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

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, (2006). [2] J.S. Kim, et al., Fractality in ocmplex networks: Critical and supercritical skeletons, (cond-mat/ ). II. Self-similarity in SF networks [1] J.S. Kim, Block-size heterogeneity and renormalization in scale-free networks, (cond-mat/ ).

Networks are everywhere Introduction node, link, & degree Network Introduction

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 ~ lnN 3.Percolation transition at p=1/N

1-α1-α 2-α2-α 4-α4-α 3-α3-α 5-α5-α 6-α6-α 8-α8-α 7-α7-α Scale-free network: the static model Goh et al., PRL (2001). The number of vertices is fixed as N. Two vertices are selected with probabilities p i p j.

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

Random sequential packing: 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. 1 Nakamura (1986), Evans (1987) I-2. Box-counting

Fractal scaling d B = 4.1 WWW Box mass inhomogeneity

Log Box Size Log Box Number dBdB Fractal dimension d B Box-covering method: I-2. Box-counting

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-3. Purposes

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).

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 What is the skeleton ? Kim, Noh, Jeong PRE (2004) I-5. Skeleton Skeleton is an optimal structure for transport in a given network.

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

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

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

Test of the mean branching number: b WWWmetabolic yeast Internet BA Static skeleton random

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

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

Test of the mean branching number: b WWWmetabolicyeast wwwmetabolicYeast PIN Original Networks Cluster-growing Exponential Power law Box-covering Power law skeletons Cluster-growing ExponentialPower law Box-covering Power law random skeleton Supercritical Critical

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

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 b for the skeleton of the network generated from a SC tree

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 II. Self-similarity in SF networks

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

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

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

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

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