PYSANUM

PYSANUM

SEMINARI PASSATI

Pisan Young Seminars in Applied and NUmerical Mathematics

Ciclo di seminari informali di analisi numerica e matematica applicata rivolto agli studenti.

L’obiettivo degli incontri è di presentare in maniera accessible argomenti di ricerca di analisi numerica e coinvolgere gli studenti interessati. I seminari avranno una prima parte introduttiva e saranno accessibili anche a chi non ha dimestichezza con l’argomento. Si terranno principalmente in italiano, in linea con il tono informale del ciclo.
Sono incoraggiati a partecipare studenti della magistrale e studenti della triennale che abbiano familiarità con i contenuti del corso di Calcolo Scientifico.

Organizzato da dottorandi dell’Università di Pisa e della Scuola Normale Superiore.

Prossimi Seminari

11.00 – 25 Febbraio 2025
 Francesca Santucci (IMT Alti Studi Lucca)
Aula Riunioni, Dipartimento di Matematica
 Spatial Patterns of Laplacian Eigenmodes in Complex Networks
Complex systems consist of many interacting components, typically exhibiting features such as self-organization, critical transitions, or emergent phenomena.
Network science is particularly useful for studying these kinds of systems, modeling them as graphs, mathematical objects made of nodes (representing the system’s components) and links (representing their interactions).
Beyond graph theory, a more physics-inspired perspective offers additional tools (borrowed, in particular, from statistical physics) for analyzing networks.
Following this approach, we will explore the tight connection between dynamical processes on networks and the topology underneath them.
We will focus in particular on processes governed by the Graph Laplacian, denoted as $\hat{L}$, evolving over time as $\boldsymbol x(\tau)=e^{-\tau \hat{L}}\boldsymbol x(0)$. This equation is valid for diffusion, but also provides a first order description of more general dynamics near the steady state.
We will see the diffusion-based theoretical framework known as the Laplacian Renormalization Group, accounting for the communicability properties of a network and able to unveil its multiscale hierarchical nested structures.
Key tools in this framework, suggesting intriguing physical analogies, include the Graph Density Matrix, which quantifies diffusion streamlines between pairs of nodes, and the Graph Entropy, which measures the heterogeneity of the system’s state after a given diffusion time, and can be considered as an order parameter for second order transitions between diffusion aggregation phases.
We will show that such transitions identify classes of links based on diffusion timescales, corresponding to specific diffusivity modes in the Fourier space and revealing the graph’s topological organization. Finally, we will explain the deep connection between nested modules, Laplacian eigenmodes, and diffusion time scales in hierarchical modular graphs.