Vertices of large degrees in preferential attachment graph models

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing either in September 2016 (funded students) or at any point in the academic year (self-funded students).

Over the past 15-20 years, there has been much interest in models of random graphs generated by “preferential attachment" processes. In such a process, new vertices arrive sequentially and join to existing vertices depending on their current degree: vertices of higher degree are more likely to be selected as neighbours. Random graphs of this type have been used as models of the internet, as well as other computer and social networks. The distribution of the number of vertices of each xed degree at any stage is well understood, but rather less is known about the number of vertices of large degree. This has been studied in detail only for the simplest possible preferential attachment model, where each arriving vertex selects exactly one neighbour, with probability proportional to the current degree. There is a great deal of scope for a student to investigate extensions and generalisations of the known results. The project would suit a student with good knowledge and strong technical skills in probability, along with familiarity with the fundamental results and techniques of discrete mathematics.

This project will be supervised by Malwina Luczak. A potential subproject here would be to extend the results of [BL2012] to the Cooper-Frieze model.

Further details can be found in the project abstract: http://www.maths.qmul.ac.uk/sites/default/files/phd%20projects%202015/Prob/PhDproject-1-2.pdf

The application procedure is described on the School website. For further enquiries please contact Prof. Malwina Luczak ([Email Address Removed]).

This project is eligible for several sources of full funding for the 2016/17 academic year, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs. Applicants interested in the full funding will have to participate in a highly competitive selection process. The best candidates will be eligible to receive a prestigious Ian Macdonald Postgraduate Award of £1000, for which you will be considered alongside your application. The application deadline for full funding is January 31st 2016.

There is also 50% funding scheme available for students who are able to find the matching 50 % of the cost of their studies. Competition for these half-funded slots will be less intensive, and eligible students should mention their willingness to be considered for them in their application. The application deadline for 50 % funding is January 31st 2016.

This project can be also undertaken as a self-funded project, either through your own funds or through a body external to Queen Mary University of London. Self-funded applications are accepted year-round.