## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 5, Number 4 (1995), 1217-1235.

### Differential Equations for Random Processes and Random Graphs

#### Abstract

General criteria are given to ensure that in a family of discrete random processes, given parameters exhibit convergence to the solution of a system of differential equations. As one application we consider random graph processes in which the maximum degree is bounded and show that the numbers of vertices of given degree exhibit this convergence as the total number of vertices tends to infinity. Two other applications are to random processes which generate independent sets of vertices in random $r$-regular graphs. In these cases, we deduce almost sure lower bounds on the size of independent sets of vertices in random $r$-regular graphs.

#### Article information

**Source**

Ann. Appl. Probab., Volume 5, Number 4 (1995), 1217-1235.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177004612

**Digital Object Identifier**

doi:10.1214/aoap/1177004612

**Mathematical Reviews number (MathSciNet)**

MR1384372

**Zentralblatt MATH identifier**

0847.05084

**JSTOR**

links.jstor.org

**Subjects**

Primary: 05C80: Random graphs [See also 60B20]

**Keywords**

Random graph random process random regular graph independent set differential equations $d$-process

#### Citation

Wormald, Nicholas C. Differential Equations for Random Processes and Random Graphs. Ann. Appl. Probab. 5 (1995), no. 4, 1217--1235. doi:10.1214/aoap/1177004612. https://projecteuclid.org/euclid.aoap/1177004612