Dynamic graphs provide a flexible data abstraction for modelling many sorts of real-world systems, such as transport, trade, and social networks. Graph neural networks (GNNs) are powerful tools allowing for different kinds of prediction and inference on these systems, but getting a handle on uncertainty, especially in dynamic settings, is a challenging problem. In this work we propose to use a dynamic graph representation known in the tensor literature as the unfolding, to achieve valid prediction sets via conformal prediction. This representation, a simple graph, can be input to any standard GNN and does not require any modification to existing GNN architectures or conformal prediction routines. One of our key contributions is a careful mathematical consideration of the different inference scenarios which can arise in a dynamic graph modelling context. For a range of practically relevant cases, we obtain valid prediction sets with almost no assumptions, even dispensing with exchangeability. In a more challenging scenario, which we call the semi-inductive regime, we achieve valid prediction under stronger assumptions, akin to stationarity. We provide real data examples demonstrating validity, showing improved accuracy over baselines, and sign-posting different failure modes which can occur when those assumptions are violated.

Stability for dynamic network embeddings ensures that nodes behaving the same at different times receive the same embedding, allowing comparison of nodes in the network across time. We present attributed unfolded adjacency spectral embedding (AUASE), a stable unsupervised representation learning framework for dynamic networks in which nodes are attributed with time-varying covariate information. To establish stability, we prove uniform convergence to an associated latent position model. We quantify the benefits of our dynamic embedding by comparing with state-of-the-art network representation learning methods on three real attributed networks. To the best of our knowledge, AUASE is the only attributed dynamic embedding that satisfies stability guarantees without the need for ground truth labels, which we demonstrate provides significant improvements for link prediction and node classification.

Quantifying uncertainty in networks is an important step in modelling relationships and interactions between entities. We consider the challenge of bootstrapping an inhomogeneous random graph when only a single observation of the network is made and the underlying data generating function is unknown. We address this problem by considering embeddings of the observed and bootstrapped network that are statistically indistinguishable. We utilise an exchangeable network test that can empirically validate bootstrap samples generated by any method. Existing methods fail this test, so we propose a principled, distribution-free network bootstrap using k-nearest neighbour smoothing, that can pass this exchangeable network test in many synthetic and real-data scenarios. We demonstrate the utility of this work in combination with the popular data visualisation method t-SNE, where uncertainty estimates from bootstrapping are used to explain whether visible structures represent real statistically sound structures.

SUMMARY: Multi-locus sequence typing (MLST) is a widely used method of characterization of bacterial isolates. It has been applied to over 50,000 isolates in over 50 different species. Here, we present a coalescent method to jointly simulate MLST data and the clonal genealogy that gave rise to the sample. AVAILABILITY AND IMPLEMENTATION: SimMLST was implemented in C++ and Qt4 for the graphical user interface. It is distributed under the terms of the GNU General Public License. Source code and binaries for Windows and Linux are available from (http://go.warwick.ac.uk/SimMLST). A user guide and a technical description of the algorithm are provided with the program.