EPSRC Reference: 
EP/W033585/1 
Title: 
Stochastic processes on random graphs with clustering 
Principal Investigator: 
Wang, Dr M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical & Physical Sciences 
Organisation: 
University of Sussex 
Scheme: 
New Investigator Award 
Starts: 
01 October 2022 
Ends: 
30 September 2025 
Value (£): 
254,240

EPSRC Research Topic Classifications: 
Logic & Combinatorics 
Mathematical Analysis 
Statistics & Appl. Probability 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Random graphs are mathematical tools for studying the properties of the complex networks that are ubiquitous in our everyday life (social networks, the Internet, the World Wide Web, among others). In the case of social networks, two common features about their geometries stand out: heterogeneity and clustering. The former refers to the presence of a small number of highly connected nodes (i.e. the "influencers"), and the latter is often used to explain the phenomenon that friends of my friends are also my friends. Recent years have witnessed growing interests on random graph models that share realworld network features and particular attention has been paid to the impact of heterogeneity on the networks. Compared with heterogeneity, mathematically rigorous studies on clustering have been relatively few, and it is the aim of the proposed research programme to provide one such study. The main object under scrutiny, the socalled model of random intersection graphs, has a simple yet flexible mechanism to produce clustering. By conducting parallel studies on these graphs in different clustering regimes and by comparing the results from these studies, the programme can produce convincing evidence that clustering impacts various properties of the networks. More precisely, the research questions that will be studied pertain to the following aspects of the networks:
1) Macroscopic structures as the graph size increases to infinity.
2) Largetime behaviours of certain dynamic processes (percolation and contact process) on the graphs.
Like many other random graph models, the random intersection graph experiences a drastic change in the component sizes as its edge density increases. This phenomenon is often referred to as a phase transition. If we take a sequence of increasingly large random intersection graph, each frozen at the precise point of the phase transition, it is expected that we will see interesting behaviours emerge. The first part of the programme aims to give a detailed description on the macroscopic structures of these graphs, relying upon a "zoomingout" procedure that runs roughly as follows. As the graphs grow larger and larger, we shrink the edge lengths therein in a suitable way so that the changing pattern of the graphs stabilises and a "limit object" appears. By identifying the suitable scale of edge lengths and the limit object, we will be able to gain valuable information on the sequence of graphs itself.
In the second part of the programme, we will look at how the particular structures of the random intersection graphs influence the two stochastic processes, percolation and contact process, running on them. The percolation process is connected to the aforementioned phase transition. By looking at this process, we will be able to discern the patten in which the components in the graphs merge with each other and form a giant component. Contact process is another classical probabilistic model and has been used to model the spread of computer virus on a network. For both processes, we expect to see distinct behaviours in the different clustering regimes of the graphs. It is the aim of the programme to confirm this expectation as well as to give a detailed description on the largetime behaviours of these processes in each regime.

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Organisation Website: 
http://www.sussex.ac.uk 