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Networks in Time and Space

Periodic Reporting for period 3 - NETS (Networks in Time and Space)

Reporting period: 2019-04-01 to 2020-09-30

This project concerns developing the analysis methods for spatio-temporal networks, or the temporal evolution of interrelationships. The generation of a network process therefore requires modelling complex interrelationships across time. This is challenging because static networks (single snapshots of interrelationships) are just being understood from a formal mathematical point-of-view. The advantage of observing multiple networks is that you have more data, and so know more. Analysis of spatio-temporal networks is challenging due to the discrete or combinatorial nature of networks that needs to be combined with a spatio-temporal model for the network structure. This latter structure in continuous time-space, modelled using tools from harmonic analysis, must be aligned with discrete modelling, using techniques from combinatorics.

The understanding of time-varying networks is important to society because they are all around us: social networks that influence the political discourse, networks of species that need to be understood in the context of climate change, as well as brain networks that can give insights into the health of individuals. The lack of rigorous analysis methods mean we fail to understand this type of data. Developing this methodology can therefore impact the design of algorithms that monitor social networks, our ecological monitoring, as well as treatment of individuals with for example neuro-degenerative diseases.

To solve the problems of of modelling the temporal evolution of interrelationships, the objectives of the projects are to:
* To build models for temporally-varying interactions, and develop their methods of inference (WP 1 & 2).
* To build better models of events (WP 3)
* To build improved multivariate spatial models, in order to improve the network models (WP 4).
* To adapt the developed methods to impact problems in ecology, neuroscience and geosciences (all WPs).
In order to build better multivariate network models we have started the project by focusing on temporal evolution. As of yet we have extended our understanding of multiple phenomena across time to include complicated bivariate phenomena with intricate phase and amplitude shifts (Sykulski et al (2016,2017)); and applied this in understanding pain perception. We have also extended our understanding for very rapidly changing temporal processes, and summarized them in scenarios where analysis was previously impossible (Guillaumin et al., 2017). We have better developed our understanding of efficient approximations of the likelihood (Sykulski et al 2018 accepted in Biometrika).

Simultaneously we have developed a very general understanding of the structure of networks. Again the challenge is to develop summaries and statistical characterization that is invariant under permutation. Our advances correspond to describing the topology of a given network (Maugis et al (2017)), with the view towards network comparison (Maugis et al (2017)). Our contributions include the fast counting of features, or aspects of the data that are informative (Maugis et al (2017)). We have also derived which features are most canonical in comparing network structure, and studied their evolution in a time evolving network (Maugis et al (2017)).

Part of the project is understanding point processes, as networks are often constructed from observations of points. The group has developed new understanding of high dimensional point processes (Rajala et al (2018, JRSSc)). Existing high dimensional models corresponds to scenarios where the model and observations are both very complex, but is not a setting used for spatial point processes. We have developed new methods that automatically discard those interactions that seem to carry less weight, corresponding to a mathematical Occam’s razor, finding the simplest explanation of the data. This (for the first time) extends the high dimensional understanding to the spatial point process setting. We further developed our understanding of interactions in Rajala et al (2018, J of Ecology). This is important for practical applications because in ecology we wish to explore when a set of species is behaving erratically, or part of a unit.

Finally our understanding of complex algorithms impacts civil society. Olhede and Wolfe (2018) summarises and treats how to balance algorithmic complexity, and understanding its impact on applications.
State of the art temporal modelling would not be able to resolve the high frequency methods developed by (Guillaumin et al., 2017). Adopting previously applied methodology would mean either a loss of accuracy or stability in the estimation technology. This is therefore a first in deconvolving complex temporal evolution. Sykulski et al (2016,2017) provided novel understanding of highly complex bivariate (paired) relationships between temporal variables.

Most developments of network analysis techniques are based on making generating mechanisms very constrained. Maugis et al (2017) answered a long outstanding question from Milo et al (2004) on what features such be used for network comparison under a very general set of assumptions for network generation. The other two manuscripts (Maugis et al (2017), Maugis et al (2017)) developed specialized understanding of the numerical implementation of counting those features, or their statistical analysis.

Finally our developments of high dimensional data analysis for spatial point processes opens a very new field for high dimensional data analysis (Rajala et al (2018a, 2018b)). We have built on these techniques to develop scale-based and multivariate point process understanding, which will further simplify data analysis, and enable our understanding of massive collections of types of point processes. This has allowed us to question how we evaluate ecological hypotheses of clustering versus random distribution.

We expect to further develop our framework for in the temporal setting, building formal model generating mechanism.

We expect our development of spatial modelling, to advance and encompass greater model complexity and model heterogeneity.
nonparametric time-varying network summaries