A graph $G$ is locally irregular if for every edge $(u,v) \in E(G)$, the vertices $u$ and $v$ have distinct degrees. The locally irregular edge-colouring problem is a graph decomposition problem in which the edges of a graph are coloured such that each colour induces a locally irregular subgraph. If a graph $G$ can be edge-coloured such that each colour induces a locally irregular subgraph, then $G$ is said to be decomposable; otherwise, $G$ is exceptional. Graph neural networks (GNNs) are a class of deep learning models designed to operate on graph-structured data and are typically applied to tasks involving rich node features. In this work, we investigate and benchmark several popular GNN architectures for the classification of decomposable and exceptional graphs using minimal feature input. We compare models with different input representations, and provide a detailed analysis of their learning behaviour and the resulting graph embeddings. We show that the performance of many message passing GNNs degrades in the absence of node features in many graph-related problems. However, little work has been done to show why that is the case. Our results show that this is not the case in general and modern GNNs are able to learn meaningful structural representations that enable effective classification of these graphs to a certain extent. Furthermore, we observe some generalization to out-of-distribution graphs, suggesting both the promise and current limitations of GNNs as tools for pattern discovery in theoretical graph problems.

We consider the dynamics of an idealized cross-diffusion model of biological invasion in which the diffusion of an invading population is inhibited by a resident population, which is in turn degraded by the former. We formulate and numerically solve the problem that describes the self-similar smoothing of an initially piecewise-constant invading population. In the limit, the height ahead of the initial discontinuity vanishes and the speed of propagation also vanishes, but only logarithmically slowly. This result is confirmed by a matched asymptotic analysis. The singular nature of this limit indicates that the system does not permit the existence of compactly supported solutions that exhibit a moving front or interface, which has important implications for the simulation of more complex models that also feature this degenerate diffusive mechanism.

We illustrate how collecting data from rejected horticultural produce may give more information about the prevalence of product infestation by pests or diseases than if we only sampled accepted (consignment) produce. Our results show that sampling from rejected product can have different outcomes depending on the selectivity of grading (the rejection process) and relative sensitivity of the inspection of rejected fruit. If rejected produce is more likely to be infested and inspection sensitivity is as high as or higher than for packed fruit, then the rejection process has the following impact: (1) it selectively prevents some infested fruit from entering the consignment, lowering the expected consignment prevalence, and hence, (2) it lowers the chance of the consignment being rejected based on the consignment sample alone. Sampling the reject fruit and finding no infestation (3) further lowers the inferred prevalence in the consignment, but (4) increases the chance of a consignment being rejected if there is no tolerance for detecting infestation within the reject bin. If rejected produce is less or as likely to be infested, then the benefits from (1), (2) and (3) may no longer occur. In many situations, however, sampling from the reject bin can provide greater confidence in product integrity without the loss of packed product to destructive sampling.