Alessandra Bianchi


Dipartimento di Matematica
Università degli Studi di Padova


Via Trieste, 63 - 35121 Padova, Italy
Room: Ufficio 512, quinto piano
Phone: +39 049 827 1406
Fax:     +39 049 827 1428

e-mail: alessandra.bianchi@unipd.it

 



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TEACHING

  • 2021/2022

    • Calcolo delle Probabilità, Corso di Laurea Magistrale in Statistica, I semestre.

    • Introduzione ai modelli probabilistici, Scuola Galileiana di Padova, III trimestre

    • Probabilità e Statistica, Corso di Laurea Triennale in Matematica, II semestre.

    • Random Graphs and Networks, PhD course in Mathematics, January/February 2022.

      Course of the Doctoral Program in Mathematical Sciences, A.A. 2021/22, January-February.
      (Link to the page of the Doctoral School)

      Course requirements:

      Basic knowledge of probability theory: discrete random variables, finite and countable probability spaces,
      convergence of random variables, convergence theorems (law of large number, central limit theorem).

      Detailed program and references (pdf)

      Examination and grading:
      Seminar

      LECTURES

      • Lecture 1 - January 25, 2022 - pdf and video ( part 1 part 2 )
        Introduction to real-world networks; Graph setting; Common properties of networks and formalism.

      • Lecture 2 - January 26, 2022 - pdf and video
        Random graph setting. Uniform and binomial model: comparison and asymptotic equivalence.
        Monotonicity and thresholds in Erdös Rényi random graph.

      • Lecture 3 - January 27, 2022 - pdf and video
        Thresholds for small subgraphs containment (triangles versus k-vertex cycles; edges, wedges and k-vertex trees).
        Critical window around a threshold: Poisson paradigm.

      • Lecture 4 - February 01, 2022 - pdf and video
        Threshold for connectivity. Sparse and dense regimes in Erdös Rényi random graphs.
        Galton-Watson Branching Processes: construction and extinction probability.

      • Lecture 5 - February 02, 2022 - pdf and video
        Extinction probability in Galton-Watson Branching Processes (proof).
        Size of the total progeny and exploration process in Galton-Watson Branching Processes.

      • Lecture 6 - February 03, 2022 - pdf and video
        Phase transition in Erdös Rényi random graph: existence of a giant component (statement) and analysis of
        the sub-critical regime. Tools: Exploration process of the Erdös Rényi random graph. Chernoff bounds.

      • Lecture 7 - February 08, 2022 - pdf and video
        Super-critical regime in Erdös Rényi random graph: existence of a giant component and small-world property.
        Tool: local convergence (in probability) for a sequence of random graphs.

      • Lecture 8 - February 09, 2022 - pdf and video
        Inhomogeneous Random Graphs (IRG): two-types random graphs and general (finite types) setting; distribution of the vertex-degree. Multi-type branching processes: construction and basical notation.

      • Lecture 9 - February 10, 2022 - pdf and video
        Survival probabilities in multi-type branching processes.
        Main results about IRG: local structure; phase transition and giant component; small world property.
        Generalized Random Graphs: assumptions on vertex-weights (conditions for a scale-free behavior) and main results.

      • Lecture 10 - February 15, 2022 - pdf and video
        Configuration model: construction with uniform matchings; simplicity probability and uniform models.
        Assumptions on the degree sequence: sparse regime and scale-free property.
        Unimodular branching processes: definition and survival probability.

      • Lecture 11 - February 16, 2022 - pdf and video
        Main results about configuration model: local structure; phase transition and giant component; small world property.
        Preferential attachment model: construction with intermediate updating of the degrees and basical properties.

      • Lecture 12 - February 17, 2022 - pdf and video
        Main results about preferential attachment model: local structure; connectivity; small world property.
        Further directions: clustering coefficient, spatial random graphs and scale-free percolation, related topics.


    • 2020/2021

    • 2019/2020

    • 2018/2019

    • 2017/2018

    • 2014-2017

    • 2012-2014

      • Statistica, Corso di Laurea in Biologia Molecolare, II semestre.


    TEACHING in Bologna


COLLABORATORS

Gianmarco Bet
Anton Bovier
Stefano Bregni
Massimo Campanino
Francesca Collet
Irene Crimaldi
Giampaolo Cristadoro
Sander Dommers
Marco Ferrari
Alexandre Gaudillière
Cristian Giardinà
Dima Ioffe
Andreas Knauf
Marco Lenci
Marilena Ligabó
Elena Magnanini
Paolo Milanesi
Françoise Pène
Gaia Pozzoli
Samuele Stivanello


LINKS

  • Probability Group of Padova

  • DAI Seminar

  • Mathematics ArXiv

  • MathSciNet Full Search

  • About Padova