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Week 4 – Random Graphs Dr. Anthony Bonato Ryerson University AM8002 Fall 2014

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Complex Networks2 Random graphs Paul ErdősAlfred Rényi

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Complex Networks3

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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 5 123 4

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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

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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

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A.a.s. an event A n 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.

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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

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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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11 Degrees and diameter Theorem 4.2: A.a.s. the degree of each vertex of G in G(n,p) equals concentration: binomial distribution

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Markov’s inequality Theorem 4.3 (Markov’s inequality) For any non-negative random variable X and t > 0, we have that

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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

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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 f x|y (x|y)=Pr[X=x | Y=y] note that for a fixed y, f x|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]

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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

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Definition a martingale is a sequence (X 0,X 1,...,X t ) of random variables over a given probabiltiy space such that for all i > 0, E[X i | X 0,X 1,...,X i-1 ] = X i-1

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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 X i denote the gamblers bankroll after i flips then (X 0,X 1,...,X t ) is a martingale, since: E[X i | X 0,X 1,...,X i-1 ] = 1/2(X i-1 +1)+1/2(X i-1 -1) = X i

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Doob martingales let A, Z 1,..., Z t be random variables define X 0 = E[A], X i = E[A| Z 1,..., Z i ] for 1 ≤ i ≤ t can be shown that (X 0,X 1,...,X t ) is a martingale; called the Doob martingale Idea: A = f(Z 1,..., Z t ) is some function f, with X 0 = E[A] and X t = A each Z i is “revealed” more and more until we know everything and hence, A

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Azuma-Hoeffding inequality Theorem 4.5 Let (X 0,X 1,...,X t ) be a martingale such that |X i+1 – X i | ≤ c for all i (c-Lipschitz condition). Then for all λ > 0, concentration inequality

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Example: vertex colouring let A = χ(G(n,p)), and let Z i 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

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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)))!

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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

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