## Publications

36 results found

Gosztolai A, Arnaudon A, 2021, Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature., *Nat Commun*, Vol: 12

Describing networks geometrically through low-dimensional latent metric spaces has helped design efficient learning algorithms, unveil network symmetries and study dynamical network processes. However, latent space embeddings are limited to specific classes of networks because incompatible metric spaces generally result in information loss. Here, we study arbitrary networks geometrically by defining a dynamic edge curvature measuring the similarity between pairs of dynamical network processes seeded at nearby nodes. We show that the evolution of the curvature distribution exhibits gaps at characteristic timescales indicating bottleneck-edges that limit information spreading. Importantly, curvature gaps are robust to large fluctuations in node degrees, encoding communities until the phase transition of detectability, where spectral and node-clustering methods fail. Using this insight, we derive geometric modularity to find multiscale communities based on deviations from constant network curvature in generative and real-world networks, significantly outperforming most previous methods. Our work suggests using network geometry for studying and controlling the structure of and information spreading on networks.

Peach R, Arnaudon A, Barahona M, 2021, Relative, local and global dimension in complex networks

<jats:title>Abstract</jats:title> <jats:p>Dimension is a fundamental property of objects and the space in which they are embedded. Yet ideal notions of dimension, as in Euclidean spaces, do not always translate to physical spaces, which can be constrained by boundaries and distorted by inhomogeneities, or to intrinsically discrete systems such as networks. To take into account locality, finiteness and discreteness, dynamical processes can be used to probe the space geometry and define its dimension. Here we show that each point in space can be assigned a relative dimension with respect to the source of a diffusive process, a concept that provides a scale-dependent definition for local and global dimension also applicable to networks. To showcase its application to physical systems, we demonstrate that the local dimension of structural protein graphs correlates with structural flexibility, and the relative dimension with respect to the active site uncovers regions involved in allosteric communication. In simple models of epidemics on networks, the relative dimension is predictive of the spreading capability of nodes, and identifies scales at which the graph structure is predictive of infectivity.</jats:p>

Peach R, Arnaudon A, Schmidt J,
et al., 2021, HCGA: Highly comparative graph analysis for network phenotyping, *Patterns*, Vol: 2, ISSN: 2666-3899

Networks are widely used as mathematical models of complex systems across many scientific disciplines. Decades of work have produced a vast corpus of research characterizing the topological, combinatorial, statistical, and spectral properties of graphs. Each graph property can be thought of as a feature that captures important (and sometimes overlapping) characteristics of a network. In this paper, we introduce HCGA, a framework for highly comparative analysis of graph datasets that computes several thousands of graph features from any given network. HCGA also offers a suite of statistical learning and data analysis tools for automated identification and selection of important and interpretable features underpinning the characterization of graph datasets. We show that HCGA outperforms other methodologies on supervised classification tasks on benchmark datasets while retaining the interpretability of network features. We exemplify HCGA by predicting the charge transfer in organic semiconductors and clustering a dataset of neuronal morphology images.

Gosztolai A, Arnaudon A, 2021, Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature, Publisher: arXiv

Defining the geometry of networks is typically associated with embedding inlow-dimensional spaces such as manifolds. This approach has helped designefficient learning algorithms, unveil network symmetries and study dynamicalnetwork processes. However, the choice of embedding space is network-specific,and incompatible spaces can result in information loss. Here, we define adynamic edge curvature for the study of arbitrary networks measuring thedeformation between pairs of evolving dynamical network processes on differenttimescales. We show that the curvature distribution exhibits gaps atcharacteristic timescales indicating bottleneck-edges that limit informationspreading. Importantly, curvature gaps robustly encode communities until thephase transition of detectability, where spectral clustering methods fail. Weuse this insight to derive geometric modularity optimisation and demonstrate iton the European power grid and the C. elegans homeobox gene regulatory networkfinding previously unidentified communities on multiple scales. Our worksuggests using network geometry for studying and controlling the structure ofand information spreading on networks.

Palacios J, Zisis E, Coste B, et al., 2020, BlueBrain/NeuroM: New function extract_dataframe that extract morphometrics as a dataframe

Neuronal Morphology Analysis Tool

Peach R, Arnaudon A, Barahona M, 2020, Graph centrality is a question of scale, *Physical Review Research*, Vol: 2, ISSN: 2643-1564

Classic measures of graph centrality capture distinct aspects of node importance, from the local (e.g., degree) to the global (e.g., closeness). Here we exploit the connection between diffusion and geometry to introduce a multiscale centrality measure. A node is defined to be central if it breaks the metricity of the diffusion as a consequence of the effective boundaries and inhomogeneities in the graph. Our measure is naturally multiscale, as it is computed relative to graph neighbourhoods within the varying time horizon of the diffusion. We find that the centrality of nodes can differ widely at different scales. In particular, our measure correlates with degree (i.e., hubs) at small scales and with closeness (i.e., bridges) at large scales, and also reveals the existence of multi-centric structures in complex networks. By examining centrality across scales, our measure thus provides an evaluation of node importance relative to local and global processes on the network.

Peach RL, Arnaudon A, Barahona M, 2020, Semi-supervised classification on graphs using explicit diffusion dynamics, *Foundations of Data Science*, Vol: 2, Pages: 19-33, ISSN: 2639-8001

Classification tasks based on feature vectors can be significantly improved by including within deep learning a graph that summarises pairwise relationships between the samples. Intuitively, the graph acts as a conduit to channel and bias the inference of class labels. Here, we study classification methods that consider the graph as the originator of an explicit graph diffusion. We show that appending graph diffusion to feature-based learning as a posteriori refinement achieves state-of-the-art classification accuracy. This method, which we call Graph Diffusion Reclassification (GDR), uses overshooting events of a diffusive graph dynamics to reclassify individual nodes. The method uses intrinsic measures of node influence, which are distinct for each node, and allows the evaluation of the relationship and importance of features and graph for classification. We also present diff-GCN, a simple extension of Graph Convolutional Neural Network (GCN) architectures that leverages explicit diffusion dynamics, and allows the natural use of directed graphs. To showcase our methods, we use benchmark datasets of documents with associated citation data.

Bock A, Arnaudon A, Cotter C, 2019, Selective Metamorphosis for Growth Modelling with Applications to Landmarks, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG

Arnaudon A, Holm DD, Sommer S, 2019, Stochastic Metamorphosis with Template Uncertainties

In this paper, we investigate two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincaré theory. In the metamorphosis of images, the Lie group of diffeomorphisms deforms a template image that is undergoing its own internal dynamics as it deforms. This type of deformation allows more freedom for image matching and has analogies with complex uids when the template properties are regarded as order parameters. The first stochastic perturbation we consider corresponds to uncertainty due to random errors in the reconstruction of the deformation map from its vector field. We also consider a second stochastic perturbation, which compounds the uncertainty of the deformation map with the uncertainty in the reconstruction of the template position from its velocity field. We apply this general geometric theory to several classical examples, including landmarks, images, and closed curves, and we discuss its use for functional data analysis.

Saravanan M, Arnaudon A, 2018, Engineering solitons and breathers in a deformed ferromagnet: Effect of localised inhomogeneities, *PHYSICS LETTERS A*, Vol: 382, Pages: 2638-2644, ISSN: 0375-9601

Arnaudon A, Holm DD, Sommer S, 2018, A Geometric Framework for Stochastic Shape Analysis, *Foundations of Computational Mathematics*, ISSN: 1615-3375

We introduce a stochastic model of diffeomorphisms, whose action on a varietyof data types descends to stochastic evolution of shapes, images and landmarks.The stochasticity is introduced in the vector field which transports the datain the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework forshape analysis and image registration. The stochasticity thereby models errorsor uncertainties of the flow in following the prescribed deformation velocity.The approach is illustrated in the example of finite dimensional landmarkmanifolds, whose stochastic evolution is studied both via the Fokker-Planckequation and by numerical simulations. We derive two approaches for inferringparameters of the stochastic model from landmark configurations observed atdiscrete time points. The first of the two approaches matches moments of theFokker-Planck equation to sample moments of the data, while the second approachemploys an Expectation-Maximisation based algorithm using a Monte Carlo bridgesampling scheme to optimise the data likelihood. We derive and numerically testthe ability of the two approaches to infer the spatial correlation length ofthe underlying noise.

Arnaudon A, Holm D, Sommer S, 2018, String methods for stochastic image and shape matching, *Journal of Mathematical Imaging and Vision*, Vol: 60, Pages: 953-967, ISSN: 0924-9907

Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the large deformation diffeomorphic metric mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite-temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.

Arnaudon A, Takao S, 2018, Networks of coadjoint orbits: from geometric to statistical mechanics

A class of network models with symmetry group $G$ that evolve as aLie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example $G=SO(3)$ and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and the position-coupled case reduces to a new system that has a broken symmetry group $SO(3)/SO(2)$ similar to the heavy top. In the latter system, we numerically observed an interesting synchronisation-like phenomenon for a certain class of initial conditions. Adding a type of noise and dissipation that preserves the coadjoint orbit of the network model, we found that the invariant measure is given by the Gibbs measure, from which the notion of temperature is defined. We then observed a surprising `triple-humped' phase transition in the heavy top-like lattice model, where the spins switched from one equilibrium position to another before losing magnetisation as we increased the temperature. This work is only a first step towards connecting geometric mechanics with statistical mechanics and several interesting problems are open for further investigation.

Arnaudon A, 2018, Structure preserving noise and dissipation in the Toda lattice, *Journal of Physics A: Mathematical and Theoretical*, Vol: 51, ISSN: 1751-8113

In this paper, we use Flaschka's change of variables of the open Toda latticeand its interpretation in term of the group structure of the LU factorisationas a coadjoint motion on a certain dual of Lie algebra to implement a structurepreserving noise and dissipation. Both preserve the structure of coadjointorbit, that is the space of symmetric tri-diagonal matrices and arise as a newtype of multiplicative noise and nonlinear dissipation of the Toda lattice. Weinvestigate some of the properties of these deformations and in particular thecontinuum limit as a stochastic Burger equation with a nonlinear viscosity.This work is meant to be exploratory, and open more questions that we cananswer with simple mathematical tools and without numerical simulations.

Arnaudon A, Ganaba N, Holm DD, 2018, The stochastic energy-Casimir method, *Comptes Rendus Mécanique*, Vol: 346, Pages: 279-290, ISSN: 1631-0721

Akylzhanov R, Arnaudon A, 2018, Contractions of group representations via geometric quantisation

We propose a general framework to contract unitary dual of Lie groups viaholomorphic quantization of their co-adjoint orbits. The sufficient condition for the contractability of a representation is expressed via cocycles on coadjoint orbits. This condition is checked explicitly for the contraction of ${\mathrm SU}_2$ into $\mathbb{H}$. The main tool is the geometric quantization. We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix.

Arnaudon A, De Castro AL, Holm DD, 2018, Noise and Dissipation on Coadjoint Orbits, *Journal of Nonlinear Science*, Vol: 28, Pages: 91-145, ISSN: 0938-8974

Arnaudon A, López MC, Holm DD, 2018, Un-reduction in field theory, *Letters in Mathematical Physics*, Vol: 108, Pages: 225-247, ISSN: 0377-9017

Ribeiro Castro A, 2017, Noise and dissipation in rigid body motion, Publisher: Springer

Using the rigid body as an example, we illustrate some features of stochastic geometric mechanics. These features include: (i) a geometric variational motivation for the noise structure involving Lie-Poisson brackets and momentum maps , (ii) stochastic coadjoint motion with double bracket dissipation , (iii) description and its stationary solutions , (iv) random dynamical systems , random attractors and SRB measures connected to statistical physics.

Arnaudon A, Gibbon JD, 2017, Integrability of the hyperbolic reduced Maxwell-Bloch equations for strongly correlated Bose-Einstein condensates, *Physical Review A*, Vol: 96, ISSN: 2469-9926

Sommer S, Arnaudon A, Kuhnel L, et al., 2017, Bridge simulation and metric estimation on landmark manifolds, Information Processing in Medical Imaging 2017, Publisher: Springer Verlag, Pages: 571-582, ISSN: 0302-9743

We present an inference algorithm and connected Monte Carlo based estimationprocedures for metric estimation from landmark configurations distributedaccording to the transition distribution of a Riemannian Brownian motionarising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric.The distribution possesses properties similar to the regular Euclidean normaldistribution but its transition density is governed by a high-dimensional PDEwith no closed-form solution in the nonlinear case. We show how the density canbe numerically approximated by Monte Carlo sampling of conditioned Brownianbridges, and we use this to estimate parameters of the LDDMM kernel and thusthe metric structure by maximum likelihood.

Arnaudon A, Holm DD, Pai A, et al., 2017, A stochastic large deformation model for computational anatomy, Information Processing in Medical Imaging

In the study of shapes of human organs using computational anatomy, variations are found to arise from inter-subject anatomical differences, disease-specific effects, and measurement noise. This paper introduces a stochastic model for incorporating random variations into the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. By accounting for randomness in a particular setup which is crafted to fit the geometrical properties of LDDMM, we formulate the template estimation problem for landmarks with noise and give two methods for efficiently estimating the parameters of the noise fields from a prescribed data set. One method directly approximates the time evolution of the variance of each landmark by a finite set of differential equations, and the other is based on an Expectation-Maximisation algorithm. In the second method, the evaluation of the data likelihood is achieved without registering the landmarks, by applying bridge sampling using a stochastically perturbed version of the large deformation gradient flow algorithm. The method and the estimation algorithms are experimentally validated on synthetic examples and shape data of human corpora callosa.

Arnaudon A, Holm DD, Ivanov RI, 2017, G-Strands on symmetric spaces., *Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences*, Vol: 473, ISSN: 1471-2946

We study the G-strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group G, and we treat in more detail examples with symmetric space SU(2)/S(1) and SO(4)/SO(3). The latter model simplifies to an apparently new integrable nine-dimensional system. We also study the G-strands on the infinite dimensional group of diffeomorphisms, which gives, together with the Sobolev norm, systems of 1+2 Camassa-Holm equations. The solutions of these equations on the complementary space related to the Witt algebra decomposition are the odd function solutions.

Arnaudon A, 2016, On a lagrangian reduction and a deformation of completely integrable systems, *Journal of Nonlinear Science*, Vol: 26, Pages: 1133-1160, ISSN: 1432-1467

We develop a theory of Lagrangian reduction on loop groups for completely integrable systems after having exchanged the role of the space and time variables in the multi-time interpretation of integrable hierarchies. We then insert the Sobolev norm H1H1 in the Lagrangian and derive a deformation of the corresponding hierarchies. The integrability of the deformed equations is altered, and a notion of weak integrability is introduced. We implement this scheme in the AKNS and SO(3) hierarchies and obtain known and new equations. Among them, we found two important equations, the Camassa–Holm equation, viewed as a deformation of the KdV equation, and a deformation of the NLS equation.

Revaz Y, Arnaudon A, Nichols M,
et al., 2016, Computational issues in chemo-dynamical modelling of the formation and evolution of galaxies, *Astronomy & Astrophysics*, Vol: 588, ISSN: 1432-0746

Chemo-dynamical N-body simulations are an essential tool for understanding the formation and evolution of galaxies. As the number of observationally determined stellar abundances continues to climb, these simulations are able to provide new constraints on the early star formaton history and chemical evolution inside both the Milky Way and Local Group dwarf galaxies. Here, we aim to reproduce the low α-element scatter observed in metal-poor stars. We first demonstrate that as stellar particles inside simulations drop below a mass threshold, increases in the resolution produce an unacceptably large scatter as one particle is no longer a good approximation of an entire stellar population. This threshold occurs at around 103M⊙, a mass limit easily reached in current (and future) simulations. By simulating the Sextans and Fornax dwarf spheroidal galaxies we show that this increase in scatter at high resolutions arises from stochastic supernovae explosions. In order to reduce this scatter down to the observed value, we show the necessity of introducing a metal mixing scheme into particle-based simulations. The impact of the method used to inject the metals into the surrounding gas is also discussed. We finally summarise the best approach for accurately reproducing the scatter in simulations of both Local Group dwarf galaxies and in the Milky Way.

Arnaudon A, 2016, On a deformation of the nonlinear Schrödinger equation, *Journal of Physics A: Mathematical and Theoretical*, Vol: 49, ISSN: 1751-8113

We study a deformation of the nonlinear Schrödinger (NLS) equation recently derived in the context of deformation of hierarchies of integrable systems. Although this new equation has not been shown to be completely integrable, its solitary wave solutions exhibit typical soliton behaviour, including near elastic collisions. We will first focus on standing wave solutions which can be smooth or peaked, then with the help of numerical simulations we will study solitary waves, their interactions and finally rogue waves in the modulational instability regime. Interestingly, the structure of the solution during the collision of solitary waves or during the rogue wave events is sharper and has larger amplitudes than in the classical NLS equation.

Arnaudon A, Lopez MC, Holm DD, 2015, Covariant un-reduction for curve matching, MFCA 2015, Publisher: MICCAI Society, Pages: 95-106

The process of un-reduction, a sort of reversal of reduction by the Lie groupsymmetries of a variational problem, is explored in the setting of fieldtheories. This process is applied to the problem of curve matching in theplane, when the curves depend on more than one independent variable. Thissituation occurs in a variety of instances such as matching of surfaces orcomparison of evolution between species. A discussion of the appropriateLagrangian involved in the variational principle is given, as well as someinitial numerical investigations.

Kühnel L, Arnaudon A, Sommer S, Differential geometry and stochastic dynamics with deep learning numerics

In this paper, we demonstrate how deterministic and stochastic dynamics onmanifolds, as well as differential geometric constructions can be implementedconcisely and efficiently using modern computational frameworks that mixsymbolic expressions with efficient numerical computations. In particular, weuse the symbolic expression and automatic differentiation features of thepython library Theano, originally developed for high-performance computationsin deep learning. We show how various aspects of differential geometry and Liegroup theory, connections, metrics, curvature, left/right invariance, geodesicsand parallel transport can be formulated with Theano using the automaticcomputation of derivatives of any order. We will also show how symbolicstochastic integrators and concepts from non-linear statistics can beformulated and optimized with only a few lines of code. We will then giveexplicit examples on low-dimensional classical manifolds for visualization anddemonstrate how this approach allows both a concise implementation andefficient scaling to high dimensional problems.

Zisis E, Keller D, Kanari L, et al., Architecture of the Neuro-Glia-Vascular System

<jats:title>Abstract</jats:title><jats:p>Astrocytes connect the vasculature to neurons and mediate the supply of nutrients and biochemicals. They also remove metabolites from the neurons and extracellular environment. They are involved in a growing number of physiological and pathophysiological processes. Understanding the biophysical, physiological, and molecular interactions in this neuro-glia-vascular ensemble (NGV) and how they support brain function is severely restricted by the lack of detailed cytoarchitecture. To address this problem, we used data from multiple sources to create a data-driven digital reconstruction of the NGV at micrometer anatomical resolution. We reconstructed 0.2 mm<jats:sup>3</jats:sup> of rat somatosensory cortical tissue with approximately 16000 morphologically detailed neurons, its microvasculature, and approximately 2500 morphologically detailed protoplasmic astrocytes. The consistency of the reconstruction with a wide array of experimental measurements allows novel predictions of the numbers and locations of astrocytes and astrocytic processes that support different types of neurons. This allows anatomical reconstruction of the spatial microdomains of astrocytes and their overlapping regions. The number and locations of end-feet connecting each astrocyte to the vasculature can be determined as well as the extent to which they cover the microvasculature. The structural analysis of the NGV circuit showed that astrocytic shape and numbers are constrained by vasculature’s spatial occupancy and their functional role to form NGV connections. The digital reconstruction of the NGV is a resource that will enable a better understanding of the anatomical principles and geometric constraints which govern how astrocytes support brain function.</jats:p><jats:sec><jats:title>Table of contents</jats:title><jats:sec><jats:title>Main points</jats:title><jats:list list-ty

Arnaudon A, Barp A, Takao S, Irreversible Langevin MCMC on Lie Groups

It is well-known that irreversible MCMC algorithms converge faster to theirstationary distributions than reversible ones. Using the special geometricstructure of Lie groups $\mathcal G$ and dissipation fields compatible with thesymplectic structure, we construct an irreversible HMC-like MCMC algorithm on$\mathcal G$, where we first update the momentum by solving an OU process onthe corresponding Lie algebra $\mathfrak g$, and then approximate theHamiltonian system on $\mathcal G \times \mathfrak g$ with a reversiblesymplectic integrator followed by a Metropolis-Hastings correction step. Inparticular, when the OU process is simulated over sufficiently long times, werecover HMC as a special case. We illustrate this algorithm numerically usingthe example $\mathcal G = SO(3)$.

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