Fractality vs selfsimilarity in scalefree networks B Kahng
Fractality vs self-similarity in scale-free networks B. Kahng Seoul Nat’l Univ. , Korea & CNLS, LANL Jin S. Kim, K. -I. Goh, G. Salvi, E. Oh and D. Kim The 2 nd KIAS Conference on Stat. Phys. , 07/03 -06/06
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).
Introduction Network • node, link, & degree Networks are everywhere
Random graph model by Erdős & Rényi [Erdos & Renyi 1959] Put an edge between each vertex pair with probability p 1. Poisson degree distribution 2. D ~ ln. N 3. Percolation transition at p=1/N
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 7 -α 3 -α Two vertices are selected with probabilities pi pj. Goh et al. , PRL (2001).
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 I-1. Fractality
I-2. Box-counting Random sequential packing: Nakamura (1986), Evans (1987) 1. At each step, a node is selected randomly and served as a seed. 2. Search the network by distance from the seed and assign newly burned vertices to the new box. 3. Repeat (1) and (2) until all nodes are assigned their respective boxes. 4. is chosen as the smallest number of boxes among all the trials. 3 1 2 4
I-2. Box-counting Fractal scaling d. B = 4. 1 Box mass inhomogeneity WWW
I-2. Box-counting Box-covering method: Log Box Number d. B Log Box Size Fractal dimension d. B
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.
I-4. Origin 1. Disassortativity, by Yook et al. , PRE (2005) 2. 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).
I-5. Skeleton What is the skeleton ? Kim, Noh, Jeong PRE (2004) 1. For a given network, loads (BCs) on each edge are calculated. 2. Generate a spanning tree by following the descending order of edge loads (BCs). Skeleton is an optimal structure for transport in a given network.
I-6. Fractal scalings of the original network, skeleton, and random ST original skeleton random Fractal structures
Fractal scalings of the original network, skeleton, and random ST original skeleton random Non-fractal structures
I-7. Branching tree Network → Skeleton → Tree → Branching tree Mean branching number If then the tree is subcritical If then the tree is supercritical
Test of the mean branching number: <m>b skeleton WWW metabolic yeast random Internet BA Static
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
I-9. Supercritical branching tree For the supercritical branching tree Cluster-size distribution behaves similarly to but with exponential cutoff.
Test of the mean branching number: <m>b WWW skeleton metabolic yeast random www metabolic Yeast PIN Original Networks Cluster-growing Exponential Power law Box-covering Power law skeletons Cluster-growing Exponential Power law Box-covering Power law Supercritical Critical
I-10. 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: i) SF with the degree exponent g. ii) Fractal for q~0 and non-fractal for q>>0.
Networks generated from a critical branching tree Critical branching tree + local shortcuts + global shortcuts fractal Non-fractal
Fractal scaling and mean branching ratio for the fractal model
Networks generated from a supercritical branching tree Supercritical branching tree + local shortcuts + global shortcuts Fractal+small world Non-fractal
Fractal scaling and <m>b for the skeleton of the network generated from a SC tree
II. Self-similarity in SF networks 1. The distribution of renormalized-degrees under coarse-graining is studied. 2. Modules or boxes are regarded as super-nodes 3. Module-size distribution 4. How is h involved in the RG transformation ? Coarse-graining process
Random and clustered SF network: (Non-fractal net) Analytic solution
Derivation
h and q act as relevant parameters in the RG transformation
For fractal networks, WWW and Model
For a nonfractal network, the Internet Self-similar
Scale invariance of the degree distribution for SF networks Jung et al. , PRE (2002)
The deterministic model is self-similar, but not fractal ! Fractality and self-similarity are disparate in SF networks.
Summary I Fractal networks Skeleton + Local shortcuts Branching tree Fractal model Yeast PIN Critical WWW Supercritical [1] Goh et al. , PRL 96, 018701 (2006). [2] J. S. Kim et al. , cond-mat/0605324.
Summary II 1. h and q transformation. act as relevant parameters in the RG 2. Fractality and self-similarity are disparate in SF networks. [1] J. S. Kim et al. , cond-mat/0605587.
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