## Dynamics on Graphs

*By Rick Durrett, Duke U.*

An extensive rewrite of the 2007 book Random Graph Dynamics. The new version studies
a wide varitey of dynamics on a small number of graphs, primarily the configuration
model and inhomogeneous random graphs, which of course contain Erdos-Renyi and random
regular graphs as special cases. *'s indicate sections that are not yet written.

Jan 25, 2023
1.1. Branching Processes

1.2. Cluster growth as an epidemics

1.3. Cluster growth as a random walk

1.4. Diameter of the giant component

1.5. CLT for the giant component

1.6. Combinatorial approach

1.7. Critical regime

1.8. Critical exponents

1.9. Scaling theory

1.10 Threshold for connectivity

Feb 18, 2023
2.1. Configuration model

2.2. Limiting degree distribution approach

2.3. Distances: finite variance

2.4. Distances: power laws 2 < γ < 3

2.5. Subcritical cluster sizes

2.6. Critical regime

2.7. Percolation

Feb 18, 2023
3.1. Finitely many types

3.2. Motivating examples

3.3. Welcome to the machinge

3.4. Results for the survival probability

3.5. Survival probability for examples

3.6. Subcritical cluster sizes

March 6, 2023
4.1. SIR epidemics with fixed infection times

4.2. SIR epidemics with general infection times

4.3. SIR on the complete graph

4.4. Miller-Volz equations

March 6, 2023
5.1. Trees

5.2. Power-law random graphs

5.3. Results for the star graph

5.4. Prolonged persistence for power-laws

5.5. Galton-Watson trees

March 11, 2023
6.1. Spectral gap

6.2. Conductance

6.3. First degree distribution, min degree 3

6.4. Only degrees 2 and 3

*6.5. Random regular graphs (Lubetsky and Sly)

### 7. Voter model, coalescing random walk

### 8. Dynamics on evolving graphs