(1) Study the existence of hubs in dynamical networks that incorporate dynamics of both addition and removal.
The main focus of the fellowship has been to analyse hubs in mathematical models of dynamical networks that incorporate dynamics of addition and removal. Here, I have collaborated with the fellowship supervisor, external researchers, and done research in solo projects. We have studied two different mathematical models of dynamical networks: super-linear preferential attachment trees and preferential attachment trees with vertex death. This objective has been reached in the following research projects:
(a) For super-linear preferential attachment trees, we have analysed the existence of hubs. This has led to two scientific articles where we provide sufficient conditions under which hubs can appear in these models. This largely generalised known results in the literature, which focussed on special cases only. Moreover, the analysis carried out leveraged a novel perspective that simplified analyses compared to earlier work.
(b) For preferential attachment trees with vertex death, we have analysed the (non-)existence of hubs. This has led to two scientific articles. In the first article, we provide sufficient conditions under which hubs cannot exist; the second article focusses on necessary and sufficient conditions under which hub do exist. Here, we are the first to study the (non-)existence of hubs in such models, which has shown these models exhibit much richer behaviour compared to classical preferential attachment models without vertex death.
On-going work seeks to combine the research in projects (a) and (b) to study a model known as super-linear preferential attachment with vertex death. Here, we again expect richer behaviour compared to the super-linear preferential attachment model. We expect this to lead to at least one more scientific article in the next 5 years.
(2) Study the interconnectedness of dynamical networks that incorporate dynamics of addition and removal.
For this objective we studied the preferential attachment tree with vertex death model, as in project (1a). The aim is to understand the emergence of large-scale connected structures in this tree model. To this end, we have identified the local weak limit of the model, which serves as an approximation of the typical structure of the network around a node. To understand large-scale connected structures, we further need to understand if and to what extent properties of the local weak limit translate to the entire network. This is currently on-going work. We expect this to lead to at least one more scientific article in the next 5 years.
(3) Study the spread of information on networks over time.
For this objective I have studied a long-range competition model on boxes of the hypercubic lattice. Here, two types of infections are placed at distinct vertices of the hypercubic box of volume n. Each type spreads to other unoccupied vertices at a rate that depends on the distance between the occupied and unoccupied vertex. These long-range rates may be different for both infections, and are allowed to depend on n, the volume of the box. This project focussed on the `weak spatial dependence' regime, in which the dependence of the rates on the distance between vertices is weak.
We have analysed sufficient and necessary conditions under which coexistence occurs. This means that, asymptotically, both types of information reach a positive proportion of all vertices. In the case that coexistence does not occur, we also provide precise results on the asymptotic size of the type that reaches only a negligible proportion of all vertices. Furthermore, the project has identified possibilities for further research in regimes where the spatial dependence is strong. We expect this to lead to at least one more scientific article in the next 5 years.
