Networks in Time and Space

Project Information

NETS

Grant agreement ID: 682172

DOI

10.3030/682172

Project closed

EC signature date 26 April 2016

Start date 1 May 2016

End date 30 April 2023

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 587 602,00

EU contribution

€ 1 587 602,00

Coordinated by

ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Switzerland

Periodic Reporting for period 5 - NETS (Networks in Time and Space)

Reporting period: 2022-05-01 to 2023-04-30

Summary of the context and overall objectives of the project

This project developed analysis methods for spatio-temporal networks and data, and the temporal evolution of interrelationships. The generation of a network process requires modelling complex interrelationships across time. This is challenging because static networks 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 combinatorial nature of networks that needs to be combined with a spatio-temporal model for the network structure.

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 objectives of the projects were:
* To build models for temporally-varying interactions, and develop their methods of inference (WP 1 & 2). Achieved results are published in Suveges and Olhede (2023), Lunagomez et al (2022), Chandna et al (2022), Maugis et al (2020).
* To build better models of events (WP 3) and better methods for their analysis; this has been implemented in Rajala et al (2018), Rajala et al (2023), as well as Martin et al (2023).
* To build improved multivariate spatial models and design methods of their estimation (WP 4); this has been implemented in Guillaumin et al (2022).
* To adapt the developed methods to impact problems in ecology and neuroscience (all WPs); results published in Chang et al (2016), Rajala et al (2019), Rupawala et al (2023).

Publications:
S. Chandna et al, Local linear graphon estimation using covariates, Biometrika, 109(3), 721-734, 2022.

P. Chang et al, The development of nociceptive network activity in the somatosensory cortex of freely moving rat pups, Cerebral Cortex, 26(12), 4513-4523, 2016.

A. P. Guillaumin et al, The Debiased Spatial Whittle Likelihood, J. Roy. Stat Soc. B, 84(4), 1526-1557, 2022.

S. Lunagomez et al, Modeling Network populations via graph distances, J. Am. Stat Assoc., 116(536), 2023-2040, 2021.

J. S. Martin et al, Multivariate geometric anisotropic Cox processes, Scand. J. of Stat., 2023.

P. A. Maugis et al, Testing for equivalence of network distribution using subgraph count. J. CGS, 29(3), 455-465, 2020.

T. Rajala et al, When do we have the power to detect biological interactions in spatial point patterns? J. of Ecology, 107(2), 711-721, 2019.

T. Rajala et al, Detecting multivariate interactions in spatial point patterns with Gibbs models and variable selection, J. Roy. Stat. Soc. C, 67(5), 1237-1273, 2018.

T. Rajala et al, What is the Fourier Transform of a Spatial Point Pattern?, IEEE Trans. Info. Theory, 2023.

M. Rupawala et al. A developmental shift in habituation to pain in human neonates, Current Biology, 33(8), 1397-1406, 2023.

M. Suveges and S. C. Olhede, Networks with correlated edge processes, J. Roy. Stat. Soc. A, 2023.

A. Sykulski et al, The debiased Whittle likelihood, Biometrika, 106(2), 251-266, 2019.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Work on the project has progressed as expected. We have derived ways of modelling correlated networks in time, thus capturing the properties of dependent edges. This allows us to describe scenarios where an observed system is exhibiting a structured behavior over time, which itself may be inhomogeneous in time. We developed methods of estimating such a patterned form of behaviour, and showed its utility when studying contacts in a hospital network. Using group models to describe human interactions was also applied to the nation-scale data set of corporate VAT interactions in Bulgaria, where the group effects were shown to be informative.

Stepping beyond simple parametric models, we built additional models for multiple networks, including treating the graph like a stochastic shape. Inference in a Bayesian setting was determined. Tests non-reliant on nodal ordering were also developed. Methods to understand the sensory network obtained from EEG recordings were also developed, and applied to prematurely born infant data to understand nociception.

To study spatial patterns we developed new methodology for analysis of data taking the form of incidences. We developed high dimensional models, and estimation methods. Further methods were developed to study point patterns alone, developing the first modern non-parametric estimation techniques based on tapering. Additional methods were proposed for visualization. Estimation methods were applied to rainforest data from forest ecology.

Analysis methods for spatial patterns in terms of real-valued observations regularly observed in space were proposed, after the corresponding temporal problem had been addressed. We also obtained lower order errors in terms of boundary patterns for temporal observations, and corrected for those lower order effects both for temporal and spatial data. Irregular spatial domain and missing observations were all tackled in one framework. Analysis methods were developed that are both computationally efficient (nlog(n), with n observations), and also without the lowest order bias term. Extensions to spatio-temporal processes have been proposed and developed.

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

State of the art in 2016 had studied some evolutionary group models. Most of those were solely time-varying, and did not capture any dependence structure of interest. We developed models and analysis methods for temporally varying and dependent processes. We also clarified the blockmodel approximation to an underlying graph limit model, by both determining the effect of a smoothness assumption in the wrong dimension, as well as if the network is contaminated by nuisance structure corresponding to uncharacterizable edge dependence. This dependence was set to mimic a network collected in terms of its edges over time. Neither characterising flexible and complex dependence nor showing robustness to it, was possible prior to the work of the proposal.

Further methods to characterise networks over time were formed in a Bayesian setting, and connected to statistical shape models. Methods to estimate the underlying entropy of graph limits were proposed as well as various testing procedures. Applications were made to help the Bulgarian tax authority detect VAT fraud using network structure, and to characterise the network response of the brain to pain. All of these developments advanced the state of the art.

We developed methods to model and estimate high dimensional multitype point patterns. We also developed a practical Fourier theory for point patterns, where we showed properties of the Fourier transforms. These were two `firsts’ that advanced state of the art significantly.

Finally we developed spatial analysis further, advancing analysis techniques beyond spatial domain based analysis. Fourier domain methods were used to arrive at likelihood methods that are computationally efficient (nlog(n)), and also without the lowest order bias term. This outperformed state-of-the-art in 2016 significantly. We also developed methods for spatio-temporal phenomena.

nonparametric time-varying network summaries