# Course Description: Complex Networks 2010

**outline**

many different systems, indeed many different aspects of different systems, are described as â€œa complex biological networkâ€�. some of these models are bewilderingly dissimilar from each other, so much that it may seem they are studied through completely different disciplines or methods. this course (and lecture notes) are meant as a pocket map to the vast land of such complex networks. the aim is not to study many particular examples of complex networks in detail, but rather to acquire a sense of where various examples of practical importance fit in the theoretical landscape and with respect to each other.

the main themes of the course are: topology, metrics, dynamics and algorithmics. biological networks consist of many, dissimilar agents; each agent interacts with a complex subset of the whole system. topology deals with neighbourhood: who interacts with who, who is connected to who. metrics deals with quantities that are no longer disembodied, but live on topological entities, the players and their connections. dynamics deals with how either topology or metric information imbibed in the graph change with time. algorithmics is required because we cannot carry out any calculations naively in finite time, a certain know-how of computational methods is required.

the course will meet twice weekly, for one theory and one practice session. practice shall be carried out at the libraryâ€™s computer room in the matlab language. for their final evaluation, students will present some important paper in their own dealing with complex networks, including a recalculation of all relevant quantitative plots using the tools learned.

a crash session on matlab will be provided for students with some knowledge of programming in other languages. however, some programming experience and basic calculus and linear algebra are expected. **topology: **

graph theory. adjacency, directed+undirected; local vs. global structure: connected components, multiple connectivity, number of loops (tr(M^n)), minimal path-djikstra, diameter, girth, clustering, cliques.

special graphs: planar graphs, trees, Euler theorem, basic lattices, Erdos, Sheidegger, triangulations,

special graphs ii: small world, scale free. randomization of graphs/Maslov. **metrics: **

random resistor networks. embeddings. djikstra ii. laplacian. **dynamics: **

dynamics of the graph itself (e.g. synaptogenesis/neurogenesis) vs. dynamics of the quantities (e.g. synaptic plasticity)

dynamics on a graph: Markov processes on graphs, e.g. the population dynamics of ion channels on the graph of transitions between states. chemical kinetics. perceptrons. classical plasticity.

dynamics of the graph: growth models.

metrical dynamics: activity-dependent neurogenesis. **systems/examples**

c. elegans neural net. vasculature. ecosystems: foodwebs. ancestry structure, phylogenetics. wordnet. social networks, sexual networks. distribution of surnames (galton). chemical kinetics of enzymatic cascades. signaling. gene induction. protein interaction maps. emerging network vs. underlying network (e.g. phone network). retina.

network inference

phylogenetics, autoregressive, markov random fields.