Random graphs (a droplet) Y. Kohayakawa (So Paulo) NeuroMatFirst - - PowerPoint PPT Presentation

▶

Dec 09, 2023 7 likes •351 views

Random graphs (a droplet) Y. Kohayakawa (So Paulo) NeuroMatFirst Workshop IME/USP 20 January 2014 Partially supported by CNPq (477203/2012-4, 308509/2007-2) and by FAPESP (2013/07699-0, 2013/03447-6) Random graphs Aim 1 Aim of talk A

SLIDE 1

Random graphs (a droplet)

Y. Kohayakawa (São Paulo)

NeuroMat—First Workshop IME/USP 20 January 2014

Partially supported by CNPq (477203/2012-4, 308509/2007-2) and by FAPESP (2013/07699-0, 2013/03447-6)

SLIDE 2

Random graphs Aim

Aim of talk

A glimpse of the theory of random graphs

SLIDE 3

Random graphs Aim

Aim of talk

A glimpse of the theory of random graphs ⊲ The Erd˝

s–Rényi random graph

SLIDE 4

Random graphs Aim

Aim of talk

A glimpse of the theory of random graphs ⊲ The Erd˝

s–Rényi random graph

⊲ A directed variant

SLIDE 5

Random graphs Outline

Outline of the talk

⊲ Preliminaries

SLIDE 6

Random graphs Outline

Outline of the talk

⊲ Preliminaries

Graphs

SLIDE 7

Random graphs Outline

Outline of the talk

⊲ Preliminaries

Graphs

⊲ The Erd˝

s–Rényi random graph

SLIDE 8

Random graphs Outline

Outline of the talk

⊲ Preliminaries

Graphs

⊲ The Erd˝

s–Rényi random graph

⊲ The phase transition

SLIDE 9

Random graphs Outline

Outline of the talk

⊲ Preliminaries

Graphs

⊲ The Erd˝

s–Rényi random graph

⊲ The phase transition ⊲ A version for directed graphs

SLIDE 10

Random graphs Graphs

Graphs

SLIDE 11

Random graphs Graphs

Graphs

⊲ Graph: G = (V, E)

SLIDE 12

Random graphs Graphs

Graphs

⊲ Graph: G = (V, E)

V: set of vertices

SLIDE 13

Random graphs Graphs

Graphs

⊲ Graph: G = (V, E)

V: set of vertices
E: set of edges ( = unordered pairs of vertices)

SLIDE 14

Random graphs Graphs

A graph

SLIDE 15

Random graphs Graphs

A graph

By V. Krebs, from http://www.orgnet.com/Erdos.html

SLIDE 16

Random graphs Random graphs

Random graphs

SLIDE 17

Random graphs Random graphs

Random graphs

⊲ Erd˝

s and Rényi (1959, 1960): systematic study of random graphs.

SLIDE 18

Random graphs Random graphs

Random graphs

⊲ Erd˝

s and Rényi (1959, 1960): systematic study of random graphs.

ER model: G(n, m) =

SLIDE 19

Random graphs Random graphs

Random graphs

⊲ Erd˝

s and Rényi (1959, 1960): systematic study of random graphs.

ER model: G(n, m) = G on [n] and m edges, chosen uniformly at random

SLIDE 20

Random graphs Random graphs

Random graphs

⊲ Erd˝

s and Rényi (1959, 1960): systematic study of random graphs.

ER model: G(n, m) = G on [n] and m edges, chosen uniformly at random ⊲ Uniform model on

([n]

2 )

m

SLIDE 21

Random graphs Random graphs

Random graphs

⊲ Erd˝

s and Rényi (1959, 1960): systematic study of random graphs.

ER model: G(n, m) = G on [n] and m edges, chosen uniformly at random ⊲ Uniform model on

([n]

2 )

m

⊲ G(n, p): binomial variant; 0 ≤ p = p(n) ≤ 1

SLIDE 22

Random graphs Phase transition in G(n, p)

Phase transition: G(n, p)

SLIDE 23

Random graphs Phase transition in G(n, p)

Phase transition: G(n, p)

Theorem 1 (Łuczak (1990), building on Bollobás (1984)). Let np = 1 + ε, where ε = ε(n) → 0 but n|ε|3 → ∞, and k0 = 2ε−2 log n|ε|3.

SLIDE 24

Random graphs Phase transition in G(n, p)

Phase transition: G(n, p)

Theorem 1 (Łuczak (1990), building on Bollobás (1984)). Let np = 1 + ε, where ε = ε(n) → 0 but n|ε|3 → ∞, and k0 = 2ε−2 log n|ε|3. (i) If nε3 → −∞, then G(n, p) a.a.s. contains no component of order greater than k0. Moreover, a.a.s. each component of G(n, p) is either a tree, or contains precisely one cycle.

SLIDE 25

Random graphs Phase transition in G(n, p)

Phase transition: G(n, p)

Theorem 1 (Łuczak (1990), building on Bollobás (1984)). Let np = 1 + ε, where ε = ε(n) → 0 but n|ε|3 → ∞, and k0 = 2ε−2 log n|ε|3. (i) If nε3 → −∞, then G(n, p) a.a.s. contains no component of order greater than k0. Moreover, a.a.s. each component of G(n, p) is either a tree, or contains precisely one cycle. (ii) If nε3 → ∞, then G(n, p) a.a.s. contains exactly one component of

rder greater than k0. This component a.a.s. has (2+o(1))εn vertices.

SLIDE 26

Random graphs Directed graphs

Directed graphs

SLIDE 27

Random graphs Directed graphs

Directed graphs

⊲ Directed graph: D = (V, E)

SLIDE 28

Random graphs Directed graphs

Directed graphs

⊲ Directed graph: D = (V, E)

V: set of vertices

SLIDE 29

Random graphs Directed graphs

Directed graphs

⊲ Directed graph: D = (V, E)

V: set of vertices
E: set of arcs/directed edges = ordered pairs of distinct vertices
E ⊂ (V)2

SLIDE 30

Random graphs Directed graphs

Directed graphs

⊲ Directed graph: D = (V, E)

V: set of vertices
E: set of arcs/directed edges = ordered pairs of distinct vertices
E ⊂ (V)2

⊲ Binomial directed graph: D(n, p)

SLIDE 31

Random graphs Phase transition in D(n, p)

Phase transition: D(n, p)

SLIDE 32

Random graphs Phase transition in D(n, p)

Phase transition: D(n, p)

Theorem 2 (Łuczak and Seierstad (2009); Łuczak (1990); Karp (1990)). Let np = 1 + ε with ε = ε(n) → 0.

SLIDE 33

Random graphs Phase transition in D(n, p)

Phase transition: D(n, p)

Theorem 2 (Łuczak and Seierstad (2009); Łuczak (1990); Karp (1990)). Let np = 1 + ε with ε = ε(n) → 0. (i) If ε3n → −∞, then a.a.s. every strong component in D(n, p) is either a vertex or a cycle of length O(1/|ε|).

SLIDE 34

Random graphs Phase transition in D(n, p)

Phase transition: D(n, p)

Theorem 2 (Łuczak and Seierstad (2009); Łuczak (1990); Karp (1990)). Let np = 1 + ε with ε = ε(n) → 0. (i) If ε3n → −∞, then a.a.s. every strong component in D(n, p) is either a vertex or a cycle of length O(1/|ε|). (ii) If ε3n → ∞, then a.a.s. D(n, p) contains a unique complex compo- nent, of order (4 + o(1))ε2n, whereas every other strong component is either a vertex or a cycle of length O(1/ε).