## 2009 Probability Seminar - 10

- Date: 04/15/2009

University of British Columbia

Branching processes and real world networks

The aim of this talk is to highlight the usefulness of continuous time

branching process theory in understanding refined asymptotics about

various random network models. We shall exhibit their usefulness in two

different contexts:

(1) First passage percolation: Consider a connected network and suppose

each edge in the network has a random positive edge weight.

Understanding the structure and weight of the shortest path between

nodes in the network is one of the most fundamental problems studied in

modern probability theory. In the modern context these problems take an

additional significance with the minimal weight measuring the cost of

sending information while the number of edges on the optimal path

(hopcount) representing the actual time for messages to get between

vertices in the network. In the context of the configuration model of

random networks we shall show how branching processes allow us to find

the limiting distribution of the minimal weight path as well as

establishing a general central limit theorem for the hopcount with

matching means and variances.

(2) Spectral distribution of random trees: Many models of random trees

(including general models embedded in continuous time branching

processes) satisfy a general form of convergence locally to limiting

infinite objects. In this context we find via soft arguments, the

convergence of the spectral distribution of the adjacency matrix to a

limiting (model dependent) non random distribution. For any $\gamma$ we

also find a sufficient condition for there to be a positive mass at

$\gamma$ in the limit.

Joint work with Remco van der Hoftsad, Gerard Hooghiemstra, Steve Evans

and Arnab Sen.

3:00pm-4:00pm, WMAX 216