Periodic Reporting for period 1 - DynaNet (Mathematical random graph models for real-world dynamical networks)
Periodo di rendicontazione: 2023-09-01 al 2025-08-31
Networks are everywhere around us and play an important role in our lives. Think of the neurons in a brain, viruses like COVID-19 spreading through the population (a social network), fake news on social media networks like Instagram influencing elections, or computer viruses attacking digital infrastructure via computer networks. Mathematics can contribute to the understanding of such real-world networks, in terms of their structure and and behaviour, by analysing mathematical models of such networks. Due to the abstract level of mathematics, such analyses are widely applicable, e.g. computer viruses spreading through computer networks and viruses spreading through the population. Over the last two decades, evolving random graph models have proved to be one of the most successful and popular models for explaining the emergence of empirically observed features in many real-world networks, such as well-connectedness (e.g. every two Facebook users are separated by a chain of about 4 to 5 befriended users), and the existence of few extremely well-connected nodes called hubs in networks (e.g. millions of webpages link towards Google). So far, only evolving models incorporating the addition of vertices and edges to the graph have been studied in detail. Such a dynamical construction, however, is far from realistic: Real-world networks allow for both addition as well as deletion of vertices and edges. Examples include social media networks (users create/delete accounts and connections); biological networks like the brain (neural plasticity, the brain can grow/shrink); and the Internet (devices can break down and be removed from the network). Very little attention, however, has been dedicated to the study of mathematical models that incorporate such realistic dynamics.
The overall objectives of this project are to address these shortcomings, by:
(1) Studying the existence of hubs in dynamical networks that incorporate dynamics of both addition and removal,
(2) Studying the interconnectedness of dynamical networks that incorporate dynamics of both addition and removal,
(3) Studying the spread of information on networks over time.
The overall objectives of this project are to address these shortcomings, by:
(1) Studying the existence of hubs in dynamical networks that incorporate dynamics of both addition and removal,
(2) Studying the interconnectedness of dynamical networks that incorporate dynamics of both addition and removal,
(3) Studying the spread of information on networks over time.
(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.
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.
Results:
(1a) The findings in these articles provide sufficient conditions for the existence of hubs in super-linear preferential attachment models on a much more general level compared to the models studied in the existing literature.
(1b) The findings in these articles are the first of their kind, as we are the first to study the (non-)existence of hubs in preferential attachment trees with vertex death. The analysis carried out in these articles identifies novel regimes in which we analyse behaviour that cannot be observed in models without death.
(3) This has lead to 1 scientific article. The findings in this article go beyond the state of the art, as they improve on existing tools used for the analysis of competition models. In particular, our work shows that couplings between branching processes and the competition process can be leveraged to a greater extent and hence provide a precise understanding of the behaviour of the competition model in a more general setting. This includes parameter regimes that had not been studied previously.
Impact:
- Scientific Impact:
* Projects (1a) and (1b) have provided various directions for further research and have created contacts with other experts in the field with whom I intend to collaborate.
* Project (3) provides insights for further research for long-range competition models on spatial random graphs, such as long-range percolation and age-dependent random connection models.
- Economic and Societal Impact:
The impact of this fellowship on an economic and societal level is currently limited. The research does not have a direct real-world application as of yet, nor is it in a sufficiently advanced stage to e.g. influence policies. Key needs in this direction to ensure further uptake, in particular to other scientific areas with a stronger focus on applications (e.g. statistics and physics), are further research to solidify the theoretical framework and dissemination to scientists in these more applied areas.
(1a) The findings in these articles provide sufficient conditions for the existence of hubs in super-linear preferential attachment models on a much more general level compared to the models studied in the existing literature.
(1b) The findings in these articles are the first of their kind, as we are the first to study the (non-)existence of hubs in preferential attachment trees with vertex death. The analysis carried out in these articles identifies novel regimes in which we analyse behaviour that cannot be observed in models without death.
(3) This has lead to 1 scientific article. The findings in this article go beyond the state of the art, as they improve on existing tools used for the analysis of competition models. In particular, our work shows that couplings between branching processes and the competition process can be leveraged to a greater extent and hence provide a precise understanding of the behaviour of the competition model in a more general setting. This includes parameter regimes that had not been studied previously.
Impact:
- Scientific Impact:
* Projects (1a) and (1b) have provided various directions for further research and have created contacts with other experts in the field with whom I intend to collaborate.
* Project (3) provides insights for further research for long-range competition models on spatial random graphs, such as long-range percolation and age-dependent random connection models.
- Economic and Societal Impact:
The impact of this fellowship on an economic and societal level is currently limited. The research does not have a direct real-world application as of yet, nor is it in a sufficiently advanced stage to e.g. influence policies. Key needs in this direction to ensure further uptake, in particular to other scientific areas with a stronger focus on applications (e.g. statistics and physics), are further research to solidify the theoretical framework and dissemination to scientists in these more applied areas.