arXiv subjects

Luca Rossi

Publications and source records attributed to Luca Rossi.

At least 37 records · Page 2Linked to original sources

arXiv2025

Reaction-diffusion model for a population structured in phenotype and space I -- Criterion for persistence

We consider a reaction-diffusion model for a population structured in phenotype. We assume that the population lives in a heterogeneous periodic environment, so that a given phenotypic trait may be more or less fit according to the spatial location. The model features spatial mobility of individuals as well as mutation. We first prove the well-posedness of the model. Next, we derive a criterion for the persistence of the population which involves the generalised principal eigenvalue associated with the linearised elliptic operator. This notion allows us to handle the possible lack of coercivity of the operator. We then obtain a monotonicity result for the generalised principal eigenvalue, in terms of the frequency of spatial fluctuations of the environment and in terms of the spatial diffusivity. We deduce that the more heterogeneous is the environment, or the higher is the mobility of individuals, the harder is the persistence for the species. This work lays the mathematical foundation to investigate some other optimisation problems for the environment to make persistence as hard or as easy as possible, which will be addressed in the forthcoming companion paper.

math.AP↗

arXiv2025

Generating Graphs via Spectral Diffusion

In this paper, we present GGSD, a novel graph generative model based on 1) the spectral decomposition of the graph Laplacian matrix and 2) a diffusion process. Specifically, we propose to use a denoising model to sample eigenvectors and eigenvalues from which we can reconstruct the graph Laplacian and adjacency matrix. Using the Laplacian spectrum allows us to naturally capture the structural characteristics of the graph and work directly in the node space while avoiding the quadratic complexity bottleneck that limits the applicability of other diffusion-based methods. This, in turn, is accomplished by truncating the spectrum, which, as we show in our experiments, results in a faster yet accurate generative process, and by designing a novel transformer-based architecture linear in the number of nodes. Our permutation invariant model can also handle node features by concatenating them to the eigenvectors of each node. An extensive set of experiments on both synthetic and real-world graphs demonstrates the strengths of our model against state-of-the-art alternatives.

cs.LG↗

arXiv2025

Generalized principal eigenvalues for parabolic operators in bounded domains

We introduce here new generalized principal eigenvalues for linear parabolic operators with heterogeneous coefficients in space and time. We consider a bounded spatial domain and an unbounded time interval $I$ : $I=\mathbb{R},\ \mathbb{R}^+$ or $\mathbb{R}^-$, and operators with coefficients having a fairly general dependence on space and time. The notions we introduce rely on the parabolic maximum principle and extend some earlier definitions introduced for elliptic operators [BNV]. We first show that these eigenvalues hold the key to understanding the large time behavior and entire solutions of heterogeneous Fisher-KPP type equations. We then describe the relation of these quantities with principal Floquet bundles for parabolic operators which provides further characterizations of the principal eigenvalues. These allow us to derive monotonicity properties and comparisons between generalized principal eigenvalues, as well as perturbation results and further properties involving limit operators. We show that the sign of these eigenvalues encodes different versions of the maximum principle for parabolic operators. Lastly, we explicitly compute the generalized principal eigenvalues for several classes of operators such as spatial-independent, periodic, almost periodic, uniquely ergodic or random stationary ergodic coefficients.

math.AP↗

arXiv2025

Looking into the faintEst WIth MUSE (LEWIS): Exploring the nature of ultra-diffuse galaxies in the Hydra-I cluster II. Stellar kinematics and dynamical masses

Context: This paper focuses on a class of galaxies characterised by an extremely low surface brightness: the ultra-diffuse galaxies (UDGs). We used new integral-field spectroscopic data from the ESO Large Programme Looking into the faintEst WIth MUSE (LEWIS) project. Aims: Our main goals are addressing the formation channels and investigating possible correlations of their observational properties. In particular, we derive their stellar kinematics and dynamical properties. Methods: We extract the 1D stacked spectrum inside the effective radius to obtain an unbiased measure of $σ_{\rm eff}$. To derive the spatially-resolved stellar kinematics, we first apply the Voronoi tessellation algorithm to bin the spaxels in the datacube and then follow the same prescription adopted for the 1D case. In addition, we extract the velocity profiles along the galaxy's major and minor axes. Results: We find that 7 out of 18 UDGs in LEWIS show a mild rotation, 5 do not have evidence of any rotation, and the remaining 6 UDGs are unconstrained cases. This is the first large census of velocity profiles for UDGs. On average, UDGs in LEWIS are characterised by low values of $σ_{\rm eff}$, comparable with available values from the literature. In the Faber-Jackson relation plane, we found a group of UDGs consistent with the relation within the errorbars, whereas outliers are objects with non-negligible rotation components. UDGs and LSBs in LEWIS have larger dark matter content than dwarf galaxies with similar total luminosity. We do not find clear correlations between the derived properties and the local environment. Conclusions: Based on the stellar kinematics, two classes of UDGs are found in the Hydra I cluster: the rotating and non-rotating systems. This result, combined with other structural properties, can help to discriminate between the several formation scenarios proposed for UDGs.

astro-ph.GA↗

arXiv2024

A New Lotka-Volterra Model of Competition With Strategic Aggression -- Civil Wars When Strategy Comes Into Play

In this monograph, we introduce a new model in population dynamics that describes two species sharing the same environmental resources in a situation of open hostility. The interactions among these populations are described not in terms of random encounters, but via the strategic decisions of one population that can attack the other according to different levels of aggressiveness. This leads to a non-variational model for the two populations at war, taking into account structural parameters such as the relative fit of the two populations with respect to the available resources and the effectiveness of the attack strikes of the aggressive population. The analysis that we perform is rigorous and focuses on the dynamical properties of the system, by detecting and describing all the possible equilibria and their basins of attraction. Moreover, we will analyze the strategies that may lead to the victory of the aggressive population, i.e. the choices of the aggressiveness parameter, in dependence of the structural constants of the system and possibly varying in time in order to optimize the efficacy of the attacks, which take to the extinction in finite time of the defensive population. The model that we present is flexible enough to include also technological competition models of aggressive companies releasing computer viruses to set a rival company out of the market.

math.AP↗

arXiv2024

Spreading, flattening and logarithmic lag for reaction-diffusion equations in R^N: old and new results

This paper is concerned with the large-time dynamics of bounded solutions of reaction-diffusion equations with bounded or unbounded initial support in R N. We start with a survey of some old and recent results on the spreading speeds of the solutions and their asymptotic local one-dimensional symmetry. We then derive some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. Lastly, we reclaim some known results about the logarithmic lag between the position of the solutions and that of planar or spherical fronts expanding with minimal speed, for almost-planar or compactly supported initial conditions. We then prove some new logarithmic-in-time estimates of the lag of the position of the solutions with respect to that of a planar front, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. These estimates entail in particular that the same lag as for compactly supported initial data holds true for a class of unbounded initial supports. The paper also contains some related conjectures and open problems.

math.AP↗

arXiv2024

Backstepping control for the sterile mosquitoes technique: stabilization of extinction equilibrium

The control of a mosquito population using the sterile insect technique is considered. Building on a model-based approach, where the control input is the release rate of sterilized males, we propose a non-negative backstepping control law capable of globally stabilizing the extinction equilibrium of the system. A simulation study supports and validates the theoretical findings, showing the efficacy of the approach both on a reduced model, used for control design, and on a complete model of the mosquito population dynamics.

math.OC↗

arXiv2024

Graph Neural Networks in Vision-Language Image Understanding: A Survey

2D image understanding is a complex problem within computer vision, but it holds the key to providing human-level scene comprehension. It goes further than identifying the objects in an image, and instead, it attempts to understand the scene. Solutions to this problem form the underpinning of a range of tasks, including image captioning, visual question answering (VQA), and image retrieval. Graphs provide a natural way to represent the relational arrangement between objects in an image, and thus, in recent years graph neural networks (GNNs) have become a standard component of many 2D image understanding pipelines, becoming a core architectural component, especially in the VQA group of tasks. In this survey, we review this rapidly evolving field and we provide a taxonomy of graph types used in 2D image understanding approaches, a comprehensive list of the GNN models used in this domain, and a roadmap of future potential developments. To the best of our knowledge, this is the first comprehensive survey that covers image captioning, visual question answering, and image retrieval techniques that focus on using GNNs as the main part of their architecture.

cs.CV↗

arXiv2024

GNN-LoFI: a Novel Graph Neural Network through Localized Feature-based Histogram Intersection

Graph neural networks are increasingly becoming the framework of choice for graph-based machine learning. In this paper, we propose a new graph neural network architecture that substitutes classical message passing with an analysis of the local distribution of node features. To this end, we extract the distribution of features in the egonet for each local neighbourhood and compare them against a set of learned label distributions by taking the histogram intersection kernel. The similarity information is then propagated to other nodes in the network, effectively creating a message passing-like mechanism where the message is determined by the ensemble of the features. We perform an ablation study to evaluate the network's performance under different choices of its hyper-parameters. Finally, we test our model on standard graph classification and regression benchmarks, and we find that it outperforms widely used alternative approaches, including both graph kernels and graph neural networks.

cs.LG↗

arXiv2024

Biological invasions and epidemics with nonlocal diffusion along a line

The goal of this work is to understand and quantify how a line with nonlocal diffusion given by an integral enhances a reaction-diffusion process occurring in the surrounding plane. This is part of a long term programme where we aim at modelling, in a mathematically rigorous way, the effect of transportation networks on the speed of biological invasions or propagation of epidemics. We prove the existence of a global propagation speed and characterise in terms of the parameters of the system the situations where such a speed is boosted by the presence of the line. In the course of the study we also uncover unexpected regularity properties of the model. On the quantitative side, the two main parameters are the intensity of the diffusion kernel and the characteristic size of its support. One outcome of this work is that the propagation speed will significantly be enhanced even if only one of the two is large, thus broadening the picture that we have already drawn in our previous works on the subject, with local diffusion modelled by a standard Laplacian. We further investigate the role of the other parameters, enlightening some subtle effects due to the interplay between the diffusion in the half plane and that on the line. Lastly, in the context of propagation of epidemics, we also discuss the model where, instead of a diffusion, displacement on the line comes from a pure transport term.

math.AP↗

arXiv2023

MUSTANG: Multi-Stain Self-Attention Graph Multiple Instance Learning Pipeline for Histopathology Whole Slide Images

Whole Slide Images (WSIs) present a challenging computer vision task due to their gigapixel size and presence of numerous artefacts. Yet they are a valuable resource for patient diagnosis and stratification, often representing the gold standard for diagnostic tasks. Real-world clinical datasets tend to come as sets of heterogeneous WSIs with labels present at the patient-level, with poor to no annotations. Weakly supervised attention-based multiple instance learning approaches have been developed in recent years to address these challenges, but can fail to resolve both long and short-range dependencies. Here we propose an end-to-end multi-stain self-attention graph (MUSTANG) multiple instance learning pipeline, which is designed to solve a weakly-supervised gigapixel multi-image classification task, where the label is assigned at the patient-level, but no slide-level labels or region annotations are available. The pipeline uses a self-attention based approach by restricting the operations to a highly sparse k-Nearest Neighbour Graph of embedded WSI patches based on the Euclidean distance. We show this approach achieves a state-of-the-art F1-score/AUC of 0.89/0.92, outperforming the widely used CLAM model. Our approach is highly modular and can easily be modified to suit different clinical datasets, as it only requires a patient-level label without annotations and accepts WSI sets of different sizes, as the graphs can be of varying sizes and structures. The source code can be found at https://github.com/AmayaGS/MUSTANG.

cs.CV↗

arXiv2022

Asymptotic one-dimensional symmetry for the Fisher-KPP equation

Let $u$ be a solution of the Fisher-KPP equation $$ \partial_t u=Δu+f(u),\quad t>0,\ x\in\mathbb{R}^N. $$ We address the following question: does $u$ become locally planar as $t\to+\infty$ ? Namely, does $u(t_n,x_n+\cdot)$ converge locally uniformly, up to subsequences, towards a one-dimensional function, for any sequence $((t_n,x_n))_{n\in\mathbb{N}}$ in $(0,+\infty)\times\mathbb{R}^N$ such that $t_n\to+\infty$ as $n\to+\infty$ ? This question is in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. The answer depends on the initial datum $u_0$ of $u$. It is known to be affirmative when the support of $u_0$ is bounded or when it lies between two parallel half-spaces. Instead, the answer is negative when the support of $u_0$ is "V-shaped". We prove here that $u$ is asymptotically locally planar when the support of $u_0$ is a convex set (satisfying in addition a uniform interior ball condition), or, more generally, when it is at finite Hausdorff distance from a convex set. We actually derive the result under an even more general geometric hypothesis on the support of $u_0$. We recover in particular the aforementioned results known in the literature. We further characterize the set of directions in which $u$ is asymptotically locally planar, and we show that the asymptotic profiles are monotone. Our results apply in particular when the support of $u_0$ is the subgraph of a function with vanishing global mean.

math.AP↗

arXiv2022

Spreading sets and one-dimensional symmetry for reaction-diffusion equations

We consider reaction-diffusion equations $\partial_tu=Δu+f(u)$ in the whole space $\mathbb{R}^N$ and we are interested in the large-time dynamics of solutions ranging in the interval $[0,1]$, with general unbounded initial support. Under the hypothesis of the existence of a traveling front connecting $0$ and $1$ with a positive speed, we discuss the existence of spreading speeds and spreading sets, which describe the large-time global shape of the level sets of the solutions. The spreading speed in any direction is expressed as a Freidlin-Gärtner type formula. This formula holds under general assumptions on the reaction and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We then investigate the local properties of the level sets at large time. Some flattening properties of the level sets of the solutions, if initially supported on subgraphs, will be presented. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we state some asymptotic local one-dimensional and monotonicity symmetry properties for the elements of the $Ω$-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we present some logarithmic-in-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. Some related conjectures and open problems are also listed.

math.AP↗

arXiv2022

Spreading speeds and spreading sets of reaction-diffusion equations

This paper deals with the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in $\mathbb{R}^N$. We prove a variational formula for the spreading speeds in any direction, and we also describe the asymptotic shape of the level sets of the solutions at large time. The Freidlin-Gärtner type formula for the spreading speeds involves newly introduced notions of bounded and unbounded directions of the initial support. The results hold for a large class of reaction terms and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We also prove some results of independent interest on some conditions guaranteeing the spreading of solutions with large initial support and the link between these conditions and the existence of traveling fronts with positive speed. The proofs use a mix of ODE and PDE methods, as well as some geometric arguments. The results are sharp and counterexamples are shown when the assumptions are not all fulfilled.

math.AP↗

arXiv2021

Coupled reaction-diffusion equations on adjacent domains

We consider a reaction-diffusion system for two densities lying in adjacent domains of $\mathbb{R}^N$. We treat two configurations: either a cylinder and its complement, or two half-spaces. Diffusion and reaction heterogeneities for the two densities are considered, and an exchange occurs through the separating boundary. We study the long-time behavior of the solution, and, when it converges to a positive steady state, we prove the existence of an asymptotic speed of propagation in some specific directions. Moreover, we determine how such a speed qualitatively depends with respect to several parameters appearing in the model. In the case $N=2$, we compare such properties to those studied in [6-9] for a model with a line representing a road of fast diffusion at the boundary of a half-plane, which can be seen as a singular limit of the problem studied here.

math.AP↗

arXiv2021

Modeling the propagation of riots, collective behaviors, and epidemics

This paper is concerned with a family of Reaction-Diffusion systems that we introduced in [15], and that generalizes the SIR type models from epidemiology. Such systems are now also used to describe collective behaviors.In this paper, we propose a modeling approach for these apparently diverse phenomena through the example of the dynamics of social unrest. The model involves two quantities: the level of social unrest, or more generally activity, u, and a field of social tension v, which play asymmetric roles. We think of u as the actually observed or explicit quantity while v is an ambiant, sometimes implicit, field of susceptibility that modulates the dynamics of u. In this article, we explore this class of model and prove several theoretical results based on the framework developed in [15], of which the present work is a companion paper. We particularly emphasize here two subclasses of systems: tension inhibiting and tension enhancing. These are characterized by respectively a negative or a positivefeedback of the unrest on social tension. We establish several properties for these classes and also study some extensions. In particular, we describe the behavior of the system following an initial surge of activity. We show that the model can give rise to many diverse qualitative dynamics. We also provide a variety of numerical simulations to illustrate our results and to reveal further properties and open questions.

math.AP↗

arXiv2021

Community Detection in Multiplex Networks

A multiplex network models different modes of interaction among same-type entities. In this article we provide a taxonomy of community detection algorithms in multiplex networks. We characterize the different algorithms based on various properties and we discuss the type of communities detected by each method. We then provide an extensive experimental evaluation of the reviewed methods to answer three main questions: to what extent the evaluated methods are able to detect ground-truth communities, to what extent different methods produce similar community structures and to what extent the evaluated methods are scalable. One goal of this survey is to help scholars and practitioners to choose the right methods for the data and the task at hand, while also emphasizing when such choice is problematic.

cs.SI↗

arXiv2020

Propagation of epidemics along lines with fast diffusion

It has long been known that epidemics can travel along communication lines, such as roads. In the current COVID-19 epidemic, it has been observed that major roads have enhanced its propagation in Italy. We propose a new simple model of propagation of epidemics which exhibits this effect and allows for a quantitative analysis. The model consists of a classical $SIR$ model with diffusion, to which an additional compartment is added, formed by the infected individuals travelling on a line of fast diffusion. Exchanges between individuals on the line and in the rest of the domain are taken into account. A classical transformation allows us to reduce the proposed model to a system analogous to one we had previously introduced [5] to describe the enhancement of biological invasions by lines of fast diffusion. We establish the existence of a minimal spreading speed and we show that it may be quite large, even when the basic reproduction number $R_0$ is close to $1$. More subtle qualitative features of the final state, showing the important influence of the line, are also proved here.

math.AP↗