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

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

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

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 W  1/2 (no matter how you reorder the nodes)

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

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

probability that nodes i and j are connected: The limit: 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

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

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

Label node born in step k, connecting to {1,…,m}, by (k/n, m/k) - Nodes with label (x 1, y 1 ) and (x 2, y 2 ) ( x 1 < x 2 ) are connected iff - Labels are uniformly distributed in the unit square 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) /( 2m+n) 2 Allow multiple edges!!! Repeat until we insert edges.

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

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

Can we construct a sequence converging to 1 - xy? Method 1: W -random graph x 1,…,x n,…: independent points from [0,1] connect x i and x j with probability 1 - x i x j 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