NNTuck: A factor model of multilayer network interdependence

I. Aguiar, D. Taylor, and J. Ugander, A factor model of multilayer network interdependence, arXiv preprint arXiv:2206.01804, (2022).

This work is on the arxiv and can be found here.

The tools associated with this work, along with a tutorial jupyter notebook can be found here. Please let me know if you have any questions at all.

– Contributed talk at the SIAM Mathematics of Data Science conference, September 2022
– Contributed talk at the Statistical Inference for Network Models Satellite, June 2021

Multilayer networks describe the rich ways in which nodes are related through different types of connections in separate layers. These multiple relationships are naturally represented by an adjacency tensor that can be interpreted using techniques from multilinear algebra. In this work we propose the use of the nonnegative Tucker decomposition (NNTuck) with KL-divergence as an expressive factor model for multilayer networks that naturally generalizes existing methods for stochastic block models of multilayer networks. We show how the NNTuck provides an intuitive factor-based perspective on layer dependence and enables linear-algebraic techniques for analyzing dependence with respect to specific layers. Algorithmically, we show that using expectation maximization (EM) to maximize this log-likelihood under the NNTuck model is step-by-step equivalent to tensorial multiplicative updates for the nonnegative Tucker decomposition under a KL loss, extending a previously known equivalence from nonnegative matrices to nonnegative tensors. Using both synthetic and real-world data, we evaluate the use and interpretation of the NNTuck as a model of multilayer networks. The ability to quantify dependencies between layers has the potential to inform survey instruments for collecting social network data, identify redundancies in the structure of a network, and indicate relationships between disparate layers. Therefore, we propose a definition of layer dependence based on using a likelihood ratio test to evaluate three nested models: the layer independent, layer dependent, and layer redundant NNTucks. We show how these definitions, paired with analysis of the factor matrices in the NNTuck, can be used to understand and interpret layer dependence in different contexts.