need to be verfied on the base of experimental data. It is clear that there is a hidden relationship between the conceptual network framework and the way data are created or processed. On this day we like to discuss relationships between graphical models as used in statistics, parameter estimation methods and non-linear dynamical processes defined on networks. The day is closely related to a previous symposium workshop: **Information extraction from complex data sets**, (INF).