Week 4 – Random Graphs

Dr. Anthony BonatoRyerson University

AM8002Fall 2014

Complex Networks 2

Random graphs

Paul Erdős Alfred Rényi

Complex Networks 3

Complex Networks 5

G(n,p) random graph model(Erdős, Rényi, 63)

• p = p(n) a real number in (0,1), n a positive integer

• G(n,p): probability space on graphs with nodes {1,…,n}, two nodes joined independently and with probability p

51 2 3 4

Formal definition

• n a positive integer p a real number in [0,1]• G(n,p) is a probability space on labelled graphs

with vertex set V = [n] = {1,2,…,n} such that

• NB: p can be a function of n– today, p is a constant

)|(|2)|(| )1()Pr(

GEn

GE ppG

Properties of G(n,p)

• consider some graph G in G(n,p)• the graph G could be any n-vertex graph, so not

much can be said about G with certainty

• some properties of G, however, are likely to hold

• we are interested in properties that occur with high probability when n is large

A.a.s.

• an event An happens asymptotically almost surely (a.a.s.) in G(n,p) if it holds there with probability tending to 1 as n→∞

Theorem 4.1. A.a.s. G in G(n,p) is diameter 2.

• just say: A.a.s. G(n,p) has diameter 2.

First moment method

• in G(n,p), all graph parameters:

|E(G)|, γ(G), ω(G), …

become random variables

• we focus on computing the averages of these parameters or expectation

Discussion

Calculate the expected number of edges in G(n,p).

• use of expectation when studying random graphs is sometimes referred to as the first moment method

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Degrees and diameter

Theorem 4.2: A.a.s. the degree of each vertex of G in G(n,p) equals

• concentration: binomial distribution

pnonpnOpn ))1(1()log(

Markov’s inequality

Theorem 4.3 (Markov’s inequality)

For any non-negative random variable X and t > 0, we have that

./][]Pr[ tXEtX

Chernoff bound

Theorem 4.4 (Chernoff bound)

Let X be a binomially distributed random variable on G(n,p) with E[x] = np. Then for ε ≤ 3/2 we have that

]).[3

exp(2]][||][Pr[|2

XEXEXEX

Martingales

• let X and Y be random variables on the same probability space

• the conditional mass function of X given Y = y is defined by

fx|y(x|y)=Pr[X=x | Y=y]

• note that for a fixed y, fx|y(x|y) is a function of x

• the conditional expection of X when Y=y is given by its expectation

• let g(x) = E[X | Y=y]; g is the conditional expectation of X on Y, written E[X|Y]

x

yx yxxfyYXE ),(]|[ |

Intuition

• E[X|Y] is the expected value of X assuming Y is known

• note that E[X|Y] is a random variable– precise value depends on the value of Y

Definition

• a martingale is a sequence (X0,X1,...,Xt) of random variables over a given probabiltiy space such that for all i > 0,

E[Xi| X0,X1,...,Xi-1] = Xi-1

Example

• a gambler starts with $100• she flips a fair coin t times; when the coin

is heads, she wins $1; tails, she loses $1.• let Xi denote the gamblers bankroll after i

flips• then (X0,X1,...,Xt) is a martingale, since:

E[Xi | X0,X1,...,Xi-1] = 1/2(Xi-1+1)+1/2(Xi-1-1)

= Xi

Doob martingales

• let A, Z1,..., Zt be random variables

• define X0 = E[A], Xi = E[A| Z1,..., Zi ] for 1 ≤ i ≤ t

• can be shown that (X0,X1,...,Xt) is a martingale; called the Doob martingale

• Idea: A = f(Z1,..., Zt ) is some function f, with X0 = E[A] and Xt = A

• each Zi is “revealed” more and more until we know everything and hence, A

Azuma-Hoeffding inequality

Theorem 4.5 Let (X0,X1,...,Xt) be a martingale such that |Xi+1 – Xi| ≤ c for all i (c-Lipschitz condition).

Then for all λ > 0,

• concentration inequality

).2/exp(2]|Pr[| 220 ctXX t

Example: vertex colouring

• let A = χ(G(n,p)), and let Zi contains the information on the presence/absence of edges ij with j < i

• Doob martingale here is called the vertex-exposure martingale– reveal one vertex at a time

Concentration of chromatic number

Theorem 4.6 For G in G(n,p) and all real λ >0,

• hence, χ(G(n,p)) is concentrated around its expectation; proved before anyone knew E(χ(G(n,p)))!

).2/exp(2]|)]([)(Pr[| 2 nGEG

Complex Networks 23

Aside: evolution of G(n,p)

• think of G(n,p) as evolving from a co-clique to clique as p increases from 0 to 1

• at p=1/n, Erdős and Rényi observed something interesting happens a.a.s.:– with p = c/n, with c < 1, the graph is disconnected with all

components trees, the largest of order Θ(log(n))– as p = c/n, with c > 1, the graph becomes connected with a giant

component of order Θ(n)

• Erdős and Rényi called this the double jump• physicists call it the phase transition: it is similar to

phenomena like freezing or boiling

Complex Networks 24