arXiv subjects

Ian Gallagher

Publications and source records attributed to Ian Gallagher.

14 recordsLinked to original sources

arXiv2026

The ASE-LSE Disagreement Landscape: An End-to-End Characterisation of Extremes and Structural Drivers

Two of the most widely used methods for analysing graph data, Adjacency Spectral Embedding and Laplacian Spectral Embedding, often produce different results when applied to the same graph. Yet the structural reasons behind this disagreement remain incompletely understood. This paper provides an end-to-end account of ASE-LSE latent subspace disagreement. We first prove that the two methods produce identical latent subspaces for every embedding dimension whenever the Laplacian is a scalar multiple of the adjacency matrix, and show that this scalar relationship holds if and only if the graph is either regular or bipartite biregular. This anchor result identifies a sufficient condition for perfect agreement that pins down the floor of the disagreement spectrum and supplies the baseline for the perturbation analysis. We then prove that no maximal-disagreement graph or family of graphs exists: the disagreement is always strictly below its theoretical ceiling, and we exhibit a witness family demonstrating that no finite maximum is attainable, so the disagreement landscape has no maximiser. With both endpoints established, we derive a Regularity Departure Bound whose two terms isolate degree heterogeneity and eigengap as the primary structural factors influencing disagreement in the middle regime. Empirical validation across thousands of simulated graphs confirms the mechanisms predicted by the bound: heterogeneity pushes disagreement up, eigengap suppresses it, and their joint ratio emerges as a unified predictor of ASE-LSE disagreement, suggesting when the two embeddings can be treated as interchangeable and when they cannot.

stat.ML↗

arXiv2026

Generator-based Graph Generation via Heat Diffusion

Graph generative modelling has become an essential task due to the wide range of applications in chemistry, biology, social networks, and knowledge representation. In this work, we propose a novel framework for generating graphs by adapting the Generator Matching (arXiv:2410.20587) paradigm to graph-structured data. We leverage the graph Laplacian and its associated heat kernel to define a continous-time diffusion on each graph. The Laplacian serves as the infinitesimal generator of this diffusion, and its heat kernel provides a family of conditional perturbations of the initial graph. A neural network is trained to match this generator by minimising a Bregman divergence between the true generator and a learnable surrogate. Once trained, the surrogate generator is used to simulate a time-reversed diffusion process to sample new graph structures. Our framework unifies and generalises existing diffusion-based graph generative models, injecting domain-specific inductive bias via the Laplacian, while retaining the flexibility of neural approximators. Experimental studies demonstrate that our approach captures structural properties of real and synthetic graphs effectively.

stat.ML↗

arXiv2025

Sub-Terahertz Spin Relaxation Dynamics of Boron-Vacancy Centers in Hexagonal Boron Nitride

Quantum sensors based on spin-defect relaxation have become powerful tools for detecting faint magnetic signals, yet their operation has remained largely confined to low magnetic fields and gigahertz frequencies. Extending such sensors into high-field ($> 0.3$ T) and sub-terahertz regimes would enable quantum metrology across a wide range of electromagnetic phenomena and scientific applications, but has proven challenging. Here, we demonstrate that negatively charged boron vacancies ($\mathrm{V_B^-}$) in two-dimensional hexagonal boron nitride can function as relaxation-based quantum sensors operating up to 0.2 terahertz. Their uniform spin-orientation and persistent spin-contrast at high fields enable direct measurement of intrinsic spin relaxation across previously unexplored temperature and frequency regimes. We also reveal a crossover in relaxation behavior \textemdash initially decreasing at low fields before rising at higher fields \textemdash consistent with the emergence of single-phonon-induced resonant noise that becomes significant at sub-terahertz frequencies. These results establish $\mathrm{V_B^-}$ centers as a versatile platform for quantum sensing in the sub-terahertz, high-field regime.

quant-ph↗

arXiv2025

Defect Engineering in Large-Scale CVD-Grown Hexagonal Boron Nitride: Formation, Spectroscopy, and Spin Relaxation Dynamics

Recently, numerous techniques have been reported for generating optically active defects in exfoliated hexagonal boron nitride (hBN), which hold transformative potential for quantum photonic devices. However, achieving on-demand generation of desirable defect types in scalable hBN films remains a significant challenge. Here, we demonstrate that formation of negative boron vacancy defects, VB-, in suspended, large-area CVD-grown hBN is strongly dependent on the type of bombarding particles (ions, neutrons, and electrons) and irradiation conditions. In contrast to suspended hBN, defect formation in substrate-supported hBN is more complex due to the uncontrollable generation of secondary particles from the substrate, and the outcome strongly depends on the thickness of the hBN. We identify different defect types by correlating spectroscopic and optically detected magnetic resonance features, distinguishing boron vacancies (formed by light ions and neutrons) from other optically active defects emitting at 650 nm assigned to anti-site nitrogen vacancy (NBVN) and reveal the presence of additional dark paramagnetic defects that influence spin-lattice relaxation time (T1) and zero-field splitting parameters, all of which strongly depend on the defect density. These results underscore the potential for precisely engineered defect formation in large-scale CVD-grown hBN, paving the way for the scalable fabrication of quantum photonic devices.

cond-mat.mtrl-sci↗

arXiv2025

Unsupervised Attributed Dynamic Network Embedding with Stability Guarantees

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 four 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.

stat.ML↗

arXiv2024

Energy level structure of diamond silicon vacancy centers in an off-axis magnetic field

We report the development of an experimental approach that can characterize the energy level structure of an individual silicon vacancy (SiV) center in a magnetic field and extract the key parameters for the energy level structure. This approach builds upon a theoretical model that includes effects of the static strain, dynamic Jahn-Teller coupling, and magnetic field and introduces two coupling rates, one each for the ground and the excited states, to characterize the combined effects of strain and Jahn-Teller coupling. With the use of an analytical expression for the energy level structure of the SiV center under a transverse magnetic field, these two coupling rates can be directly derived from the measurement of the frequency separation between two spin-conserved transitions in a photoluminescence excitation spectrum and the measurement of the coherent population trapping resonance related to the SiV ground spin states. Additional experimental studies on the dependence of the energy level structure on the magnetic field amplitude and direction further verify the theoretical model and reveal contributions from unequal orbital magnetic coupling for the ground and excited states.

cond-mat.mes-hall↗

arXiv2024

Coherent population trapping and spin relaxation of a silicon vacancy center in diamond at mK temperatures

We report experimental studies of coherent population trapping and spin relaxation in a temperature range between 4 K and 100 mK in a silicon vacancy (SiV) center subject to a transverse magnetic field. Near and below 1 K, phonon-induced spin dephasing becomes negligible compared with that induced by the spin bath of naturally abundant 13C atoms. The temperature dependence of the spin dephasing rates agrees with the theoretical expectation that phonon-induced spin dephasing arises primarily from orbital relaxation induced by first order electron-phonon interactions. A nearly 100-fold increase in spin lifetime is observed when the temperature is lowered from 4 K to slightly below 1 K, indicating that two-phonon spin-flip transitions play an essential role in the spin relaxation of SiV ground states.

quant-ph↗

arXiv2024

Valid Conformal Prediction for Dynamic GNNs

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.

stat.ML↗

arXiv2023

A Simple and Powerful Framework for Stable Dynamic Network Embedding

In this paper, we address the problem of dynamic network embedding, that is, representing the nodes of a dynamic network as evolving vectors within a low-dimensional space. While the field of static network embedding is wide and established, the field of dynamic network embedding is comparatively in its infancy. We propose that a wide class of established static network embedding methods can be used to produce interpretable and powerful dynamic network embeddings when they are applied to the dilated unfolded adjacency matrix. We provide a theoretical guarantee that, regardless of embedding dimension, these unfolded methods will produce stable embeddings, meaning that nodes with identical latent behaviour will be exchangeable, regardless of their position in time or space. We additionally define a hypothesis testing framework which can be used to evaluate the quality of a dynamic network embedding by testing for planted structure in simulated networks. Using this, we demonstrate that, even in trivial cases, unstable methods are often either conservative or encode incorrect structure. In contrast, we demonstrate that our suite of stable unfolded methods are not only more interpretable but also more powerful in comparison to their unstable counterparts.

cs.SI↗

arXiv2023

Intensity Profile Projection: A Framework for Continuous-Time Representation Learning for Dynamic Networks

We present a new representation learning framework, Intensity Profile Projection, for continuous-time dynamic network data. Given triples $(i,j,t)$, each representing a time-stamped ($t$) interaction between two entities ($i,j$), our procedure returns a continuous-time trajectory for each node, representing its behaviour over time. The framework consists of three stages: estimating pairwise intensity functions, e.g. via kernel smoothing; learning a projection which minimises a notion of intensity reconstruction error; and constructing evolving node representations via the learned projection. The trajectories satisfy two properties, known as structural and temporal coherence, which we see as fundamental for reliable inference. Moreoever, we develop estimation theory providing tight control on the error of any estimated trajectory, indicating that the representations could even be used in quite noise-sensitive follow-on analyses. The theory also elucidates the role of smoothing as a bias-variance trade-off, and shows how we can reduce the level of smoothing as the signal-to-noise ratio increases on account of the algorithm `borrowing strength' across the network.

cs.LG↗

arXiv2022

Spectral embedding and the latent geometry of multipartite networks

Spectral embedding finds vector representations of the nodes of a network, based on the eigenvectors of a properly constructed matrix, and has found applications throughout science and technology. Many networks are multipartite, meaning that they contain nodes of fundamentally different types, e.g. drugs, diseases and proteins, and edges are only observed between nodes of different types. When the network is multipartite, this paper demonstrates that the node representations obtained via spectral embedding lie near type-specific low-dimensional subspaces of a higher-dimensional ambient space. For this reason we propose a follow-on step after spectral embedding, to recover node representations in their intrinsic rather than ambient dimension, proving uniform consistency under a low-rank, inhomogeneous random graph model. We demonstrate the performance of our procedure on a large 6-partite biomedical network relevant for drug discovery.

stat.ME↗

arXiv2021

Spectral embedding for dynamic networks with stability guarantees

We consider the problem of embedding a dynamic network, to obtain time-evolving vector representations of each node, which can then be used to describe changes in behaviour of individual nodes, communities, or the entire graph. Given this open-ended remit, we argue that two types of stability in the spatio-temporal positioning of nodes are desirable: to assign the same position, up to noise, to nodes behaving similarly at a given time (cross-sectional stability) and a constant position, up to noise, to a single node behaving similarly across different times (longitudinal stability). Similarity in behaviour is defined formally using notions of exchangeability under a dynamic latent position network model. By showing how this model can be recast as a multilayer random dot product graph, we demonstrate that unfolded adjacency spectral embedding satisfies both stability conditions. We also show how two alternative methods, omnibus and independent spectral embedding, alternately lack one or the other form of stability.

stat.ML↗

arXiv2019

Persistent Homology of Graph Embeddings

Popular network models such as the mixed membership and standard stochastic block model are known to exhibit distinct geometric structure when embedded into $\mathbb{R}^{d}$ using spectral methods. The resulting point cloud concentrates around a simplex in the first model, whereas it separates into clusters in the second. By adopting the formalism of generalised random dot-product graphs, we demonstrate that both of these models, and different mixing regimes in the case of mixed membership, may be distinguished by the persistent homology of the underlying point distribution in the case of adjacency spectral embedding. Moreover, despite non-identifiability issues, we show that the persistent homology of the support of the distribution and its super-level sets can be consistently estimated. As an application of our consistency results, we provide a topological hypothesis test for distinguishing the standard and mixed membership stochastic block models.

math.ST↗

arXiv2019

Spectral embedding of weighted graphs

When analyzing weighted networks using spectral embedding, a judicious transformation of the edge weights may produce better results. To formalize this idea, we consider the asymptotic behavior of spectral embedding for different edge-weight representations, under a generic low rank model. We measure the quality of different embeddings -- which can be on entirely different scales -- by how easy it is to distinguish communities, in an information-theoretic sense. For common types of weighted graphs, such as count networks or p-value networks, we find that transformations such as tempering or thresholding can be highly beneficial, both in theory and in practice.

stat.ML↗