1. Alan Frieze af1p@random.math.cmu.edu Charalampos (Babis) E. Tsourakakis ctsourak@math.cmu.edu WAW 2012 22 June ‘12 WAW '121

2.  Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '122

3. 3 Internet Map [lumeta.com] Food Web [Martinez ’91] Protein Interactions [genomebiology.com] Friendship Network [Moody ’01]

4.  Modelling “real-world” networks has attracted a lot of attention. Common characteristics include:  Skewed degree distributions (e.g., power laws).  Large Clustering Coefficients  Small diameter  A popular model for modeling real-world planar graphs are Random Apollonian Networks. WAW '124

5. 5 Apollonius (262-190 BC ) Construct circles that are tangent to three given circles οn the plane.

6. WAW '126 Apollonian Gasket

7. WAW '127 Higher Dimensional (3d) Apollonian Packing. From now on, we shall discuss the 2d case.

8.  Dual version of Apollonian Packing WAW '128

9.  Start with a triangle (t=0).  Until the network reaches the desired size  Pick a face F uniformly at random, insert a new vertex in it and connect it with the three vertices of F WAW '129

10. 10

11.  Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1211

12. WAW '1212

13. WAW '1213

14. DegreeTheoremSimulation 30.40.3982 40.20.2017 50.1143 60.07140.0715 70.0476 80.03330.0332 90.02420.0243 100.01820.0179 110.01400.0137 120.01100.0111 WAW '1214

15.  Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1215

16. WAW '1216 Depth of a face (recursively): Let α be the initial face, then depth(α)=1. For a face β created by picking face γ depth(β)=depth(γ)+1. e.g.,

17. WAW '1217

18. WAW '1218

19. WAW '1219

20. WAW '1220

21. WAW '1221

22. WAW '1222

23. WAW '1223

24. WAW '1224 Broutin Devroye Large Deviations for the Weighted Height of an Extended Class of Trees. Algorithmica 2006

25.  This cannot be used though to get a lower bound: WAW '1225 Diameter=2, Depth arbitrarily large

26.  Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1226

27. WAW '1227

28. WAW '1228

29.  Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1229

30. WAW '1230 Chung LuVu Mihail Papadimitriou

31. WAW '12 31 S1S1 S2S2 S3S3 …. Star forest consisting of edges between S 1 and S 3 -S’ 3 where S’ 3 is the subset of vertices of S 3 with two or more neighbors in S 1.

32. WAW '1232 ….

33. WAW '1233

34. WAW '1234

35.  Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1235

36. WAW '1236 Conductance Φ is at most t -1/2. Conjecture: Φ= Θ(t -1/2 ) Are RANs Hamiltonian? Conjecture: No Length of the longest path? Conjecture: Θ(n)

37. Thank you! WAW '1237