Limits of randomly grown graph sequences Katalin Vesztergombi

Limits of randomly grown graph sequences Katalin Vesztergombi Eötvös University, Budapest With: Christian Borgs, Jennifer Chayes, László Lovász, Vera Sós

Convergent graph sequences Probability that random map V(F) V(G) is a hom Example: random graphs with probability 1

The limit object For every convergent graph sequence (Gn) there is a such that Lovász-Szegedy

0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 0

Half-graphs

A random graph with 100 nodes and with edge density 1/2 W 1/2

Rearranging the rows and columns

A random graph with 100 nodes and with edge density 1/2 (no matter how you reorder the nodes) W 1/2

Randomly grown uniform attachment graph At step n: - a new node is born; - any two nodes are connected with probability 1/n Ignore multiplicity of edges

A randomly grown uniform attachment graph with 200 nodes

After n steps: probability that nodes i < j are not connected: expected degree of j: expected number of edges:

The limit: probability that nodes i and j are connected: if i=xn and j=yn These are independent events for different i, j.

A randomly grown uniform attachment graph with 200 nodes Proof: By estimating the cut-distance.

Randomly grown prefix attachment graph At step n: - a new node is born; - connects to a random previous node and all its predecessors

This tends to some shades of gray; is that the limit? No, by computing triangle densities! Is this graph sequence convergent at all? A randomly grown prefix attachment graph with 200 nodes Yes, by computing subgraph densities!

This also tends to some shades of gray; is that the limit? No… A randomly grown prefix attachment graph with 200 nodes (ordered by degrees)

Label node born in step k, connecting to {1, …, m}, by (k/n, m/k) - Labels are uniformly distributed in the unit square - Nodes with label (x 1, y 1) and (x 2, y 2) (x 1< x 2) are connected iff Limit can be represented as

The limit of randomly grown prefix attachment graphs (as a function on [0, 1]2)

Preferential attachment graph on n fixed nodes At step m: any two nodes i and j are connected with probability (d(i)+1)(d(j)+1)/(2 m+n)2 Allow multiple edges!!! Repeat until we insert edges.

A preferential attachment graph with 200 fixed nodes and with 5, 000 (multiple) edges

Proof by computing t(F, Gn) A randomly grown preferential attachment graph with 200 fixed nodes ordered by degrees and with 5, 000 (multiple) edges

Can we construct a sequence converging to 1 -xy? Method 1: W-random graph x 1, …, xn, …: independent points from [0, 1] connect xi and xj with probability 1 -xi xj Works for any W

Method 2: growing in order At step n: - a new node is born, and connected to i with prob (n-i)/n - any two old nodes are connected with probability 1/n Ignore multiplicity of edges Works for any monotone decreasing W