A paper that discusses whether randomly grown graphs, as opposed to static random graphs, are truly random.

1. Abhranil Das Spring 2012

2. Real vs. Random
• Many real networks grow.
• These have highly skewed degree
distributions: often a power law.
• Random graphs usually have Poisson
distribution

3. The Model
• Start from a single vertex without edges.
• At each step a vertex is added.
• With probability δ, two vertices are joined with
an undirected edge.
• At time t: t vertices and δt (avg.) edges.
• Different from preferential attachment model.

4. Degree Evolution
Equations are approximate only in the short term.

5. Degree Distribution

6. Component Distribution Evolution

7. Component Distribution Evolution

8. Generating Function for Component
Distribution

9. Giant Component Phase Transition

10. Arbitrary Static Distribution

11. Comparison with Static Graph

12. References
Are randomly grown graphs really random?
Callaway et al, arXiv:cond-mat/0104546v2, 14 June 2001
M. Molloy and B. Reed, Random Structures and
Algorithms 6, 161 (1995).
M. Molloy and B. Reed, Combinatorics,
Probability and Computing 7, 295 (1998).