Fractality vs self-similarity in scale-free networks

1. Fractality vsself-similarity in scale-free networks B. KahngSeoul Nat’l Univ., Korea & CNLS, LANL Jin S. Kim, K.-I. Goh, G. Salvi, E. Oh and D. Kim The 2nd KIAS Conference on Stat. Phys., 07/03-06/06

2. Contents I. Fractal scaling in SF networks [1] K.-I. Goh, G. Salvi, B. Kahng and D. Kim, Skeleton and fractal scaling in complex networks, PRL 96, 018701 (2006). [2] J.S. Kim, et al., Fractality in ocmplex networks:Critical and supercritical skeletons, (cond-mat/0605324). II. Self-similarity in SF networks [1] J.S. Kim, Block-size heterogeneity and renormalization in scale-free networks, (cond-mat/0605587).

3. Introduction Introduction Network • node, link, & degree Networks are everywhere

4. Random graph model by Erdős & Rényi [Erdos & Renyi 1959] Put an edge between each vertex pair with probability p • Poisson degree distribution • D ~ lnN • Percolation transition at p=1/N

5. Scale-free network: the static model 5-α 6-α 1-α 4-α The number of vertices is fixed as N . 2-α 8-α 3-α 7-α Two vertices are selected with probabilities pi pj. Goh et al., PRL (2001).

6. I-1. Fractality I. Fractal scaling in SF networks Song, Havlin, and Makse, Nature (2005). Box-covering method: Mean mass (number of nodes) within a box: Contradictory to the small-worldness:  Cluster-growing method

7. I-2. Box-counting Random sequential packing: Nakamura (1986), Evans (1987) • At each step, a node is selected randomly and served as a seed. • Search the network by distance from the seed and assign newly burned vertices to the new box. • Repeat (1) and (2) until all nodes are assigned their respective boxes. • is chosen as the smallest number of boxes among all the trials. 3 1 2 4

8. I-2. Box-counting Box mass inhomogeneity WWW Fractal scaling dB= 4.1

9. I-2. Box-counting Log Box Number dB Log Box Size Box-covering method: Fractal dimension dB

10. I-3. Purposes Fractal complex networks www, metabolic networks, PIN (homo sapiens) PIN (yeast, *), actor network Non-fractal complex networks Internet, artificial models (BA model, etc), actor network, etc Purposes: 1. The origin of the fractal scaling. 2. Construction of a fractal network model.

11. I-4. Origin • Disassortativity, by Yook et al., PRE (2005) • Repulsion between hubs, by Song et al., Nat. Phys. (2006). Fractal network=Skeleton+Shortcuts Skeleton=Tree based on betweenness centrality Skeleton  Critical branching tree  Fractal By Goh et al., PRL (2006).

12. I-5. Skeleton What is the skeleton ? Kim, Noh, Jeong PRE (2004) • For a given network, loads (BCs) on each edge are calculated. • Generate a spanning tree by following the descending order of edge loads (BCs). Skeleton Skeleton is anoptimal structure for transport in a given network.

14. I-6. Fractal scalings random original skeleton Fractal scalings of the original network, skeleton, and random ST Fractal structures

15. random original skeleton Fractal scalings of the original network, skeleton, and random ST Non-fractal structures

16. I-7. Branching tree If then the tree is subcritical Network → Skeleton → Tree → Branching tree Mean branching number If then the tree is critical If then the tree is supercritical

18. Test of the mean branching number: <m>b skeleton yeast WWW metabolic random Static BA Internet

19. I-8. Critical branching tree For the critical branching tree M is the mass within the circle Goh PRL (2003), Burda PRE (2001) Cluster-size distribution

20. I-9. Supercritical branching tree For the supercritical branching tree Cluster-size distribution behaves similarly to but with exponential cutoff.

21. Test of the mean branching number: <m>b skeleton WWW metabolic yeast random Critical Supercritical

22. I-10. Model construction Model construction rule i) A tree is grown by a random branching process with branching probability: ii) Every vertex increases its degree by a factor p;qpki are reserved for global shortcuts, and the rest attempt to connect to local neighbors (local shortcuts). iii) Connect the stubs for the global shortcuts randomly. • Resulting network structure is: • SF with the degree exponent g. • Fractal for q~0 and non-fractal for q>>0.

24. Networks generated from a critical branching tree

25. Fractal scaling and mean branching ratio for the fractal model

26. Networks generated from a supercritical branching tree

27. Fractal scaling and <m>b for the skeleton of the network generated from a SC tree

28. II. Self-similarity in SF networks • The distribution of renormalized-degrees under coarse-graining is studied. • Modules or boxes are regarded as super-nodes • Module-size distribution • How is h involved in the RG transformation ? Coarse-graining process

29. Random and clustered SF network: (Non-fractal net) Analytic solution

30. Derivation

31. h and q act asrelevant parameters in the RG transformation

32. For fractal networks, WWW and Model

33. For a nonfractal network, the Internet  Self-similar

34. Scale invariance of the degree distribution for SF networks Jung et al., PRE (2002)

35. The deterministic model is self-similar, but not fractal ! Fractality and self-similarity are disparate in SF networks.

36. Critical Supercritical Summary I Skeleton + Local shortcuts Branching tree Fractal networks Fractal model Yeast PIN WWW [1] Goh et al., PRL 96, 018701 (2006). [2] J.S. Kim et al., cond-mat/0605324.

37. Summary II 1. h and q act asrelevant parameters in the RG transformation. 2. Fractality and self-similarity are disparate in SF networks. [1] J.S. Kim et al., cond-mat/0605587.