Sparse dynamic network with reciprocating relationships

Brief description:
Scientific research of recent years is marked by a tremendous interest in the study, the understanding and the modelling of complex networks. This is a direct result of the rapid increase in the availability and importance of network data which cover a huge part of the more general area of complex "big data" systems. Some of the most critical examples of networks in the 21st century are social networks, the internet, collaboration networks, networks of business relations and biological networks. Their applicability in modern life, brings at the forefront of scientific research the development of theory and methodology for realistic statistical network models that capture the salient properties of real-world networks.

There is a number of issues and difficulties in designing realistic networks, which consequently makes their design a highly challenging task. Most traditional representation techniques fail to capture some crucial features of real world networks as for example graph sparsity; real world network graphs usually have a finite number of edges per vertex and exhibit properties like power law degree distribution and small world phenomena which can only occur in sparse graphs. Therefore, traditional models are inherently misspecified. The development of a coherent theory and methodology is, despite intense efforts in mathematics, so far an unsolved issue.

Recently a new class of models was introduced (Caron, 2012; Caron and Fox, 2014) proposing a new graph representation based on completely random measures. Under this approach, a graph is represented by a point process on the plane which then gives a hierarchical Bayesian nonparametric construction. The construction successfully captures the desired network properties such as exchangeability, sparsity and power-law degree distribution. These properties turn out to be able to capture the structure of real world networks like Facebook, political blogspheres, protein networks, citation networks and world wide web networks. Furthermore the model has interpretable parameters, provides an urn process construction and posterior characterization. Another important aspect of the success of this approach is scalability, as the model provides scalable inference and hence is suitable for "big data" problems with up to hundreds of thousands of nodes and millions of edges.

Aims and objectives:
The objective of my project is to extend this framework in various directions. Firstly, towards a multivariate scenario which is important in order to capture overlapping community structure allowing the members of the network to be affiliated to more than one communities. Also, the property of dynamic behaviour is of interest as new connections emerge over time and then new edges appear in the network. Finally,it is very appealing to handle reciprocity in the sense that one person's actions towards another increase the probability of the same type of action being returned. This will be provided with the use of mutually exciting point processes (Hawkes, 1971; Blundell, 2012). The combination of the above will ultimate lead to a new class of models for sparse dynamic networks with reciprocating relationships which will be applied in real life networks aiming to infer their structure and capture their behaviour.

The project falls within the EPSRC statistics and applied probability research area within the mathematical sciences theme.