Probability: Reinforced Branching Processes

The aim of this course is to define the notion of reinforced branching processes and give tools to study them, starting with a crash course on continuous-time martingales.

Although reinforced branching processes are interesting objects in themselves, I chose to motivate their definition by looking at one particular case: the preferential attachment tree with fitnesses introduced by Bianconi and Barabási as a model for complex networks. This model is a discrete-time random process on the set of rooted trees, but if we embed this process in continuous time with the help of random exponential clocks, we obtain a continuous-time process called a reinforced branching process.

Reinforced branching processes can be seen as population processes with immortal particles. They are a particular case of the more general Crump-Mode-Jägers processes for which tools are available, especially when they admit a *Malthusian* parameter. We will show how to apply these methods, but will then focus on the case when there is no Malthusian parameter.

In this latter case, a phenomenon called *condensation* occurs: in terms of networks, it means that there exists a *small* set of nodes (typically sublinear in the size of the network) such that the sum of the degrees of these nodes is linear in the size of the network. If there exists one node having linear degree in terms of the size of the network, we say that *the winner takes it all* or that there is *extensive condensation*.

Our aim is to understand under what condition there is condensation, and whether this is extensive or non-extensive condensation. We will briefly introduce the main tools to study reinforced branching processes: martingales, Crump-Mode-Jägers processes, random point processes.