Related papers: Gaussian estimates for a heat equation on a networ…
We establish Gaussian-type upper bounds on the heat kernel for a continuous-time random walk on a graph with unbounded weights under an ergodicity assumption. For the proof we use Davies' perturbation method, where we show a maximal…
The problem of finding the optimal set of source nodes in a diffusion network that maximizes the spread of information, influence, and diseases in a limited amount of time depends dramatically on the underlying temporal dynamics of the…
In this work, we consider the numerical solution of an initial boundary value problem for the distributed order time fractional diffusion equation. The model arises in the mathematical modeling of ultra-slow diffusion processes observed in…
Deployment of small cell base stations (SBSs) overlaying the coverage area of a macrocell BS (MBS) results in a two-tier hierarchical small cell network. Cross-tier and inter-tier interference not only jeopardize primary macrocell…
The aim of this survey is to serve as an introduction to the different techniques available in the broad field of Aggregation-Diffusion Equations. We aim to provide historical context, key literature, and main ideas in the field. We start…
Nash and Sobolev inequalities are known to be equivalent to ultracontractive properties of heat-like Markov semigroups, hence to uniform on-diagonal bounds on their kernel densities. In non ultracontractive settings, such bounds can not…
Adaptive networks are suitable for decentralized inference tasks, e.g., to monitor complex natural phenomena. Recent research works have intensively studied distributed optimization problems in the case where the nodes have to estimate a…
The rise of Generative AI (GenAI) in recent years has catalyzed transformative advances in wireless communications and networks. Among the members of the GenAI family, Diffusion Models (DMs) have risen to prominence as a powerful option,…
Time plays an essential role in the diffusion of information, influence and disease over networks. In many cases we only observe when a node copies information, makes a decision or becomes infected -- but the connectivity, transmission…
Traditional Graph Neural Networks (GNNs) rely on message passing, which amounts to permutation-invariant local aggregation of neighbour features. Such a process is isotropic and there is no notion of `direction' on the graph. We present a…
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The…
We study the heat semigroup maximal operator associated with a well-known orthonormal system in the d-dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted $L^p$…
In this paper, we study a class of equations representing nonlinear diffusion on networks. A particular instance of our model can be seen as a network equivalent of the porous-medium equation. We are interested in studying perturbations of…
We consider metric graphs with Kirchhoff boundary conditions. We study the intrinsic metric, volume doubling and a Poincar\'e inequality. This enables us to prove a parabolic Harnack inequality. The proof involves various techniques from…
We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are…
The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical…
Network optimization is a fundamental challenge in the Internet of Things (IoT) network, often characterized by complex features that make it difficult to solve these problems. Recently, generative diffusion models (GDMs) have emerged as a…
Distributed learning is the problem of inferring a function in the case where training data is distributed among multiple geographically separated sources. Particularly, the focus is on designing learning strategies with low computational…
Several systems can be modeled as sets of interdependent networks where each network contains distinct nodes. Diffusion processes like the spreading of a disease or the propagation of information constitute fundamental phenomena occurring…
Many works have been proposed in the literature to capture the dynamics of diffusion in networks. While some of them define graphical markovian models to extract temporal relationships between node infections in networks, others consider…