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, JanuaryFebruary.
(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 realworld 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 kvertex cycles; edges, wedges and kvertex 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.
GaltonWatson Branching Processes: construction and extinction probability.

Lecture 5  February 02, 2022 
pdf and
video
Extinction probability in GaltonWatson Branching Processes (proof).
Size of the total progeny and exploration process in GaltonWatson 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 subcritical regime.
Tools: Exploration process of the Erdös Rényi random graph. Chernoff bounds.

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

Lecture 8  February 09, 2022 
pdf and
video
Inhomogeneous Random Graphs (IRG): twotypes random graphs
and general (finite types) setting; distribution of
the vertexdegree. Multitype branching processes: construction and basical notation.

Lecture 9  February 10, 2022 
pdf and
video
Survival probabilities in multitype branching processes.
Main results about IRG: local structure; phase transition and giant component; small world property.
Generalized Random Graphs: assumptions on vertexweights (conditions for a scalefree 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 scalefree 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 scalefree percolation, related topics.

2020/2021

2019/2020

2018/2019

2017/2018

20142017

20122014

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
