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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., Comment: 28 pages

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.

In this paper, we present GRASP, 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. Our permutation invariant model can also handle node features by concatenating them to the eigenvectors of each node. 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 methods. This is achieved by truncating the spectrum, which as we show in our experiments results in a faster yet accurate generative process. An extensive set of experiments on both synthetic and real world graphs demonstrates the strengths of our model against state-of-the-art alternatives.

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., Comment: Accepted for publication at BMVC 2023

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.

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., Comment: EMS Surveys in Mathematical Sciences, In press. arXiv admin note: substantial text overlap with arXiv:2105.08344

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., Comment: 20 pages, 5 figures, 5 tables

We consider reaction-diffusion equations $\partial_tu=\Delta u+f(u)$ 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-G\"artner 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 $\Omega$-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., Comment: arXiv admin note: substantial text overlap with arXiv:2105.08344; text overlap with arXiv:2207.05147

Let $u$ be a solution of the Fisher-KPP equation $$ \partial_t u=\Delta u+f(u),\quad t>0,\ x\in\mathbb{R}^N. $$ 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., Comment: arXiv admin note: text overlap with arXiv:2105.08344

The convolution operator at the core of many modern neural architectures can effectively be seen as performing a dot product between an input matrix and a filter. While this is readily applicable to data such as images, which can be represented as regular grids in the Euclidean space, extending the convolution operator to work on graphs proves more challenging, due to their irregular structure. In this paper, we propose to use graph kernels, i.e. kernel functions that compute an inner product on graphs, to extend the standard convolution operator to the graph domain. This allows us to define an entirely structural model that does not require computing the embedding of the input graph. Our architecture allows to plug-in any type of graph kernels and has the added benefit of providing some interpretability in terms of the structural masks that are learned during the training process, similarly to what happens for convolutional masks in traditional convolutional neural networks. We perform an extensive ablation study to investigate the model hyper-parameters' impact and show that our model achieves competitive performance on standard graph classification and regression datasets., Comment: IEEE Transactions on Neural Networks and Learning Systems (2024)

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-G\"artner 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.

We analyze a mean-field game model proposed by economists R.E. Lucas and B. Moll (2014) to describe economic systems where production is based on knowledge growth and diffusion. This model reduces to a PDE system where a backward Hamilton-Jacobi-Bellman equation is coupled with a forward KPP-type equation with nonlocal reaction term. We study the existence of traveling waves for this mean-field game system, obtaining the existence of both critical and supercritical waves. In particular we prove a conjecture raised by economists on the existence of a critical balanced growth path for the described economy, supposed to be the expected stable growth in the long run. We also provide nonexistence results which clarify the role of parameters in the economic model.

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., Comment: 149 pages, 15 figures. Detailed version, including a Toolbox chapter

The semantic segmentation of parts of objects in the wild is a challenging task in which multiple instances of objects and multiple parts within those objects must be detected in the scene. This problem remains nowadays very marginally explored, despite its fundamental importance towards detailed object understanding. In this work, we propose a novel framework combining higher object-level context conditioning and part-level spatial relationships to address the task. To tackle object-level ambiguity, a class-conditioning module is introduced to retain class-level semantics when learning parts-level semantics. In this way, mid-level features carry also this information prior to the decoding stage. To tackle part-level ambiguity and localization we propose a novel adjacency graph-based module that aims at matching the relative spatial relationships between ground truth and predicted parts. The experimental evaluation on the Pascal-Part dataset shows that we achieve state-of-the-art results on this task., Comment: ECCV 2020

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.

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.

We introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain, under boundary condition of the oblique derivative type. We employ these notions in the stability analysis of semilinear problems. Some of the properties we derive are new even in the Dirichlet or in the whole space cases. As an application, we show the validity of the hair-trigger effect for the Fisher-KPP equation on general, uniformly smooth domains.

We have implemented prescriptions for modelling pulsars in the rapid binary population synthesis code COMPAS. We perform a detailed analysis of the double neutron star (DNS) population, accounting for radio survey selection effects. The surface magnetic field decay timescale (${\sim}1000$\,Myr) and mass scale (${\sim}0.02$\,M$_\odot$) are the dominant uncertainties in our model. Mass accretion during common envelope evolution plays a non-trivial role in recycling pulsars. We find a best-fit model that is in broad agreement with the observed Galactic DNS population. Though the pulsar parameters (period and period derivative) are strongly biased by radio selection effects, the observed orbital parameters (orbital period and eccentricity) closely represent the intrinsic distributions. The number of radio observable DNSs in the Milky Way at present is about 2500 in our model, corresponding to approximately 10\% of the predicted total number of DNSs in the galaxy. Using our model calibrated to the Galactic DNS population, we make predictions for DNS mergers observed in gravitational waves. The DNS chirp mass distribution varies from 1.1M$_\odot$ to 2.1M$_\odot$ and the median is found to be 1.14\,M$_\mathrm{\odot}$. The expected effective spin $\chi_\mathrm{eff}$ for isolated DNSs is $\lesssim$0.03 from our model. We predict that 34\% of the current Galactic isolated DNSs will merge within a Hubble time, and have a median total mass of 2.7\,M$_\mathrm{\odot}$. Finally, we discuss implications for fast radio bursts and post-merger remnant gravitational-waves., Comment: 26 pages, Accepted for publication by MNRAS

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., Comment: 55 pages. Accepted for publication on ACM Computing Surveys in a shorter version

Bipartite networks are a well known strategy to study a variety of phenomena. The commonly used method to deal with this type of network is to project the bipartite data into a unipartite weighted graph and then using a backboning technique to extract only the meaningful edges. Despite the wide availability of different methods both for projection and backboning, we believe that there has been little attention to the effect that the combination of these two processes has on the data and on the resulting network topology. In this paper we study the effect that the possible combinations of projection and backboning techniques have on a bipartite network. We show that the 12 methods group into two clusters producing unipartite networks with very different topologies. We also show that the resulting level of network centralization is highly affected by the combination of projection and backboning applied.

We develop a novel method for measuring the similarity between complete weighted graphs, which are probed by means of discrete-time quantum walks. Directly probing complete graphs using discrete-time quantum walks is intractable due to the cost of simulating the quantum walk. We overcome this problem by extracting a commute-time minimum spanning tree from the complete weighted graph. The spanning tree is probed by a discrete time quantum walk which is initialised using a weighted version of the Perron-Frobenius operator. This naturally encapsulates the edge weight information for the spanning tree extracted from the original graph. For each pair of complete weighted graphs to be compared, we simulate a discrete-time quantum walk on each of the corresponding commute time minimum spanning trees, and then compute the associated density matrices for the quantum walks. The probability of the walk visiting each edge of the spanning tree is given by the diagonal elements of the density matrices. The similarity between each pair of graphs is then computed using either a) the inner product or b) the negative exponential of the Jensen-Shannon divergence between the probability distributions. We show that in both cases the resulting similarity measure is positive definite and therefore corresponds to a kernel on the graphs. We perform a series of experiments on publicly available graph datasets from a variety of different domains, together with time-varying financial networks extracted from data for the New York Stock Exchange. Our experiments demonstrate the effectiveness of the proposed similarity measures.

In this paper, we develop a novel Backtrackless Aligned-Spatial Graph Convolutional Network (BASGCN) model to learn effective features for graph classification. Our idea is to transform arbitrary-sized graphs into fixed-sized backtrackless aligned grid structures and define a new spatial graph convolution operation associated with the grid structures. We show that the proposed BASGCN model not only reduces the problems of information loss and imprecise information representation arising in existing spatially-based Graph Convolutional Network (GCN) models, but also bridges the theoretical gap between traditional Convolutional Neural Network (CNN) models and spatially-based GCN models. Furthermore, the proposed BASGCN model can both adaptively discriminate the importance between specified vertices during the convolution process and reduce the notorious tottering problem of existing spatially-based GCNs related to the Weisfeiler-Lehman algorithm, explaining the effectiveness of the proposed model. Experiments on standard graph datasets demonstrate the effectiveness of the proposed model., Comment: This is an extension work for journals based on the previous work of arXiv:1904.04238v1

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., Comment: 54 pages, 3 figures

We study the large time behaviour of the Fisher-KPP equation $\partial$ t u = $\Delta$u + u -- u 2 in spatial dimension N , when the initial datum is compactly supported. We prove the existence of a Lipschitz function s of the unit sphere, such that u(t, x) converges, as t goes to infinity, to U c * |x| -- c * t + N + 2 c * lnt + s $\infty$ x |x| , where U c * is the 1D travelling front with minimal speed c * = 2. This extends an earlier result of G{\"a}rtner.

This article examines the principles outlined in the General Data Protection Regulation (GDPR) in the context of social network data. We provide both a practical guide to GDPR-compliant social network data processing, covering aspects such as data collection, consent, anonymization and data analysis, and a broader discussion of the problems emerging when the general principles on which the regulation is based are instantiated to this research area., Comment: To appear on ACM Transactions on Social Computing