Communities in time-dependent networks

My name is Marya Bazzi and I recently finished my PhD in applied mathematics at the Oxford Centre for Industrial and Applied Mathematics. The main focus of my thesis is cluster (or community) detection in time-dependent networks. But before going into that, a little bit of context:

I started my studies with an undergraduate degree in pure mathematics at the American University of Beirut in Lebanon. After that, I went on to do the MSc in Mathematical Modelling and Scientific Computing at Oxford University, which I concluded with a dissertation on matrix completion. Broadly speaking, the matrix-completion problem consists of recovering unknown entries in an incomplete data matrix. This dissertation ignited my interest in research and data analysis. Before pursuing a doctoral­ degree, I wanted to gain research experience within an industrial context. I did a 6-months internship in the R&D division of Schlumberger, where I developed a massively parallel linear solver that could be used in reservoir simulation. The internship proved to be an interesting mix of coding, debugging, and linear algebra. My PhD came next.

Network science, a relatively recent research area, seemed like it catered well to my interest in both theory and its applications, in particular data analysis. A network is a way of representing a system of interacting entities. These can be friendship ties between people, transportation routes between cities, or metabolic interactions between cells [2]. Network science is the study of networks with the aim of gaining insight into the systems they represent.

For example, it can be useful to zoom out of descriptions on the microscopic scale of individual entity interactions in order to investigate coarser grained (or higher-order) structure. A community is an example of a higher-order structure and intuitively refers to a set of entities that interact more strongly with each other than they do with the rest of the network [3].

The notion of a community alone gives rise to a series of questions: What do we mean by “more strongly”? Should this depend on the application? What kind of algorithms can we use to detect communities? How accurate are they? How meaningful is their output?

I investigated questions like these throughout my PhD, with one additional difficulty: the network entities and interactions evolved in time. My project was in collaboration with HSBC bank in London, and I used financial-asset correlation networks as real-world examples (see figure).

While it is clear that connectivity patterns within and between asset classes are time-dependent, it is less clear how one can incorporate time dependency into the detection of communities. Should communities in one temporal snapshot depend on connectivity patterns in other temporal snapshots? If so, in what way? With the help of my supervisors, Mason Porter and Sam Howison, we provide some insight into these questions in [1] and use our results to detect persistent correlation structures in financial asset markets.

My experience with network science is that it is an extremely diverse research area. Network scientists include physicists, engineers, computer scientists, biologists, and mathematicians, amongst others. Their research intersects with statistical mechanics, dynamical systems, statistics, linear algebra, optimization, graph theory, and more. Applications range from sociology and communication to information dissemination, biology, and finance. The mix is enriching but also challenging — enriching because every source of input brings its own set of original ideas, and challenging because it produces a body of research which lacks a common language and theoretical foundation. Bridging gaps between scientific communities is key to the further development of network science. The exciting and growing body of research in this area to date promises a lot more impactful developments to come in the near future.

[1] M. Bazzi, M. A. Porter, S. Williams, M. McDonald, D. J. Fenn. and S. D. Howison, Community detection in temporal multilayer networks, with an application to correlation networks, to appear in Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, arXiv:1501.00040, 2015.

[2] M.E.J. Newman, Networks: An introduction, Oxford University Press, 2010.

[3] M. A. Porter, J.-P Onnela, and P. J. Mucha, Communities in networks, Notices of the American Mathematical Society, 2009.