Drainage network models in river basin modelling

Abstract: Drainage networks have been used for modelling river networks. These networks can be loosely defined as follows.

A random or deterministic set of vertices is selected. Fix a direction and each vertex connects to a unique vertex along that direction selected on the basis of a fixed set of rules. As the number of outgoing edges from each vertex is only one, it follows that the generated (random) graph can not have any cycle and each connected component of the graphmust be a directed infinite tree. Various questions related to the structure of these random graphs can be asked, e.g., whether this graph is connected or not, the number of topological ends and so on.

In this talk we describe some well known drainage network models and some recent results on their geometry. Once we understand the geometry, it reveals that it represents a random tributary structure and hence can be used in river basin modelling. It is also believed that various emperically observed scaling relations can be understood well if we can understand scaling limits of these drainage network models.

 We also describe what do we mean by scaling limits of these models and how it can be used for justification of "Hack's law", a celebrated scaling relation for geomorphologists. This is joint work with Rahul Roy and Anish Sarkar. In the end I will describe my current research projects. No prior knowledge of random graph is necessary.