Inhomogeneity and generalised bootstrap percolation in stochastic networks - 27785

Bootstrap percolation processes were introduced by the physicists Chalupa, Leath and Reich in 1979 in order to describe certain magnetic phenomena and have been studied extensively since then by probabilists, mathematical physicists and combinatorialists. This is a class of "infection" processes on graphs that are based on the following simple rule: a node becomes infected if at least a certain number of its neighbours have become infected. Especially during the last 10 to 15 years, the field has seen a huge amount of growth, in particular, regarding probabilistic and extremal aspects of
these processes.

In this project, we will study more sophisticated versions of bootstrap percolation processes, which are based on majority rules. That is, a node becomes infected if at least a certain fraction of its neighbours have become infected. These processes have a considerable number of applications in themes ranging from the spread of ideas or trends in a society to evolutionary game theory. Despite their basic nature, their rigorous study remains a major challenge.

The main goal of the project is to develop the theory of such processes on stochastic networks that exhibit various forms of inhomogeneity. This general notion captures a number of properties that have been observed in real-world networks, such as social or biological networks. In fact, inhomogeneity is a key feature of real-world networks, which reflects the diversity within a population. The project will explore the effects that various forms of inhomogeneity have on the way these processes evolve, leading to a deeper understanding of phenomena that are not observed in "homogeneous" structures. Its results will give new insight on mechanisms that underpin many societal and economic processes. The study of these processes has become increasingly important during the last 10 years, mainly due to the enormous growth of economic activity over internet platforms as well as the increasing impact of social media.

The diffusion of ideas and behaviours within a society has been among the most fundamental themes of sociology and the marketing science. The study of these mechanisms is of particular importance nowadays, due to the rapid development of social networks such as Twitter or Facebook and their influence on web economy.
A recent study performed by the Boston Consulting Group showed that the value of Web economy in G20 countries will nearly double by 2016. According to the same study, the United Kingdom has the largest Internet economy, as a
percentage of its gross domestic product, among the G20 group of industrialised countries. The development of the Internet over the last 20 years and, subsequently, of all those networks that use it as their platform have provided scientists with a pool of data which can be and have been used to acquire a much clearer picture of how a society, viewed as a set of individuals that have some form of interaction with each other, is structured.

One of the main objectives of the proposed project is the modeling and the analysis of the word-of-mouth spread of information over the nodes of a network that represents a cohort of individuals. This is the fundamental mechanism that underlies the economic activity taking place over social networks.

The systematic study of the typical features of these networks began already during the late 1990s, in parallel to the development of the Internet and the World-Wide Web. This line of research did not only come from the Internet community and social sciences, but quite surprisingly it has been observed that biological networks also exhibit similar characteristics. The theory that has emerged since then is a mixture of combinatorial methods together with sophisticated tools that probability theory provides. Thus, the core of the proposed project lies in the amalgamation of applied probability and combinatorics. This is a far-reaching interaction which can nowadays be witnessed in several branches of mathematics from number theory and analytic combinatorics to ergodic theory. This synergy lies at the very heart of the proposed project, mainly due to the nature of our research themes. It is our intention to push forward and deepen this trend, while achieving the objectives of the proposed project.

The execution of this project will lead to precise and realistic results regarding the word-of-mouth dissemination
of ideas or trends in social networks. In the long term, these will lead to the development of tools and give the insight that will make a more sophisticated modelling of such processes possible.

The project will provide additional research capacity to the combinatorics and the applied probability theory community, as it provides the necessary means for the training and development of a postdoctoral researcher. Besides the scientific development which will be attained through the direct involvement into the execution of the project, the broader professional development of the postdoctoral fellow will be achieved through teaching, participation into conferences and research seminars (e.g. giving talks), summer schools, scientific visits as well as the organisation of workshops.