1. Modelling and Searching Networks Lecture 3 – ILT model
Miniconference on the Mathematics of Computation
MTH 707
Modelling and Searching Networks Lecture 3 – ILT model
Dr. Anthony Bonato
Ryerson University

2. Key properties of complex networks
Large scale.
Evolving over time.
Power law degree distributions.
Small world properties.
Other properties are also important: densification power law, shrinking distances,…

3. Transitivity
Anthony Bonato

4. Iterated Local Transitivity (ILT) model (Bonato, Hadi, Horn, Prałat, Wang, 08)
key paradigm is transitivity: friends of friends are more likely friends
nodes often only have local influence
evolves over time, but retains memory of initial graph

5. ILT model start with a graph of order n
to form the graph Gt+1 for each node x from time t, add a node x’, the clone of x, so that xx’ is an edge, and x’ is joined to each node joined to x
order of Gt is n2t

6. G0 = C4
Anthony Bonato

7. Exercise
4.1 Why would ILT be an appropriate model for a social network such as Facebook?
4.2 What are some drawbacks of the ILT model?

8. “All models are wrong, but some are more useful.” - George Box

9. Properties of ILT model
average degree increasing to with time
average distance bounded by constant and converging, and in many cases decreasing with time; diameter does not change
clustering coefficient higher than in a random generated graph with same average degree
bad expansion: small gaps between 1st and 2nd eigenvalues in adjacency and normalized Laplacian matrices of Gt

10. et ≈ nta, where a = log(3)/log(2).
Densification
nt = order of Gt, et = size of Gt
Lemma 4.1: For t > 0,
nt = 2tn0, et = 3t(e0+n0) - nt.
→ densification power law:
et ≈ nta, where a = log(3)/log(2).

11. Exercise
4.3 Prove Lemma (Hint: use induction and draw a picture explaining how new edges are added.)

12. Average distance Theorem 4.2: If t > 0, then 2
average distance bounded by a constant, and converges; for many initial graphs (large cycles) it decreases
diameter does not change from time 0

13. Clustering Coefficient
Theorem 4.3: If t > 0, then
c(Gt) = ntlog(7/8)+o(1).
higher clustering than in a random graph G(nt,p) with same order and average degree as Gt, which satisfies
c(G(nt,p)) = ntlog(3/4)+o(1)

14. …Degree distribution generate power law graphs from ILT?
ILT model gives a binomial-type distribution

15. Exercise 4.4 Let degt(z) be the degree of z at time t.
a) Show that if x is in Gt, then degt+1(x) = 2degt(x)+1.
b) Show that degt+1(x’) = degt(x) +1.
4.5 Assuming you start with a 3-clique, find the degrees of all vertices after two additional time-steps of the ILT model (Hint: you don’t have to draw this graph!).