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

What about negative social interaction? Zachary Karate club Anthony Bonato

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

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

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

Anti-Transitivity Anthony Bonato

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

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

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?

G0 = C4 Anthony Bonato

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.

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: …

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.)

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