1. Modelling and Searching Networks Lecture 4 – ILAT model
Miniconference on the Mathematics of Computation
MTH 707
Modelling and Searching Networks Lecture 4 – ILAT model
Dr. Anthony Bonato
Ryerson University

2. What about negative social interaction?
Zachary Karate club
Anthony Bonato

3. Structural balance theory
considers triads of nodes
triads seek closure (Heider,58), (Easley,Kleinberg,10)
balanced
unbalanced
Anthony Bonato

4. Eg 1: Market graph nodes: stocks
edges: negative correlation (competition)
properties (Boginski,Butenko,Pardalos,03)
power law
small world
cliques and co-cliques
Anthony Bonato

5. Eg 2: Big Brother nodes: players edges: votes for eviction
Anthony Bonato

6. Anti-Transitivity
Anthony Bonato

7. Iterated Local Anti-Transitivity (ILAT) model (Bonato,Infeld,Pokhrel,Prałat,17)
key paradigm is anti-transitivity: “enemies of enemies are friends”
enemies: non-edges
nodes have global influence
evolves over time
Anthony Bonato

8. ILAT 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 anti-clone of x, so that xx′ is a non-edge, and x′ is joined to each node non-joined to x
Anthony Bonato

9. Exercise
5.1 Why is the ILAT model an appropriate one for the Survivor or Big Brother network? 5.2 What are some of the drawbacks of the model?

10. G0 = C4
Anthony Bonato

11. Exercise
5.3 Draw the first 2 additional time-steps if the initial graph is the 3-clique.
5.4 Repeat, but with the 3-vertex graph with no edges.

12. Degrees
Lemma 5.0 Let degt(z) be the degree of z at time t. a) Show that if x is in Gt, then degt+1(x) = nt - 1. b) Show that degt+1(x’) = nt - degt(x) - 1. Example: If the initial graph is a 3-clique, the degrees after two additional time-steps: …

13. Exercise
5.8 Derive the expression for the number of edges at time t+1: et+1 = n2t - et - nt. (Hint: draw a picture and use Lemma 5.0.)

14. Properties (BIPP,17) Theorem 5.1: Densification: et / nt = Ω( 2 𝑡 )
it can also be shown that after at least two time-steps, the diameter is 3
Anthony Bonato