1. Modelling and Searching Networks Lecture 5 – Random graphs
Miniconference on the Mathematics of Computation
MTH 707
Modelling and Searching Networks Lecture 5 – Random graphs
Dr. Anthony Bonato
Ryerson University

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 1 2 3 4 5
Complex Networks

6. 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

7. 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

8. 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 6.1. A.a.s. G in G(n,p) is diameter 2.
just say: A.a.s. G(n,p) has diameter 2.

9. 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

10. Exercise 6.1 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

11. Degrees and diameter
Theorem 6.2: A.a.s. the degree of each vertex of G in G(n,p) equals
concentration: binomial distribution

12. Markov’s inequality
Theorem 6.3 (Markov’s inequality) For any non-negative random variable X and t > 0, we have that

13. Chernoff bound
Theorem 6.4 (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

14. 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

15. Complex Networks